Curvature-Based Assessment of Left-Turn Trajectories at Urban Intersections Using Video-Extracted Vehicle Paths

Abstract

Urban intersection design generally assumes that drivers follow idealised turning paths defined by circular arcs and, in some cases, transition curves. In practice, however, observed left-turn trajectories often depart from these theoretical paths. This study proposes a curvature-based framework for quantifying such deviations at the movement level by directly comparing observed vehicle paths with theoretical design arcs derived from intersection geometry. Naturalistic traffic data were collected at five urban intersections in Thessaloniki, Greece, using elevated video cameras. Left-turn passenger-vehicle trajectories were extracted, georeferenced, and compared with corresponding theoretical paths. For each trajectory, a best-fit circular arc was estimated, and the deviation between observed and theoretical path geometry was quantified through radius- and curvature-based percentage indicators. These indicators were then aggregated at the intersection and movement level using medians, deciles and the relative shares of flatter-than-theoretical and tighter-than-theoretical trajectories. The results show that deviations from theoretical geometry are strongly movement-specific and that the strongest flattening and tightening patterns were statistically supported by movement-level Wilcoxon signed-rank tests. In some cases, drivers systematically opened the turn relative to the design path, with median curvature deviations reaching about −14% and flatter-than-theoretical shares as high as 94%. In other cases, the opposite pattern was observed, with median curvature deviations exceeding +37% and tighter-than-theoretical shares reaching 100%. Other movements remained close to the theoretical path or displayed substantial internal heterogeneity. Overall, the proposed framework offers a practical and interpretable way to screen left-turn movements for systematic departure from design intent. This is important because it allows the analysis to move from individual path overlays to a movement-level geometric reading that can support consistency checks, intersection review and future integration with speed- and conflict-based safety analyses. These results should nonetheless be regarded as exploratory and descriptive: neither the circle-fitting residuals nor the coordinate-level geometric accuracy of the extracted trajectories were formally validated in the present study, and the reported RDP/CDP values are, therefore, not intended for use as precision-survey quantities or as a stand-alone design basis.

1. Introduction

1.1. Background

The design of at-grade urban intersections is a fundamental task in Highway and Traffic Engineering (HTE). Geometric design guidelines, such as those issued by national road authorities and by the American Association of State Highway and Transportation Officials, provide recommended radii, lane widths and channelisation layouts for turning movements [1,2,3]. For left-turn manoeuvres in particular, these guidelines typically represent vehicle paths using circular arcs, sometimes combined with transition curves, chosen to satisfy constraints on side friction, driver comfort and available space [1,4,5].

Implicit in such design practice is the assumption that drivers will adopt turning paths that closely follow the intended geometric layout. However, naturalistic trajectory studies have repeatedly shown that drivers adapt their paths in response to lane markings, opposing traffic, sight distance and their own comfort or speed preferences [6,7,8]. At traditional intersections, left-turn drivers may cut across lane markings to open up the curve, or, conversely, may follow tighter paths when constrained by adjacent lanes or channelising islands [9,10,11]. At small roundabouts, circulating and exiting trajectories often differ substantially from the nominal circular geometry, with observed path radii and speeds linked to both safety and operational performance [5,6,12].

In parallel, advances in computer vision (CV) and Intelligent Transportation Systems (ITSs) have enabled the large-scale extraction of vehicle trajectories from roadside or drone video, providing a rich empirical basis for analysing how drivers actually move through intersections beyond aggregated measures, such as average delay or queue length.

Most applications of high-resolution trajectory data to date have focused on conflict-based surrogate safety measures (SSM), such as Time to Collision (TTC) and Post-Encroachment Time (PET), often implemented through tools like the Surrogate Safety Assessment Model (SSAM) [13,14,15,16]. These measures are typically computed either from simulated trajectories or from trajectories extracted from cameras and are then related to crash data for validation [16,17]. Comparatively less attention has been devoted to the geometric consistency between design and observed paths at intersections: that is, how closely observed turning trajectories follow the curves implied by design guidelines, and how any systematic deviations might relate to safety and comfort.

1.2. Research Gap and Objectives

The emerging availability of high-resolution trajectory data opens an opportunity to systematically compare the geometry of observed vehicle paths with the theoretical geometry implied by design. For left-turn movements at urban intersections, this comparison is particularly relevant: deviations in curvature are directly linked to lateral acceleration, lane occupation and potential encroachment into opposing or pedestrian flows, and may, therefore, have implications for both safety and operational performance [18,19,20].

Several strands of recent research have addressed related aspects of turning manoeuvres, from empirical trajectory analysis at signalised intersections and roundabouts to Automated Driving Systems (ADSs)/Advanced Driver Assistance Systems (ADASs) trajectory planning, as reviewed in detail in Section 2.

Despite this progress, the existing literature exhibits three notable gaps:

• Limited direct linkage between design curves and observed trajectories. Many studies have described observed turning paths qualitatively or reported representative radii for selected trajectories [5,6,21], but few have quantified the distribution of curvature deviations across all vehicles and movements at a set of real intersections, explicitly benchmarking against the geometric design arcs.
• Lack of simple, transferable indicators. There is no widely adopted set of indicators that summarises, in a compact and interpretable form, how closely drivers follow the design geometry, nor that distinguishes between systematic “flattening” (drivers opening the curve relative to design) and “tightening” (drivers following sharper paths than intended). Existing safety indicators are mostly conflict-based and do not explicitly encode geometric consistency [13,14,15,16].
• Scarce multi-site evidence in real urban networks. Most available work has been based on single sites, experimental layouts, or a limited number of approaches, which constrains the generalisability of findings and the ability to compare patterns across different intersection types and configurations [7,18,22]. Large-scale drone datasets, such as inD and DRIFT, provide valuable naturalistic trajectories, but they do not directly offer design-versus-observed curvature comparisons for the left-turn movements examined here [23,24].

Against this background, the present paper formulates and addresses the following research questions:

• To what extent do observed left-turn vehicle trajectories at urban intersections conform to the theoretical circular arcs defined by intersection design?
• How do curvature deviations between observed and design paths vary across different intersections, left-turn movements and, where applicable, left-turn lanes?
• Can curvature-based indicators derived from video-extracted trajectories serve as practical tools for geometric consistency checks and as inputs for the calibration of microscopic traffic simulation models?

In this sense, the proposed framework differs from existing trajectory studies in three ways. First, it does not describe observed paths only qualitatively or through dispersion measures, but benchmarks them directly against the corresponding theoretical design arc. Second, it does not focus primarily on conflict-based surrogate safety indicators or speed-profile modelling, but on geometric consistency at the movement level. Third, it applies this comparison across multiple real urban intersections, rather than at a single site or within a purely simulation-based setting.

1.3. Study Approach and Contributions

To answer these questions, the study uses video recordings from five urban intersections in Thessaloniki, Greece, covering signalised, unsignalised and roundabout configurations. For each site, vehicle trajectories performing left-turn manoeuvres are extracted through a Deep Learning (DL)-based detection and tracking pipeline, consistent with recent vision-based frameworks for vehicle trajectory estimation from street-level or aerial videos [25,26,27,28]. The image-space trajectories are transformed onto the roadway plane by means of planar homography, as commonly adopted in trajectory datasets and intersection studies [26,29]. A least-squares circle fitting procedure is then applied to the curved segment of each trajectory to estimate an effective turning radius and the corresponding curvature.

On the basis of these estimates, two indicators are defined: the Radius Difference Percentage (RDP) and the Curvature Difference Percentage (CDP) between each observed trajectory and the design arc for the corresponding movement. These indicators are aggregated at the level of the intersection, movement and lane to characterise systematic patterns of “flattening” and “tightening” relative to design. By relying on simple measures derived from the full trajectory, the framework seeks to balance interpretability for practitioners with the richness of trajectory-level information.

This paper offers three main contributions:

• It introduces a curvature-based framework that directly links observed left-turn trajectories to the underlying design geometry using simple, interpretable indicators, complementing both traditional geometric design checks and conflict-based safety indicators [13,14,16].
• It presents multi-site empirical evidence from five urban intersections, demonstrating how curvature deviations vary across intersection types and movements under naturalistic operating conditions, relating these patterns to geometric features, such as radii, lane widths and channelisation [6,8].
• It discusses how the proposed indicators may support geometric review and may provide useful input for the calibration of microscopic traffic simulation (MTS) and, in the longer term, for trajectory design in Automated Driving Systems (ADSs) at intersections [18,30,31].

