Limits of Transferring User-Defined Quantity Takeoff Rules in 2D CAD and 3D BIM Using Semantic Vertices

Featured Application

Automating the transfer of user-defined dimensions across geometrically identical objects.

Abstract

In construction projects, dimensioning automation is now implemented with relatively high precision in both 2D CAD and 3D BIM environments when conditions are clearly defined. For identical or similar objects with the same attributes, dimension-based quantity takeoff formulas can be automated, and such automated formulas can be repeatedly applied as long as the geometric form and attributes remain unchanged. However, this automation is feasible only within limited environments and under pre-defined rules. Once the geometry is slightly altered or geometric identity is disrupted, directly applying the existing automated mechanism becomes structurally constrained. This is because (1) quantity takeoff formulas are difficult to standardize universally, (2) object orientation is often difficult to determine consistently, and (3) even minor geometric changes can alter the meaning of dimensions, making automatic interpretation problematic. Accordingly, this study aims to systematically and experimentally analyze the practical limits of transferring semantic-vertex-based quantity takeoff formulas. To this end, a single isolated footing is adopted as a common reference object, and the limits of user-defined dimension-based automation in 2D CAD and 3D BIM environments are evaluated through three core comparisons: 1. 2D–3D reliability comparison: consistency of final quantity results within an acceptable tolerance range for the isolated footing; 2. User dependency assessment (3D-focused): the extent to which quantity takeoff formulas in 3D BIM depend on user judgment; 3. User-defined dimension transfer limits: the practical limits of transferring user-defined dimensions between geometrically identical isolated footings in both 2D and 3D environments. Through this analysis, this study empirically confirms that geometric variation is a key factor structurally constraining the transferability of user-defined quantity takeoff formulas. Furthermore, as future work, it proposes directions for linking automated dimension generation with quantity takeoff formulations.

1. Introduction

Dimensioning in construction projects provides fundamental information for quantity takeoff, cost estimation, and construction quality control [1]. Dimensions defined during the design phase serve as the basis for subsequent quantity calculation formulas and cost estimation procedures; therefore, how dimensions are defined and interpreted directly shapes the overall reliability and direction of quantity takeoff processes.

In traditional 2D CAD-based practice, users typically interpret dimensions displayed on drawings and then apply selected quantity calculation formulas. In this workflow, dimensional reference points are not explicitly defined at the object level, and both drawing interpretation and quantity estimation rely heavily on user experience and judgment [1]. Even when the same drawings are used, different users may adopt different calculation formulas and interpretation procedures, leading to identical final quantities but substantially different underlying processes [1]. This indicates that variability in quantity takeoff is not solely a result of geometric differences but is also strongly influenced by how users define, interpret, and operationalize dimensions in practice.

In contrast, 3D model-based environments such as Building Information Modeling (BIM) integrate object geometry and attributes, offering the potential for more structured and consistent quantity takeoff processes. Monteiro and Martins compared quantity estimation in 2D CAD and BIM environments and emphasized that although BIM enables automation, the modeling criteria established in early design stages are critically important [2]. Similarly, Wahab and Wang analyzed real project data and reported that BIM-based quantity takeoff can improve efficiency and reliability in quantity verification compared to traditional 2D approaches [3]. However, prior studies have primarily focused on efficiency and accuracy, while controlled investigations explicitly addressing how dimensional criteria are defined and interpreted remain limited.

Despite the increasing adoption of 3D modeling for quantity takeoff in practice, dimension creation and input still rely heavily on manual work [4]. Although certain processes can be partially automated, dimensions are often generated and entered individually based on user interpretation, even under identical geometric and working conditions. Consequently, dimensioning procedures are repeatedly performed, and the practical benefits of automation remain constrained in real construction projects [5].

A fundamental reason for this limitation is that quantity calculation formulas are not purely geometric computations but are governed by organizational policies and practical rules. Different organizations and practitioners adopt distinct criteria for opening deductions, rebar anchorage and splicing, construction tolerances, and measurement conventions. As a result, two users may arrive at the same final quantities for identical objects yet rely on different dimensional definitions and calculation logics.