The remainder of this paper is organised as follows. Section 2 reviews previous work on intersection geometry, observed turning trajectories and trajectory-based safety analysis, including minimum-jerk and optimal-control approaches to turning path modelling. Section 3 describes the study sites, data collection, and trajectory extraction procedures and details the curvature-based indicators. Section 4 presents the empirical results across the five intersections and discusses the observed patterns of deviation between design and observed paths. Section 5 summarises the main findings, outlines implications for design and safety assessment, and proposes directions for future research.

2. Literature Review

2.1. Geometric Design of Left-Turn Manoeuvres

Geometric design guidance for urban intersections has traditionally been developed within a deterministic framework, in which idealised vehicle paths are defined and then used to dimension lane widths, radii and channelising islands. The American Association of State Highway and Transportation Officials (AASHTO) policy, for example, specifies minimum design radii for turning movements based on design speed, side friction factors and superelevation, and illustrates recommended swept paths for passenger cars, buses and trucks [1]. Similar principles have been adopted at the national level, where recent NCHRP reports provide detailed guidance and syntheses of roundabout design and practice [2,3]. In these documents, left-turn manoeuvres are typically represented by one or more circular arcs that connect the approach and exit legs while keeping the vehicle within the intended lane discipline.

In addition to traditional circular-arc designs, several authors have explored the use of transition curves and spiral-based geometry to better reflect driver comfort criteria. Research on roundabouts, for instance, has highlighted the role of entry, circulating and exit radii in controlling operating speeds and lateral acceleration [4,5]. Šurdonja et al. developed models for the effective path radius at the roundabout centre, explicitly linking geometric layout to vehicle speed and safety performance [6]. These studies have demonstrated that geometric parameters can influence driver behaviour; however, they remain largely based on assumed driver paths rather than systematically measured trajectories.

Recent guidance and design-oriented work have emphasised the need to accommodate the swept paths of trucks and buses, while at the same time avoiding excessive radii that may encourage high-speed turning and increase risk for vulnerable road users [1,2,3]. These findings create a tension between operational efficiency and safety, particularly for left-turn manoeuvres in constrained urban environments. A key question, therefore, is whether drivers actually follow the design elements of the intersections implied by the selected radii, or whether they adapt their paths in ways that alter the intended balance between speed, comfort and conflict risk.

2.2. Empirical Observations of Turning Trajectories

Earlier empirical work on turning behaviour relied on limited observational or pilot-study evidence to describe representative vehicle paths and turning behaviour [10,11,21]. With the advent of higher-resolution cameras and automated image processing, it has become possible to collect much larger samples of trajectories under naturalistic conditions.

At roundabouts, several authors have analysed vehicle paths to understand how drivers negotiate entry, circulating and exit manoeuvres. Leonardi et al. studied human driving at roundabouts in the context of autonomous vehicle integration, again highlighting the dispersion in trajectories relative to idealised circular paths [12]. For signalised intersections, research has considered both left- and right-turn movements. Wolfermann et al. modelled speed profiles of turning vehicles and noted that drivers’ lateral positioning and path choice can vary considerably even within the same lane group [22]. Zhao et al. examined path dispersion at signalised intersections and related it to intersection geometry, traffic control, and traffic conditions, finding that heterogeneity in trajectories can be substantial [8]. Related modelling work by Dias et al. similarly characterised trajectory variation in turning vehicles at signalised intersections [32]. Observed and design-oriented studies suggest that drivers may open up a curve when adjacent space is available, whereas, in other situations, they may be forced into tighter paths by opposing flows, parked vehicles, or channelising islands [1,2,3,22].

Despite these insights, most existing studies have either reported a limited number of representative trajectories or focused on speed and lane position separately, without directly quantifying how the curvature of observed paths compares to the curvature implied by design. As a result, the literature provides only partial answers to the question of whether and to what extent drivers follow the design geometry of left-turn manoeuvres at real-world intersections.

2.3. Video-Based Trajectory Extraction and Datasets

The ability to capture and process large numbers of vehicle trajectories has been greatly enhanced by advances in computer vision (CV) and Deep Learning (DL). Modern trajectory extraction pipelines typically employ object detection networks, such as variants of the You Only Look Once (YOLO) architecture, combined with multi-object tracking algorithms (e.g., DeepSORT), to follow vehicles across video frames [25,26,27]. The resulting image-space trajectories are then projected onto the road plane using planar homography or related calibration techniques, yielding metric position–time histories in Cartesian coordinates [26,29].

In parallel, several large-scale datasets of naturalistic road user trajectories have been developed. The inD dataset, for instance, provides high-resolution trajectories extracted from drone videos at four German intersections, including detailed lane geometry and road user interactions [23]. Further datasets have extended this concept to motorways, off-ramps and complex junctions [24,33,34]. Other work has focused on developing robust pipelines for georeferenced trajectory extraction from high-altitude drone footage or fixed roadside cameras, addressing issues such as occlusions, re-identification and trajectory smoothing [28,29].

While these datasets and methods provide the technical foundation for large-scale trajectory analysis, most applications to intersections have emphasised conflict detection and surrogate safety analysis rather than comparison between observed and design geometry. Moreover, publicly available datasets often lack detailed insight into the design radii and theoretical paths used for each movement, which complicates the task of benchmarking observed curvature against design values. This underlines the need for studies that explicitly combine design drawings or geometric models with video-based trajectories at specific intersections.

2.4. Trajectory-Based Safety and Comfort Indicators

Trajectory-level data have been widely used to derive surrogate safety measures (SSM) that serve as proxies for crash risk. The Surrogate Safety Assessment Model (SSAM), developed by the Federal Highway Administration, computes indicators such as Time to Collision (TTC), Post-Encroachment Time (PET) and Deceleration Rate to Avoid the Crash (DRAC) from simulated or observed trajectories, and has been applied extensively to evaluate the safety performance of intersections and other facilities [13,14,15,16]. The following studies have proposed enhancements to these measures and examined their relationship with observed crash patterns [17]. More recently, trajectory data have also been used in real-time risk-oriented applications beyond conventional conflict screening, including multi-dimensional driving risk warning frameworks for heavy-duty vehicles. This broader trajectory-based safety perspective further highlights the growing value of high-resolution trajectory information for operational risk assessment [35].

Beyond conflict-based indicators, trajectory data have been used to study driving comfort and vehicle dynamics. Speed profiles and lateral acceleration along turning paths have been modelled to assess the consistency between design speed, operating speed and comfort thresholds [4,22]. Minimum-jerk and related formulations have been proposed as models for human-preferred trajectories, capturing the tendency of drivers to avoid abrupt changes in acceleration [20,30,31]. These approaches highlight that the curvature of the path, combined with speed, is central to understanding both safety and comfort.

However, the majority of trajectory-based safety studies have treated curvature only implicitly, through kinematic variables such as speed and acceleration, rather than explicitly comparing observed and design curvature—the indicator gap already noted in Section 1.2.

2.5. Modelling and Planning of Turning Paths

In the context of Automated Driving Systems (ADSs) and Advanced Driver Assistance Systems (ADASs), substantial research has been devoted to trajectory prediction and planning for vehicles negotiating intersections. Optimal control formulations and model predictive control schemes often generate turning trajectories that minimise a weighted combination of travel time, control effort, and jerk, subject to vehicle dynamics and collision-avoidance constraints [36,37]. Other approaches seek to emulate human driving styles by fitting parametric trajectory models, such as triclothoidal or polynomial curves, to observed data [30,31].

These developments illustrate an emerging convergence between traditional geometric design, which prescribes nominal curves for turning movements, and control-oriented trajectory planning, which optimises vehicle paths in real time—in both cases, curvature plays a central role. However, the interaction between geometric design radii and the trajectories that human drivers actually choose in everyday operation remains only partially investigated.

Overall, the literature points in the same direction, but from different angles. Geometric design studies define the nominal path that a vehicle is expected to follow. Recent trajectory-extraction studies have shown how drivers actually move through intersections, and trajectory-based safety and planning studies use such information either to assess conflicts or to optimise motion in automated systems [16,26,37]. What is still missing, however, is a practical way to compare these two worlds directly at the movement level across real urban intersections. This is the point where the present study is positioned. It combines video-extracted trajectories with geometry-based theoretical paths and uses simple curvature-deviation indicators to examine where left-turn movements remain close to design, where they systematically depart from it, and where the observed patterns may justify closer geometric review.

3. Study Sites and Methodology

3.1. Study Area and Selected Intersections

The empirical analysis is based on vehicle trajectories collected at five urban intersections in Thessaloniki, Greece. The intersections were selected purposively within the STREET21 project in order to cover a mix of common urban layouts and operating conditions, including signalised, unsignalised, and roundabout configurations, substantial left-turn demand, variation in lane arrangement and channelisation, and camera positions that allowed for reliable trajectory extraction over the full turning manoeuvre. For the purposes of this study, the intersections are denoted as K01–K05 and correspond to:

• K01—Egnatia Street/3rd September Avenue.
• K02—30th October Street/3rd September Avenue.
• K03—Mpotsari Street/Chalkidis Street.
• K04—Tsaldari Street/Kougiami Street.
• K05—3rd September Avenue/Agiou Dimitriou Street.