Therefore, even when users largely agree on the “identity” of objects, the quantity rules applied to those objects often differ, and the direct reproduction and transfer of such rules to other identical objects are currently very limited. This issue becomes more critical as object geometries grow more complex and multiple calculation formulas become feasible. Consequently, neither 2D CAD nor current BIM-based 3D objects can fully guarantee the reproducibility and transferability of user-defined quantity rules.

In particular, the reproducibility and transferability of quantity rules for identical objects depend on how “identity” is defined. When objects share the same geometry under identical geometric conditions, the transfer of user-defined rules is relatively straightforward. However, for similar geometries under similar conditions, rule transfer may be possible but typically depends on user judgment. Moreover, when object geometry is partially modified, rule transfer may or may not be feasible, requiring case-by-case evaluation.

Accordingly, the central research question of this study can be formulated as follows:

If the transfer of user-defined quantity rules is possible when object identity is ensured, how should object identity be defined?

In current industry practice, identity judgment for dimension automation relies predominantly on user interpretation or limited object attribute information. Starting from this problem recognition, this study assumes that user-defined quantity mechanisms should fundamentally be grounded in geometric identity rather than solely on object attributes or IFC properties. Even when identity can be inferred from attributes, dimensional information is still often redefined independently of quantity rules, which limits systematic automation.

To clarify this limitation, this study first conducts a proof-of-concept (PoC) experiment using two basic models: (1) a cube penetrated by a circular cylinder along the Y-axis and (2) a cube penetrated by an elliptical cylinder along the Y-axis. Through this experiment, this study conceptually examines:

• How geometric identity and similarity can be defined;
• How user-defined quantity rules may differ;
• Why the transfer of such rules cannot be fully guaranteed.

Subsequently, independent foundation concrete and reinforcement models are analyzed from three comparative perspectives to empirically validate the limitations of transferring user-defined quantity rules. Based on these findings, this study proposes new directions for automating dimension generation and quantity takeoff.

1.1. Research Objective

The objective of this study is to identify the conditions under which user-defined quantity rules can be reproduced and transferred based on geometric identity and clarify the inherent limitations of such transfers. Furthermore, this study demonstrates that geometric identity and geometric similarity do not constitute equivalent conditions for automation in dimensioning and quantity takeoff and discusses the implications for large-scale and repetitive object automation. To address this objective, the present study applies a dimension definition approach based on semantic vertices. A semantic vertex is an explicitly defined key vertex in object geometry that functions as a reference for dimensioning and functional interpretation, providing an object-level mechanism that links dimension generation with quantity takeoff in a traceable manner [6].

1.2. IFC Location in This Study

This study does not aim to replace or modify the IFC (Industry Foundation Classes) schema itself. Instead, the IFC is treated as an intermediate model within a broader workflow. Specifically, the proposed method remeshes IFC geometry and subsequently conducts geometric exploration on the remeshed model. Accordingly, the approach remains fundamentally IFC-based, employing IFC in a complementary manner rather than bypassing it. This positioning clarifies that this study does not undermine IFC integrity but instead expands experimental capabilities for automated dimensioning and transferability analysis within existing IFC environments.

1.3. General Definition of Geometric Identity

In this study, geometric identity is defined independently of any specific modeling method or BIM attribute. Two objects are considered geometrically identical if, after removing differences in coordinate systems, position, and orientation, they exhibit the same three-dimensional shape and their topological relationships among faces, edges, and vertices can be mapped one-to-one. Conversely, when shapes are similar but such correspondence is only partial or measurement results differ, the objects are classified as geometrically similar rather than identical.

1.4. Acceptable Tolerance Ranges

The ‘acceptable tolerance ranges’ used in this study do not refer to dimensional tolerances defined in construction quality standards or permissible errors in quantity takeoff according to national regulations. Rather, they represent a comparative tolerance used to evaluate differences in results when the same object is measured using two different methods (2D CAD-based and 3D BIM-based). In other words, the tolerance range is not intended to assess the absolute accuracy of each method but to provide a relative benchmark for interpreting method-to-method discrepancies. Therefore, this tolerance does not represent an error relative to a true value but rather the deviation between results produced by two methods that rely on different information structures and calculation procedures.