All five junctions are urban intersections on arterial or major collector streets, with multiple approach lanes and distinct left-turn lanes on at least one approach. The sites differ in terms of:

• Intersection size and corner radii;
• Number and width of approach lanes;
• Installed signalised systems;
• Presence or absence of channelised islands and auxiliary lanes;
• Surrounding land use and traffic demand patterns.

This heterogeneity allows us to investigate whether the relationship between design and observed curvature is consistent across a range of typical European urban intersection layouts.

3.2. Video Data Collection

Field data collection was carried out as part of the STREET21 on-field survey campaign. At each intersection, high-resolution video cameras were mounted on extendable tripods at elevated positions, providing an unobstructed view of all approaches and conflict areas and covering the complete left-turn paths across the intersection.

For each site, both the morning and the afternoon peak periods were recorded on a typical weekday under normal signal timings and traffic conditions. However, for the purposes of the present analysis, only trajectories from the morning peak period were used. During the afternoon peak, the ambient light levels were lower and, despite the same camera configuration, the contrast between vehicles and background was insufficient for reliable detection and tracking across the full turning manoeuvre. To avoid introducing bias from tracking errors in low-light conditions, all trajectories from the afternoon peak were excluded from the dataset.

Basic site information (geometric layout, lane configuration, signal control, and traffic counts) was documented from design drawings and field observations. For each intersection, the turning movements of interest were the left-turn movements between specific entry and exit legs, coded consistently across sites using the K0x notation and ordered by arm indices (e.g., movement 2→1 at K02).

3.3. Trajectory Extraction and Georeferencing

The raw video recordings were processed using a computer vision pipeline combining object detection, multi-object tracking and geometric transformation to real-world coordinates.

Object detection

Each video frame was processed using a state-of-the-art real-time object detector (YOLO family), which identifies vehicles and returns, for each frame, the pixel coordinates of bounding boxes (and, where applicable, segmentation masks).

Multi-object tracking

Detected vehicles were then linked across consecutive frames using a tracker of the DeepSORT-type. The tracker assigns a persistent unique ID to each vehicle and outputs a time-ordered sequence of pixel-level positions for that ID. This yields image-space trajectories for each recorded vehicle.

Temporal resampling

The raw videos were recorded at 25 frames per second, corresponding to a frame interval of 0.04 s. For analysis, the trajectories were resampled at a fixed time step of 0.2 s by keeping every fifth frame. This technique reduces data size and noise while preserving sufficient resolution for curvature estimation and speed reconstruction.

Geometric calibration and coordinate transformation

For each camera position in each of the five intersections, a set of ground control points was defined on the intersection layout, e.g., stop lines, crosswalk corners, and lane markings, with known coordinates in a local metric coordinate system. Using corresponding pixel coordinates from the video frames and ground coordinates from the surveyed layout, a Perspective-n-Point (PnP)/homography solution was computed. This yields the transformation

$$(u_t,v_t)\hspace{0.25em} \to \hspace{0.25em} (x_t,y_t)$$

mapping every pixel-level trajectory point to planar world coordinates in metres. The transformation was verified visually (an overlay of transformed trajectories on the intersection layout) and numerically (reprojection residuals of the control points). The result of this step is a set of georeferenced vehicle trajectories for each tracked vehicle crossing the intersection.

The present paper focuses on the geometric interpretation of the extracted trajectories rather than on standalone benchmarking of the object-detection and tracking pipeline. In addition to the visual overlay and control-point reprojection checks described above, a movement-level quality review of the extracted trajectories was also performed using the raw trajectory records. More specifically, track-level detection confidence, temporal continuity after resampling, and retained trajectory counts were examined across the analysed left-turn movements. Across the analysed movements, median track-level confidence values were generally high (approximately 0.78–0.94, with a central value around 0.87), temporal continuity after resampling remained stable, and the retention of movement-classified trajectories after preprocessing was typically high. Although these checks do not constitute a full external validation against independently surveyed vehicle positions, they provide additional quantitative support for the internal consistency of the extracted movement-level trajectory dataset used in the present study.

3.4. Selection and Preprocessing of Left-Turn Trajectories

From the full set of trajectories at each site, we extracted those corresponding to left-turn movements of interest, defined by their origin and destination legs (e.g., movement 2→1 at K02). Path classification was performed by analysing the start and end positions of each trajectory relative to the coded approach lanes and exit lanes.

For each intersection K01–K05, the following numbers of left-turn trajectories were retained for analysis (unique vehicles):

• K01: 532 trajectories across movements 1→4, 2→1 and 4→3;
• K02: 621 trajectories across movements 1→2 and 2→1 (including the leftmost lane at movement 2→1);
• K03: 138 trajectories across movements 1→2 and 2→1;
• K04: 352 trajectories across movements 1→1, 1→4, 2→1, 3→2, 4→3 and 4→4;
• K05: 676 trajectories across movements 2→1, 3→2 and 4→3.

In total, the dataset used in this paper includes 2319 left-turn trajectories across the five intersections analysed, as follows in

Table 1. Number of captured left-turn trajectories per left-turnin movement across the five intersections.

Turning Movement	No of Captured Left-Turns
K01 1_4	157
K01 2_1	80
K01 4_3	295
K02 2_1 leftmost	358
K02 1_2	226
K02 2_1	37
K03 1_2	25
K03 2_1	113
K04 1_1	13
K04 1_4	194
K04 2_1	116
K04 3_2	5
K04 4_3	19
K04 4_4	5
K05 2_1	163
K05 3_2	427
K05 4_3	86
SUM	2319

:

To ensure reliable curvature estimates, we applied basic preprocessing:

• Trajectories were ordered in time and checked for continuity of the tracking ID.
• Short or fragmented trajectories were discarded when they did not preserve sufficient continuity within the predefined comparison boundaries to allow a meaningful curvature estimate over the curved part of the manoeuvre. This filtering was not based on a single global threshold in terms of frames or path length, because the required extent depends on the geometric length of each analysed turning segment. In practice, fragmented trajectories were usually caused by temporary occlusion, whereas very short trajectories tended to produce unrealistically high fitted curvature values.
• For each retained left-turn trajectory, the portion belonging to the actual turning manoeuvre was extracted, as described next.

3.5. Definition of Theoretical Design Trajectories

For each left-turn movement at each intersection, a theoretical design trajectory was constructed directly from the intersection geometry using a symmetric definition of the entry and exit tangents with respect to the intersection point (Figure 1).

• The intersection point K was defined as the intersection of the prolongations of the centre lines of the entry and exit lanes for the left-turn movement.
• Along the exit-leg tangent, point A was identified at a certain distance from the edge of the island where vehicles straighten their trajectory.
• Along the entry-leg tangent, point B was then defined such that the distance from K to B along the entry tangent satisfies

$$\mid KB\mid =\mid KA\mid$$

In other words, the same tangential distance from the intersection point is used upstream on the entry leg and downstream on the exit leg.

• The theoretical design trajectory for the left-turn was defined as a circular arc that is tangent to the entry and exit tangents at points B and A, respectively, and connects these two tangency points.
• This symmetric construction was adopted in this study as a practical geometric simplification in order to define, in a consistent manner across movements, the curved portion over which observed and theoretical trajectories are compared. It was not intended to imply that real left-turn trajectories are inherently symmetric, nor that this construction follows a formal design standard. Rather, it provides a transparent and repeatable way to standardise the comparison extent across heterogeneous urban layouts on the basis of the available intersection geometry.
• From this construction, the theoretical turning radius and the corresponding theoretical curvature

$${\kappa }_{\mathrm{theoretical}}={\displaystyle \frac{1}{R_{\mathrm{theoretical}}}}$$

were computed for each coded left-turn movement. These values provided a consistent geometric benchmark for all trajectories of the same movement. Particularly for the roundabout (K04), the trajectories were bounded by the entry and exit points of the circulatory roadway.

3.6. Extraction of the Curved Trajectory Segment

For each observed left-turn trajectory, only the portion corresponding to the curved part of the manoeuvre between the two tangential distances was retained for analysis.

The procedure was as follows:

• Using the same local coordinate system as for the geometric design, two virtual cross-sections were defined:
  • An upstream section on the entry leg at a distance from the intersection point K along the entry-lane tangent;
  • A downstream section on the exit leg at a distance from K along the exit-lane tangent.
• For each vehicle trajectory associated with the given left-turn movement, the instants when the vehicle crossed the upstream and downstream cross-sections were identified.
• The points of the trajectory between these two crossings were extracted and treated as the curved trajectory segment used in curvature analysis; trajectory points upstream of B or downstream of A were discarded.

In this way, both the theoretical and the measured trajectories were evaluated over exactly the same geometric extent, bounded by symmetric tangential distances from the intersection point. This definition was adopted not as a claim about normative design practice, but as a repeatable comparison convention for movement-level curvature assessment. This trimming procedure was introduced to standardise the comparison domain across all analysed movements and to ensure that the observed and theoretical paths were compared over the same curvature-relevant portion of the manoeuvre. It therefore excludes longer straight approach and departure segments that are not informative for the turning-radius comparison. In this sense, the retained segment should be understood as the geometric comparison portion of the trajectory rather than as a full representation of the entire real-world manoeuvre.

3.7. Circle Fitting and Curvature Estimation of Measured Trajectories

To characterise the curvature of each observed left-turn trajectory, we approximated its curved segment with the best-fitting circular arc in the algebraic least-squares sense.

Let $(x_i,y_i)$, $i=1,\dots ,n$, denote the planar coordinates of the points of the trimmed curved segment for a given vehicle. The Kåsa circle-fitting method was applied to estimate the parameters of the circle that best fits these points [38]. In this method, the circle parameters $(x_0,y_0,R_{\mathrm{trajectory}})$—centre coordinates and radius—are obtained by solving a linear least-squares problem based on the algebraic formulation of the circle. The corresponding trajectory curvature is then computed as

$${\kappa }_{\mathrm{trajectory}}={\displaystyle \frac{1}{R_{\mathrm{trajectory}}}}.$$

The Kåsa method is non-iterative and computationally efficient, which makes it suitable for processing large numbers of trajectories. A dedicated script was implemented to apply this procedure consistently to all left-turn trajectories and to export, for each unique vehicle ID, the fitted radius, the circle centre coordinates, and the associated theoretical radius for that movement.

3.8. Curvature Deviation Indicators

To quantify how closely each observed trajectory follows the design curve, we defined two dimensionless deviation indicators expressed as percentages:

• Radius Deviation Percentage (RDP):

$$\mathrm{R}\mathrm{D}\mathrm{P}=100\times {\displaystyle \frac{R_{\mathrm{trajectory}}-R_{\mathrm{theoretical}}}{R_{\mathrm{theoretical}}}}$$

• Curvature Deviation Percentage (CDP):

$$\mathrm{C}\mathrm{D}\mathrm{P}=100\times {\displaystyle \frac{{\kappa }_{\mathrm{trajectory}}-{\kappa }_{\mathrm{theoretical}}}{{\kappa }_{\mathrm{theoretical}}}}=100\times {\displaystyle \frac{\frac{1}{R_{\mathrm{trajectory}}}-\frac{1}{R_{\mathrm{theoretical}}}}{\frac{1}{R_{\mathrm{theoretical}}}}}$$

The positive values of the RDP indicate trajectories with larger radii (flatter turns) than the design arc, while negative values correspond to tighter observed turning paths. Conversely, positive CDP values indicate higher curvature (tighter turning) than the design, and negative values indicate lower curvature. Beyond this general descriptive use, the CDP distribution was also used at the movement level to support a descriptive classification of left-turn behaviour. More specifically, the sign of the CDP indicates whether an individual observed path is flatter or tighter than the theoretical path, whereas the median CDP and the corresponding share of trajectories on the same side of zero were used to identify whether a given movement exhibits a systematic tendency toward flattening, tightening, or near-design/mixed behaviour. This classification is defined explicitly in Section 4.2 and is used there as a pragmatic screening tool for movement-level interpretation.

For each intersection and left-turn movement, the distribution of the RDP and CDP across all trajectories was computed and subsequently used in the results section to:

• Describe systematic biases (e.g., left-turn movements in which drivers consistently tighten or flatten the design arc);
• Quantify inter-driving behaviour variability;
• Explore the implications of these deviations for comfort and safety-related indicators.

In order to position the proposed indicators relative to simpler trajectory descriptors, an additional movement-level comparison was carried out with two supplementary measures: a simple radius-offset metric based on the absolute difference between the fitted and theoretical radius, and a path-dispersion metric describing the internal spread of trajectories within each movement. The comparison showed that the magnitude of the CDP is strongly associated with the simple radius-offset measure, whereas it is not equivalently associated with path dispersion. This suggests that the proposed curvature-based indicators are consistent with basic geometric deviation measures, while also capturing information that is not reducible to trajectory spread alone.

Figure 2 provides a schematic overview of the complete methodological workflow described in Section 3.2, Section 3.3, Section 3.4, Section 3.5, Section 3.6, Section 3.7 and Section 3.8. The left branch shows the video-based data pipeline—from field recording through trajectory extraction, georeferencing, and preprocessing—while the right branch shows the parallel derivation of the theoretical design trajectory from intersection geometry. The two branches converge at the circle-fitting step, from which the curvature deviation indicators (RDP and CDP) are computed and used to support movement-level classification.

4. Results and Discussion

4.1. Overview of Curvature Deviations

For each left-turn movement at the five study intersections, the distribution of the RDP and CDP was computed across all observed trajectories. These indicators summarise, in a dimensionless form, how far each observed turning path departs from the corresponding design arc:

• RDP > 0 (CDP < 0): drivers adopt flatter paths than implied by the geometric design, i.e., they use a larger effective radius;
• RDP < 0 (CDP > 0): drivers follow tighter paths, turning with a smaller effective radius than the theoretical one;
• Values close to zero indicate that the observed path is broadly consistent with the design arc.

These sign-based interpretations apply at the level of individual trajectories. However, the classification of an entire left-turn movement was not based on the sign of individual trajectories alone. Instead, movement-level interpretation was based on the combined reading of the median CDP and the predominance of trajectories on the same side of zero, as described in Section 4.2.

By examining these distributions at the level of movement (entry–exit pair) and intersection, we can identify systematic patterns that are directly linked to the underlying geometry: corner radii, lane widths, channelisation and the positioning of pedestrian crossings and stop lines. Rather than focusing on detailed inter-driver variability, the discussion below concentrates on what these patterns imply for the adequacy and safety performance of the existing geometric design.

4.2. Movement-Level Classification and Statistical Support of Curvature Deviations

For interpretive clarity, the analysed left-turn movements were grouped into three descriptive types using a pragmatic rule based on the CDP distribution. The Curvature Difference Percentage (CDP) was used as the primary indicator for this classification because it directly expresses whether the observed path has lower or higher curvature than the theoretical design arc. A movement was classified as a flattened-turn movement when the median CDP was equal to or below −10%, and at least 70% of the analysed trajectories had a CDP < 0. A movement was classified as a tightened-turn movement when the median CDP was equal to or above +10%, and at least 70% of trajectories had a CDP > 0. All remaining cases were treated as geometrically consistent or mixed. This third category includes movements with median values close to zero, as well as movements with substantial internal heterogeneity, where neither flattening nor tightening clearly predominates. These thresholds were adopted as descriptive screening criteria for movement-level interpretation and not as universal design standards. More specifically, they were selected pragmatically on the basis of the present dataset and engineering judgment in order to distinguish clearly between systematic and merely incidental deviation patterns at the movement level. To strengthen the statistical interpretation of the observed differences between theoretical and measured trajectories, a movement-level Wilcoxon signed-rank test was also applied. For each movement, the null hypothesis was that the median CDP was equal to zero, corresponding to no systematic curvature deviation from the theoretical path. The resulting p-values were interpreted together with the median CDP, the interquartile range (IQR) and the shares of flatter-than-theoretical and tighter-than-theoretical trajectories. This approach allows the movement-level typology to be supported not only by descriptive percentages, but also by inferential evidence of whether the observed curvature deviations differ statistically from the theoretical zero-deviation condition.

The results in

Table 2. Movement-level statistical assessment of CDP deviations from the theoretical path.