2. Related Work

2.1. Dimensioning and Quantity Takeoff in 2D CAD

Dimensioning and quantity takeoff in 2D CAD environments represent a traditional approach that has long been used in construction practice. In general, 2D CAD-based quantity takeoff is performed by interpreting dimensions represented on drawings and applying quantity calculation formulas selected by the user. In this process, dimensions are visually expressed as drawing elements; however, the reference points that define each dimension and their association with specific objects are not explicitly managed at the object level [1].

Despite these limitations, 2D CAD remains widely adopted in practice due to its simplicity, established workflows, and compatibility with existing standards and regulations.

Previous studies have pointed out that this 2D CAD-based quantity takeoff approach relies heavily on user experience and judgment and that the selection of reference points and quantity calculation formulas during drawing interpretation may vary among users. To mitigate these issues, various attempts have been made to standardize quantity takeoff formulas. Nevertheless, due to differences in national standards, industry practices, and internal organizational regulations, uniform interpretation and application remain practically difficult. Eastman et al. (2011) clearly explained that although standardized quantity takeoff formulas exist for 2D CAD-based workflows, the absence of object-level dimension definitions constitutes a structural limitation, making complete standardization across countries and organizations fundamentally unattainable [1].

In particular, within 2D CAD environments, even when identical final quantity takeoff results are derived from the same drawing, the underlying procedures and drawing interpretation paths applied during the quantity takeoff process may differ from one user to another. As a result, the reproducibility and verifiability of the quantity takeoff process itself are inherently limited. From the perspective of process-level automation and rule transfer, these characteristics represent structural constraints of 2D CAD-based dimensioning and quantity takeoff methods.

2.2. Quantity Takeoff and Dimensioning in 3D BIM Environments

In 3D BIM environments, the geometric information and attribute data of objects are managed in an integrated manner, which fundamentally distinguishes BIM-based quantity takeoff from traditional 2D CAD approaches. Previous studies have reported that BIM-based quantity takeoff can improve computational efficiency and verifiability through consistent object information and enhanced data interoperability [7,8,9].

Alathamneh et al. (2024) conducted a systematic literature review on BIM-based quantity takeoff and concluded that object-level integration of geometric and attribute information improves data connectivity and traceability, thereby enhancing the reliability and efficiency of quantity takeoff processes [7]. Similarly, Valinejadshoubi et al. (2024) proposed an integrated framework for the automatic extraction and visualization of quantitative data from BIM models, emphasizing the practical efficiency of automation in BIM-based quantity takeoff [8]. Pham et al. (2024) also demonstrated that BIM-based quantity takeoff outperforms conventional 2D methods in terms of time savings and reliability, empirically highlighting the advantages of 3D object-based approaches [9].

Despite these advantages, the processes of dimension creation and dimension input in BIM environments still largely rely on manual user intervention. While quantity takeoff itself can be partially automated based on object attributes, dimensions are typically generated and entered by users through manual interpretation of the model. Rashidi et al. (2024) classified BIM-based quantity takeoff methods according to RICS measurement rules into automatic, derived, and manual categories, explicitly noting that manual takeoff remains prevalent even in BIM-based workflows [10]. Furthermore, Khosakitchalert et al. (2019) reported that when BIM models are incomplete or lack sufficient detail, additional manual tasks—such as model inspection and modification—are required to ensure accurate quantity takeoff, indirectly revealing the limitations of fully automated extraction [11].

Due to this dual structure, attribute-based quantity takeoff results are often supplemented and verified using dimension-based quantity calculation formulas in practice, which has become a common approach in the domestic BIM industry. Consequently, even when identical object geometries and working conditions are given, dimension generation is repeatedly performed on a project-by-project basis, and dimension-based quantity takeoff criteria may vary depending on the user. In this respect, 3D BIM environments still exhibit limitations in terms of the reproducibility and automation of dimension-based quantity takeoff, which can be interpreted as a structural issue analogous to the limitations long identified in 2D CAD-based workflows.