Movement	N	Median CDP (%)	IQR CDP (%)	Flatter (%)	Tighter (%)	Wilcoxon p-Value
K01 1_4	157	−13.76	11.70	93.6	6.4	<0.001
K01 2_1	80	+15.77	18.09	17.5	82.5	<0.001
K01 4_3	295	+37.32	18.01	0.0	100.0	<0.001
K02 2_1 leftmost	226	−10.54	11.31	87.2	12.8	<0.001
K02 1_2	37	−8.88	22.95	64.9	35.1	0.035
K02 2_1	358	+5.71	40.35	42.7	57.3	<0.001
K03 1_2	25	−1.29	23.47	60.0	40.0	0.861
K03 2_1	113	+6.89	22.27	29.2	70.8	<0.001
K04 1_1	13	+6.84	7.59	0.0	100.0	0.001
K04 1_4	194	+3.07	5.50	25.8	74.2	<0.001
K04 2_1	116	−4.88	7.13	83.6	16.4	<0.001
K04 3_2	5	−3.42	44.51	80.0	20.0	0.225
K04 4_3	19	−28.19	22.58	94.7	5.3	<0.001
K04 4_4	5	+6.22	8.53	0.0	100.0	0.043
K05 2_1	163	−4.55	19.11	60.1	39.9	0.012
K05 3_2	427	+26.19	9.44	0.7	99.3	<0.001
K05 4_3	86	+6.95	20.31	25.6	74.4	<0.001

indicate, on a descriptive and exploratory basis, that the strongest movement-level curvature deviations were statistically different from a zero CDP under the present (non-precision-validated) trajectory dataset. Clear and statistically supported flattening was observed for K01 1 to 4 and K02 1 to 2, while statistically supported tightening was observed for K01 2 to 1, K01 4 to 3 and K05 3 to 2. These movements combined substantial median CDP values, strong predominance of trajectories on the same side of zero, and significant Wilcoxon signed-rank test results.

Other movements require a more cautious interpretation. In several cases, the Wilcoxon test indicated a statistically significant deviation from zero, but the median CDP was relatively small, or the distribution was internally heterogeneous. These movements were, therefore, interpreted as mild, borderline or mixed rather than as strong systematic departures from the design. Movements with very small samples were retained in the table for completeness but were not used as a basis for strong inferential conclusions.

Overall, the statistical results indicate, on a descriptive and exploratory basis, that curvature deviations are not uniform across intersections, but are strongly movement-specific. This confirms the need for movement-by-movement assessment rather than reliance on nominal intersection-level geometry alone.

4.3. Sensitivity Analysis of the Movement-Level Classification

The movement-level classification presented above is based on pragmatic thresholds for median CDPs and for the predominance of trajectories on the same side of the theoretical value. Because these thresholds were introduced as descriptive screening criteria rather than as formal design standards, a sensitivity analysis was carried out to examine whether the main classification patterns remained stable under alternative assumptions. The analysis was intended to assess the robustness of the strongest movement-level patterns and to identify borderline cases where the classification depends more strongly on the selected rule or trimming convention.

The sensitivity of the adopted rule was further examined under alternative median CDPs and predominance thresholds (

Table 3. Sensitivity of movement-level classification under alternative CDP threshold combinations.

Movement	Baseline Class (±10%, 70%)	Stability Across Tested Threshold Combinations	Notes
K01 1_4	Flattened	Sensitive to threshold choice	Shifts to mixed under stricter thresholds
K01 2_1	Tightened	Stable across all tested combinations	stable
K01 4_3	Tightened	Stable across all tested combinations	Stable
K02 2_1 leftmost	Mixed	Stable across all tested combinations	Stable
K02 1_2	Flattened	Sensitive to threshold choice	Shifts to mixed under stricter thresholds
K02 2_1	Mixed	Sensitive to threshold choice	Borderline; shifts from mixed under more permissive thresholds
K03 1_2	Mixed	Stable across all tested combinations	Stable
K03 2_1	Mixed	Sensitive to threshold choice	Borderline; shifts from mixed under more permissive thresholds
K04 1_1	Mixed	Sensitive to threshold choice	Borderline; shifts from mixed under more permissive thresholds
K04 1_4	Mixed	Stable across all tested combinations	Stable
K04 2_1	Mixed	Stable across all tested combinations	Stable
K04 3_2	Mixed	Stable across all tested combinations	Stable
K04 4_3	Flattened	Stable across all tested combinations	Stable
K04 4_4	Mixed	Sensitive to threshold choice	Borderline; shifts from mixed under more permissive thresholds
K05 2_1	Mixed	Stable across all tested combinations	Stable
K05 3_2	Tightened	Stable across all tested combinations	Stable
K05 4_3	Mixed	Sensitive to threshold choice	Borderline; shifts from mixed under more permissive thresholds

). The strongest flattening and tightening cases remained stable, whereas borderline movements shifted mainly between a systematic class and the mixed category.

This rule was selected in order to distinguish clearly between systematic and merely incidental deviation patterns. In practice, a negative median CDP indicates an overall tendency toward flatter-than-theoretical paths, whereas a positive median CDP indicates an overall tendency toward tighter-than-theoretical paths. The 70% predominance criterion then ensures that a movement is classified as flattened or tightened only when this tendency is shared by a clear majority of the observed trajectories. Therefore, movements with modest median deviations or with strongly mixed distributions remain in the geometrically consistent/mixed category.

• A threshold sensitivity check using alternative median CDP thresholds (±5%, ±10%, and ±15%) and predominance thresholds (60%, 70%, and 80%) further indicated that the strongest movement-level patterns remained stable, whereas borderline cases near the adopted thresholds were more sensitive to rule specification. On this basis, the ±10%/70% combination was retained as a pragmatic compromise between interpretive stability and sensitivity to systematic movement-level deviation. A targeted sensitivity check using mild asymmetric perturbations of the comparison extent likewise showed that the strongest flattening and tightening movements remained stable, whereas a borderline mixed case proved more sensitive to trimming choice.

A targeted trimming-sensitivity check was also performed on representative movements using mild asymmetric perturbations of the comparison extent (

Table 4. Sensitivity of representative movements to mild asymmetric perturbations of the comparison extent.

Movement	Baseline Median CDP (%)	Entry-Longer Median CDP (%)	Exit-Longer Median CDP (%)	Baseline Class	Classification Change?
K01 4_3	+36.85	+38.35	+36.85	Tightened	No
K04 4_3	−41.80	−31.15	−47.33	Flattened	No
K04 1_4	+2.53	−2.83	+4.86	Mixed	No
K05 2_1	−5.41	−11.75	−5.41	Mixed	Yes (to flattened under entry-longer variant)

). The strongest flattening and tightening cases remained stable, whereas one borderline mixed case shifted classification under one asymmetric variant.

4.4. Geometric Design Implications

From a design and safety perspective, the three movement types identified above through the adopted descriptive rule have different implications.

4.4.1. Movements with Systematic Flattening

In movements classified here as flattened-turn, according to the adopted descriptive rule, the effective turning radius actually used by drivers is larger than the nominal design radius. This situation typically arises in layouts with:

• Generous corner radii and wide lanes;
• Long distances between opposite stop lines;
• Absence of physical or optical constraints (islands, lane separators, or markings) that would keep vehicles near the theoretical inner edge of the turn.

Such geometry may allow drivers to maintain higher speeds through the turn. Even if the design radius itself is compliant with guidelines, the operating radius—and hence the operating speed—may in practice move toward values associated with less favourable safety conditions, especially for pedestrian conflicts on the exit crossing and for interactions with opposing or crossing vehicles. However, in the present study, this interpretation remains geometry-based and descriptive, since speed and conflict indicators were not integrated into the movement-level curvature classification.

In movements classified as flattened-turn, the curvature-based evidence can therefore be used as a screening signal for speed-management-oriented geometric review, such as:

• Reducing the effective corner radius by tightening the curb line;
• Repositioning or extending channelising islands to prevent vehicles from cutting across the lane;
• Narrowing the entry and/or exit lanes (or creating visual narrowing with markings);
• Shifting the exit crosswalk slightly upstream or re-aligning it so that pedestrians are not located in zones where flattened paths pass at higher speed.

These measures are consistent with contemporary roundabout and intersection guidance, which emphasises the role of radii and deflection in controlling speeds rather than relying solely on posted limits.

4.4.2. Movements with Systematic Tightening

In movements classified here as tightened turns, according to the adopted descriptive rule, the observed trajectories indicate that the available space for the turn may be effectively smaller than assumed, or that the intended design path is not naturally “readable” from the driver’s viewpoint. Contributing geometric factors include:

• Narrow or offset left-turn lanes;
• Restrictive corner radii combined with adjacent obstacles (islands, poles, and parking);
• Exit lanes that are laterally shifted relative to the entry lane, forcing an abrupt change in direction;
• Misalignment between the theoretical entry/exit tangents and the corridor that drivers visually perceive as the natural continuation of their lane.

Tightened-turn movements may correspond to higher lateral-acceleration demand and sharper steering inputs at a given speed, which can be uncomfortable and, for some vehicle types, operationally problematic. From an engineering point of view, such paths may also be associated with lane-keeping difficulties or reduced lateral clearance within the intersection. At the same time, these implications should be interpreted cautiously in the present paper because the current analysis did not incorporate speed, vehicle class effects, or direct conflict-based measures at the movement level.

In such movements, the curvature-based analysis can be used to motivate a closer geometric review aimed at smoother path negotiation, for example:

• Modestly increasing the effective corner radius where feasible;
• Realigning entry and exit lane centrelines to better match the envelope of the observed paths;
• Adjusting the location of stop lines and crosswalks so that critical pedestrian areas are not directly exposed to the sharpest part of the turn.

4.4.3. Geometrically Consistent Movements

In movements that remain within the geometrically consistent or mixed category, the existing geometry appears either to guide drivers broadly along the intended path or to produce no sufficiently strong unidirectional deviation from it. In such cases, the immediate priority is not necessarily to change the basic geometry, but rather to preserve the successful layout features in future resurfacing or minor works and to use these movements as reference examples when reviewing less consistent ones at the same or similar intersections.

4.5. From Curvature Indicators to Safety-Oriented Design Adjustments

A key advantage of the curvature-based approach is that it produces simple numerical indicators (RDP and CDP) that can be interpreted in relation to concrete design questions at the movement level. In practical terms, the proposed framework can be used as a screening tool to identify movements that deserve closer geometric review. In the present study, this screening was based on explicit descriptive criteria: movements with a median CDP ≤ −10% and at least 70% of trajectories flatter than theoretical were treated as systematic flattening cases, whereas movements with a median CDP ≥ +10% and at least 70% of trajectories tighter than theoretical were treated as systematic tightening cases. The remaining movements were interpreted as near-design or mixed cases.

This is useful because it provides a compact and transparent way to move from a large set of individual trajectories to a movement-level engineering reading. Instead of examining hundreds of overlaid paths separately, the analyst can identify where the observed use of the available geometry departs materially from the theoretical path and where it remains broadly consistent with design.

At the same time, the present framework should not be read as a stand-alone safety model. Curvature deviation does not by itself quantify crash risk, nor does it isolate the contribution of a single geometric variable. Rather, it highlights movements where the effective turning path differs in a systematic way from the path implied by the design, and where a closer engineering review may, therefore, be justified. In this sense, the proposed indicators can support early-stage geometric review, movement-by-movement diagnostics, and future analyses that incorporate speed, comfort, surrogate safety measures, and formal statistical testing of geometric variables.

A supplementary robustness-oriented comparison was also performed against two simpler movement-level descriptors, namely a basic radius-offset measure and a trajectory-dispersion measure. At the movement level, the magnitude of CDP showed a strong monotonic association with the simple radius-offset descriptor (Spearman ρ ≈ 0.69, p ≈ 0.002), whereas no comparably strong association was observed with path dispersion. In practical terms, this means that the proposed curvature-based indicators are not merely restating how dispersed the trajectories are within a movement. Rather, they capture the degree of systematic departure from the theoretical path, while dispersion remains a complementary descriptor of within-movement variability.

In addition, an exploratory movement-level association check was carried out between the theoretical turning radius and the magnitude of curvature deviation. More specifically, the absolute median CDP of each analysed movement was compared with the corresponding theoretical radius. This check indicated a moderate positive monotonic association (Spearman ρ ≈ 0.51, p ≈ 0.037), suggesting that movements with larger theoretical radii tended, on average, to exhibit larger absolute departures from the theoretical path. At the same time, no equally clear monotonic relation was observed for the signed median CDP, which indicates that the direction of deviation (flattening versus tightening) is not determined by radius alone. This is important because it supports the view that the geometric radius contributes to the magnitude of observed deviation, whereas the direction of that deviation remains sensitive to local movement-specific configuration, including lane arrangement, channelisation, and alignment. Taken together, these additional checks strengthen the methodological role of the proposed indicators as movement-level screening tools, rather than as stand-alone safety predictors.

5. Movement-Specific Curvature Patterns

Based on the principles of the previous chapters, this section presents the curvature analysis for each intersection (K01-K05) and left-turn movement, focusing on whether the observed trajectories tend to be flatter or tighter. The purpose of this analysis is to focus on geometry and its safety implications, rather than on detailed driver behaviour.

5.1. Intersection K01

At intersection K01, three left-turn movements were analysed: 1→4, 2→1 and 4→3 (Figure 3).

• K01, movement 1→4.

Median RDP ≈ +16%.

About 94% of trajectories have an RDP > 0.

A clear flattening tendency is evident in this movement, since drivers predominantly follow paths with a larger radius than the theoretical design arc.

• K01, movement 2→1.

Median RDP ≈ −13.6%.

About 82% of trajectories have an RDP < 0.

A consistent tightening tendency is identified in this movement, since the recorded turning paths are generally tighter than the design arc.

• K01, movement 4→3

Median RDP ≈ −27.2%.

100% of trajectories have an RDP < 0.

This movement appears to be one of the strongest tightening cases in the dataset. All recorded paths are tighter than the design, with a very large typical deviation.

Based on the above, intersection K01 exhibits strong asymmetry, since movement 1→4 is notably flattened, while movements 2→1 and especially 4→3 are much tighter than the theoretical geometry. This suggests that the arrangement of lanes, islands and corner radii allows the drivers to open their turns in one direction but constrains them in the opposite direction.

5.2. Intersection K02

Left-turn movements 1→2, 2→1 (all lanes) and 2→1 (leftmost lane) were processed in intersection K02 (Figure 4).

• K02, movement 1→2.

Median RDP ≈ +11.8%.

About 87% of trajectories have an RDP > 0, meaning that this movement shows a flattening tendency since drivers tend to consistently choose more open paths than the design arc.

• K02, movement 2→1 (all lanes).

Median RDP ≈ +9.8%.

About 65% of trajectories have an RDP > 0 and, therefore, a moderate flattening tendency is evident. Consequently, most drivers open up the turn, though less strongly than in movement 1→2.

• K02, movement 2→1 (leftmost lane).

Median RDP ≈ −5.4%.

About 57% of trajectories have an RDP < 0. On the innermost left-turn lane, the pattern shifts toward mild tightening since most of the drivers tend to follow tighter paths than the theoretical arc.

This intersection illustrates how lane allocation and lateral positioning influence turning geometry by comparing the same nominal movement (2→1), which can be either flattened when considering all lanes together or tightened in the leftmost lane.

5.3. Intersection K03

Two movements were analysed at intersection K03: 1→2 and 2→1 (Figure 5).

• K03, movement 1→2.

Median RDP ≈ +1.3%.

About 60% of trajectories have an RDP > 0. However, the computed deviations are small and roughly centred around zero, with a slight preference for very mild flattening. Therefore, this movement is close to the theoretical arc.

• K03, movement 2→1.

Median RDP ≈ −6.4%.

About 71% of trajectories have an RDP < 0. This movement shows a tightening tendency since most drivers adopt a path tighter than the design curve.

Overall, in intersection K03, two different behaviours are identified. In one movement (1→2), drivers tend to follow the design paths, and, in another movement (2→1), drivers tend to tighten their turn, which may reflect a smaller effective turning space and may point to potential issues in alignment or available width for the turn.

5.4. Intersection K04 (Single-Lane Roundabout)

Intersection K04 is a four-arm, single-lane roundabout with entries and exits on Tsaldari and Apostolou Kougiami streets (Figure 6). For the purposes of this study, six left-turn-like movements were analysed, defined by specific entry and exit approaches around the circulatory roadway: 1→1, 1→4, 2→1, 3→2, 4→3 and 4→4. All movements share the same theoretical design radius for the circulating path (approximately 11 m), but the observed trajectories show markedly different curvature patterns.

Movements 1→1, 1→4 and 4→4 exhibit systematic tightening. For these entry–exit pairs, almost all trajectories have negative RDP values, and the median RDP is clearly below zero. In geometric terms, vehicles tend to follow paths that are closer to the central island than the theoretical design arc. A plausible geometric interpretation is that, for these approaches, the available width between the splitter islands, the central island and the kerbs encourages drivers to negotiate the roundabout on tighter circulating paths than those implied by the theoretical design arc. Such tightening may be associated with lower circulating speeds, but also with sharper steering demand and reduced lateral clearance near the inner edge, especially for larger vehicles. However, in the present study, these implications should be read as a geometry-based interpretation rather than as direct proof of vehicle dynamics or a safety effect.

In contrast, movements 2→1 and 3→2 show consistent flattening. The majority of trajectories have positive RDPs, and the median RDP is clearly above zero. Here, vehicles travel closer to the outer edge of the circulating lane, opening up the path and using a larger effective radius than the design arc. A plausible geometric interpretation is that these movements provide sufficient usable space along the circulatory lane for drivers to move toward the outer edge and open up the turn, especially where the kerb lines and markings do not strongly guide them toward the inner part of the roundabout. In practical terms, such paths may offer higher speed potential, particularly near the downstream pedestrian crossings at the exits. However, this should again be understood as a geometry-based screening interpretation, since speed was not directly integrated into the present movement-level curvature analysis.

Movement 4→3 stands out as the most extreme flattening case at K04 and in the entire dataset. Almost all trajectories are significantly flatter than the design arc, with a large positive median RDP. Vehicles tend to track close to the outer kerb, thereby reducing the effective deflection around the central island relative to the theoretical path. A plausible interpretation is that the current layout—including the entry alignment from approach 4, the shape of the splitter island, and the location of the exit on approach 3—allows drivers to minimise steering input and to negotiate the movement with relatively high speed potential through the roundabout. In the present paper, however, this should be read as a geometry-based explanation of the observed curvature pattern rather than as direct proof of actual operating speed.

Intersection K04 demonstrates that, even in a compact single-lane roundabout with a constant design radius, the combination of entry alignment, splitter island geometry, and central island shape can lead to very different effective turning radii for different entry–exit pairs. Our analysis shows that some circulating movements are systematically tightened around the central island, while others are opened up toward the outer edge. This is important because it suggests that even within the same compact roundabout, entry–exit geometry can shape the effective path in very different ways. In practical terms, these movement-level patterns can be treated as screening evidence for issues related to speed management, lane-keeping, and pedestrian exposure at the exits, rather than as direct proof of a safety effect.

5.5. Intersection K05

At K05, three movements were analysed: 2→1, 3→2 and 4→3 (Figure 7).

• K05, movement 2→1.

Median RDP ≈ +4.8%.

About 60% of trajectories have an RDP > 0. This movement shows a mild flattening tendency but remains below the threshold required for systematic flattened-turn classification.

• K05, movement 3→2.

Median RDP ≈ −20.8%.

About 99% of trajectories have an RDP < 0, and, therefore, this movement appears to be one of the strongest tightening cases in the study. Virtually all vehicles follow turning paths substantially tighter than the theoretical design arc.

• K05, movement 4→3.

Median RDP ≈ −6.5%.

About 74% of trajectories have an RDP < 0.

This movement shows a tightening tendency, although less pronounced than 3→2 and closer to the mixed/near-threshold range.

In geometric terms, K05 is dominated by tightening behaviour, although the three analysed movements do not all show the same degree of systematic deviation under the adopted descriptive rule.

5.6. Summary: Where Turns Are Flatter and Where They Are Tighter, and Where Behaviour Remains Mixed

Across the analysed intersections, the curvature results reveal a clearly movement-specific pattern. However, under the descriptive rule adopted in this study, not all deviations from the theoretical path were treated in the same way. Some movements showed a strong and unidirectional tendency toward flattening or tightening, whereas others remained closer to the design or displayed substantial internal heterogeneity.

5.6.1. Movements Classified as Flattened-Turns

The clearest flattening cases were observed in movements where the median curvature deviation was sufficiently negative, and the large majority of trajectories were flatter than the theoretical path. In the present dataset, this pattern was observed most clearly in one movement at K01 and one movement at K02, where drivers systematically opened the turn relative to the design geometry. These are, therefore, the strongest examples of movement-level flattening in the analysed urban intersections.

At the roundabout site, some circulating movements also displayed pronounced flattening behaviour. In geometric terms, these cases indicate that drivers tended to track the outer part of the circulatory roadway and negotiate the movement with larger effective radii than those implied by the corresponding theoretical arcs. Among these roundabout cases, one movement stands out as the strongest flattening pattern in the full dataset.

5.6.2. Movements Classified as Tightened-Turns

Clear tightening patterns were observed in movements where the median curvature deviation was sufficiently positive and the corresponding share of tighter-than-theoretical trajectories clearly predominated. In the signalised intersections, this was especially evident in two movements at K01, one of which represents one of the strongest tightening cases in this study. At K05, the movement from the southern approach toward the eastern leg also showed very strong tightening, with the great majority of trajectories falling on the tighter side of the theoretical path.

At the roundabout site, several movements also exhibited tightening behaviour. This is important because it shows that, depending on the entry–exit pair, the same nominal roundabout geometry can lead to quite different effective path curvatures.

5.6.3. Movements Remaining Geometrically Consistent or Mixed

Other movements did not satisfy the descriptive criteria for a systematic classification. The clearest example is the combined two-lane movement at K02, where the overall distribution is broad and internally heterogeneous. This is important because it shows that, when multiple left-turn lanes are pooled together, the resulting curvature pattern may mask lane-specific behaviour and, therefore, should not automatically be interpreted as a systematic geometric problem.

Similarly, some movements displayed only mild deviation from the theoretical path and, therefore, remained below the threshold required for a systematic flattened-turn or tightened-turn classification. In these cases, the results are better interpreted as near-design or mixed behaviour rather than as strong movement-level departure from the design.

For movements represented by small samples, the observed patterns should be treated as indicative rather than definitive. These cases still provide useful descriptive signals.

This interpretation is also broadly consistent with an exploratory movement-level association check, which suggests that the magnitude of curvature deviation tends to increase with the theoretical turning radius, even though the direction of deviation remains movement-specific. In other words, the radius appears to matter for how strongly observed paths depart from design, but not by itself for whether that departure takes the form of flattening or tightening.

Overall, the movement-level curvature analysis shows that the theoretical design radii alone are not sufficient to describe how vehicles actually negotiate left turns at urban intersections. At the same time, the present results should be read as geometry-based screening evidence rather than as direct causal proof. In practice, the combination of lane layout, local alignment, island geometry, stop-line placement and pedestrian-crossing position appears to shape the effective turning path in ways that are highly movement-specific. This is exactly why a movement-by-movement reading of the curvature is more informative than relying on nominal design geometry alone.

6. Conclusions and Future Work

This study outlines a curvature-based method used to compare theoretical design paths with actual left-turn movements observed at urban intersections and a single-lane roundabout. The data came from 2319 left-turn trajectories captured during the morning rush hour at five different intersections in Thessaloniki (referred to as K01 through K05). These trajectories were pulled from street-level video footage and analysed using a computer vision pipeline combined with circle-fitting techniques to ensure accurate, georeferenced positioning.

6.1. Summary of Findings

This study examined how observed left-turn vehicle trajectories at urban intersections differ from the theoretical circular paths implied by the design geometry. Using video-extracted trajectories from five intersections in Thessaloniki, the analysis compared observed and theoretical turning curvature through two simple indicators, namely the Radius Difference Percentage (RDP) and the Curvature Difference Percentage (CDP).

The key descriptive finding of the analysis—interpreted in light of the exploratory nature of the underlying trajectory accuracy—is that curvature deviations are not uniform across intersections, but are strongly movement-specific. The same intersection may include movements with systematic flattening, systematic tightening and mixed behaviour. Therefore, the practical contribution of the proposed approach lies in its ability to identify and prioritise specific left-turn movements where the theoretical design path does not fully represent observed driver behaviour. Some movements displayed a clear tendency toward flattening, meaning that drivers systematically opened the turn relative to design, while others displayed a clear tendency toward tightening and, therefore, followed smaller effective radii than those implied by the theoretical arc. Other movements remained closer to the design or showed mixed internal behaviour. An exploratory movement-level association analysis also indicated a moderate positive association between the theoretical turning radius and the magnitude of curvature deviation, although the direction of deviation remained dependent on the local movement configuration. In order to interpret these patterns consistently, this study introduced a transparent descriptive rule based on the median CDP and the predominance of trajectories on the same side of the theoretical value. This is important because it allows the analysis to move from isolated path overlays to a movement-level reading of whether the observed behaviour is systematically flatter, tighter, or broadly near the design.

From an engineering point of view, the proposed framework offers a practical way to screen intersection movements and identify cases where the effective use of the available geometry differs materially from the design intent. At the same time, the present findings should be interpreted as geometry-based screening evidence rather than as direct proof of a safety effect.

Overall, this paper contributes a practical and interpretable curvature-based approach for comparing design and observed left-turn paths at scale across multiple intersections. Future work can build on the same recorded dataset to address additional scientific questions, including movement-level speed effects, comfort-related measures, conflict occurrence, and the statistical role of specific geometric variables in shaping real turning behaviour. This study provides a diagnostic framework for comparing designed and observed turning paths and for prioritising movements that may require closer examination in future design reviews, field assessments or safety-oriented studies.

6.2. Implications for Geometric Design and Safety

The findings of this study suggest that left-turn trajectories at urban intersections should not be assessed only in terms of nominal design geometry. In practice, the same geometric layout may be used in different ways by drivers, depending on the specific movement, lane arrangement, channelisation, and local visual guidance. This movement-specific reading is also consistent with the exploratory quantitative results of the present study, which suggest that larger theoretical radii tend to be associated with stronger absolute departures from the design, while the direction of departure depends on additional local design features. This is important because the effective path actually followed by vehicles may differ materially from the path implied by the theoretical design arc.

From a geometric point of view, systematic flattening may indicate that the layout allows drivers to negotiate the turn with a larger effective radius than intended. In practical terms, this can be treated as a screening signal that the movement may offer higher speed potential through the turn and a broader spatial spread of vehicle paths than expected from the design alone. Systematic tightening, on the other hand, may indicate a more constrained path negotiation, with higher curvature demand, sharper steering, or reduced lateral clearance within the intersection. However, these are geometry-based readings only; the present analysis did not integrate speed, vehicle-dynamics measures, or surrogate safety indicators at the movement level, and direct inferences about safety effects are, therefore, not warranted.

Accordingly, the proposed curvature-based framework can support geometric review in a practical way. Movements that show systematic flattening or tightening relative to the theoretical path can be prioritised for closer examination, whereas movements that remain near design or display mixed behaviour can be interpreted more cautiously. In this sense, the framework is particularly useful as an early-stage diagnostic tool: it helps identify where the effective use of the available geometry differs from design intent and where a more detailed engineering assessment may therefore be justified.

Overall, the practical value of the method lies in its ability to translate large sets of observed trajectories into a compact movement-level reading. This can support consistency checks, movement-by-movement review, and the prioritisation of locations for more detailed analysis. At the same time, the present discussion should be read in light of an important limitation: without direct integration of speed data, lateral acceleration, and related safety implications cannot be quantified explicitly and are, therefore, interpreted here only at the screening level. Future work could build on the same recorded dataset to examine how these curvature patterns relate to speed, comfort, surrogate safety measures, and the statistical contribution of specific geometric variables.

6.3. Methodological Contributions

Beyond the specific study sites, the proposed framework contributes a practical method for geometric safety assessment:

• It relies on georeferenced trajectories obtained from video and computer vision, which are increasingly available in both research and practice. In the present case, the extracted movement-level trajectories were also checked for internal consistency in terms of confidence, temporal continuity and retained track quality after preprocessing.
• It uses simple, interpretable indicators (RDP and CDP) that can be computed automatically for large numbers of trajectories and summarised per movement.
• It provides a direct link between observed paths and design geometry by expressing deviations in terms of radius and curvature—quantities that designers already use.

The same framework can be embedded in microscopic traffic simulation or surrogate-safety analyses by ensuring that simulated trajectories reproduce the curvature distributions observed in the field. In that way, safety evaluations based on conflicts or time-to-collision can be grounded in realistic turning paths, not only in nominal geometry.

6.4. Limitations and Directions for Future Research

The present study should be read in light of several limitations. First, the analysed dataset includes only passenger-vehicle trajectories recorded during the morning peak period. Although this provides a sufficiently rich sample for movement-level comparison, it does not capture the full temporal variability that may emerge under other demand and visibility conditions. In particular, the afternoon peak recordings were not included in the present analysis because the lower ambient light reduced the reliability of trajectory extraction.

Second, the theoretical path used for comparison was represented by a simplified circular arc and by a symmetric comparison extent defined through the condition |KB| = |KA|. This choice was made in order to standardise the geometric comparison across heterogeneous urban layouts in a transparent and repeatable way. However, this does not fully reproduce the asymmetry and local variability that may exist in real driver-specific turning paths. In particular, if the actual manoeuvre develops more gradually on one side of the turn than on the other, the symmetric comparison extent may either truncate part of the relevant curved segment or include part of a straighter segment and may, therefore, influence the estimated effective radius and the resulting curvature deviations. A targeted sensitivity check on representative movements showed that the strongest flattening and tightening cases remained stable under mild asymmetric perturbations of the comparison extent, whereas a borderline mixed case shifted classification under one asymmetric variant. This suggests that the adopted symmetric comparison range is sufficiently robust for identifying strong movement-level patterns, but that movements close to the decision thresholds should be interpreted more cautiously. Future work should examine this issue more systematically by testing asymmetric comparison ranges across a broader set of movements.

In addition, the present paper did not include a dedicated residual-based robustness assessment of the Kåsa circle-fitting results, nor a formal benchmarking of Kåsa against alternative circle-fitting methods. This is important because the sensitivity of fitted radii to residual outliers may affect movement-level curvature estimates, especially for shorter trajectories and borderline cases. Nevertheless, the methodological assessment of the overall framework was strengthened through threshold-sensitivity testing, targeted trimming-sensitivity checks, and supplementary comparison with simpler movement-level descriptors. Future work should extend this methodological assessment further by incorporating explicit residual diagnostics and alternative fitting formulations.

In line with this, it must be stated explicitly that the curvature estimates in this study have not undergone any residual validation; the reliability of the RDP and CDP values is unknown, and they shall not be used for quantitative comparison or as a design basis. This statement refers to the absolute, precision-survey level of validation. The relative, within-study comparison between movements—obtained under an identical extraction and fitting protocol—is what the present descriptive screening framework relies upon and is reported with that scope in mind.

Third, the present paper focuses on curvature-based geometric comparison and, therefore, does not integrate speed, lateral acceleration, surrogate safety measures, or formal correlation analysis between curvature indicators and individual geometric variables. This is important because such analyses would allow the same recorded dataset to be used to address additional scientific questions, including the operational and safety implications of systematic flattening or tightening patterns. In addition, the present paper does not attempt a full benchmarking of the RDP and CDP against the wider family of trajectory evaluation indicators. Nevertheless, a supplementary comparison with a simple radius-offset descriptor and a path-dispersion descriptor was carried out in order to assess whether the proposed indicators mainly reflect generic spread or more specific design-versus-observed deviation. The results suggest that curvature-based indicators are closely related to a simple geometric offset, but not reducible to trajectory dispersion alone. Future work can extend this comparison further by incorporating additional descriptors of lateral offset, path consistency and comfort-related kinematics. In addition, the movement-level classification rule adopted in the present study should be understood as provisional and dataset-specific. Its transferability and stability should be examined further on other datasets, and future work should assess how sensitive the flattened-turn/tightened-turn/mixed classification is to alternative threshold choices.

Fourth, some movements were represented by relatively small samples. These cases were retained for completeness, but they are interpreted cautiously and treated as indicative patterns rather than as a strong basis for generalisation.

Fifth, the georeferenced trajectories used in this study were not validated against independently surveyed vehicle coordinates. The homography-based transformation was verified through reprojection residuals of the ground control points and visual overlay on the intersection layout, and an additional movement-level quality review of the extracted trajectories indicated generally high track confidence, stable temporal continuity after resampling, and high retention of classified trajectories after preprocessing. However, a dedicated coordinate-level validation study was outside the scope of the present paper and remains a direction for future work. This is important because coordinate-level uncertainty may propagate into the fitted circle parameters and, therefore, into the estimated RDP and CDP values, especially for shorter trajectories or movements with tighter curvature.

Based on the control-point configuration and the spatial resolution of the video recordings, positional errors on the order of a few tens of centimetres are plausible for individual trajectory points. For movements with large theoretical radii (e.g., R_theoretical > 20 m), such positional uncertainty could translate into RDP and CDP errors well below the ±10% classification threshold applied in Section 4.2, and the movement-level classification could be expected to remain stable. For movements with smaller theoretical radii (e.g., R_theoretical < 10 m, as in the K04 circulatory roadway), the relative influence of positional uncertainty on the fitted curvature is proportionally larger, and the reported RDP and CDP values for those movements should be interpreted with additional caution. This distinction is noted here as a qualitative order-of-magnitude consideration; a formal uncertainty propagation analysis was not conducted and remains a direction for future work.

Accordingly, the geometric accuracy of trajectory coordinates has not undergone any quantitative validation; the RDP and CDP values are for exploratory illustration only and lack credibility for quantitative comparison. As above, this concerns absolute coordinate-level accuracy; movement-level comparisons within the dataset remain internally consistent because all trajectories were processed under the same calibration and extraction pipeline.

Overall, these limitations do not invalidate the descriptive contribution of this study. They define its scope more clearly: the present paper proposes a practical movement-level curvature screening framework, while future research can build on the same recorded trajectories to investigate speed effects, comfort-related measures, conflict occurrence, and the statistical contribution of specific geometric features.