2.3. Comparative Studies Between 2D CAD and 3D BIM

Previous studies comparing 2D CAD- and 3D BIM-based quantity takeoff approaches have primarily focused on aspects such as estimation accuracy, operational efficiency, and ease of result verification. Several studies have applied both 2D CAD– and BIM-based methods to identical projects and reported that BIM-based approaches can offer relative advantages in terms of process consistency and data management. For example, comparative experiments conducted using the same design drawings demonstrated that BIM-based quantity takeoff can be effective in reducing working time and improving data reusability [12]. Such comparative studies have shown that BIM-based environments can support quantity takeoff processes in a more systematic manner through object-level information management and data interoperability, thereby highlighting structural differences from traditional 2D CAD workflows [2]. However, the analytical focus of these studies has largely been limited to improvements in final quantity results or overall work efficiency, while discussions on how dimensioning criteria are defined and interpreted during the quantity takeoff process remain relatively limited.

In particular, most comparative studies have treated dimensions as representational outputs displayed on drawings or models, rather than as information units directly linked to quantity calculation rules. Consequently, few studies have explicitly set the definition and interpretation of dimensions themselves as primary comparison targets. Eastman et al. (2011) pointed out that even in BIM environments that provide object-based data structures, quantity takeoff processes may still rely on user interpretation if dimension information is not explicitly connected to quantity calculation rules [1]. This observation suggests that existing comparative studies tend to emphasize outcome-oriented performance evaluations, while insufficiently addressing differences in how dimensioning criteria are defined and logically interpreted. Therefore, although previous comparative studies between 2D CAD and 3D BIM have provided meaningful insights into quantity takeoff results and operational efficiency, studies that systematically compare the definition and interpretation of dimensions—treating them not merely as visual representations but as information directly linked to quantity calculation rules—remain limited. This research gap underscores the necessity of a new comparative approach that positions dimension definition itself as a core subject of analysis.

2.4. Research Perspective: User-Friendly Dimensioning Under Reliable Quantity Totals

This study aims to examine the existence of user-dependent, dimension-based quantity takeoff criteria under the premise that, for a single object, the total quantity calculated in both 2D CAD and 3D BIM environments remains consistent within a practically acceptable tolerance range. Rather than attempting to uniformly standardize dimensioning criteria, this study explores a user-friendly approach that allows dimension-based quantity takeoff rules to be maintained and transferred across repeated tasks. The research perspective is grounded not in the standardization of a single calculation rule but in the following question: Can dimension-based quantity takeoff criteria, derived from user knowledge and experience, be applied consistently to the same object across repeated operations?

Previous studies on automated dimensioning and quantity takeoff have primarily focused on standardizing calculation procedures and unifying computational processes in order to improve result comparability, reliability, and management efficiency. While such approaches may be effective in controlled or specific working environments, they exhibit limitations in adequately reflecting differences in user design intent and operational practices that frequently occur in real-world projects. In practical design and construction environments, dimensions are defined and interpreted differently depending on the user’s review purpose, design strategy, and representation conventions. As a result, even for the same object, the selection of dimensional reference points and the formulation of quantity calculation rules may vary, implying that a certain level of user dependency in the dimensioning process is unavoidable. This study does not regard such user dependency as an error to be eliminated but, rather, as an inherent aspect of realistic working conditions, and explicitly adopts it as a fundamental premise.

From this perspective, the present study focuses on a structural approach in which user-defined dimensioning rules are preserved at the object level and can be consistently reproduced and transferred across repeated tasks, rather than forcibly unifying dimensioning criteria. Accordingly, the objective of dimension automation in this study is not the homogenization of rules but the achievement of overall work efficiency by maintaining user-friendly quantity takeoff criteria and automating their repeated application under the condition of reliable total quantity results. This research perspective distinguishes the present study from existing standardization-oriented approaches to dimensioning and quantity takeoff and proposes an alternative direction for dimension-based automation that is directly applicable to practical working environments. Accordingly, this study conceptually frames semantic vertices as a potential structural bridge that can support the preservation and transfer of user-defined dimensioning logic across geometrically identical objects.

3. Foundations of User-Defined Dimensions and Quantity Transferability

3.1. Basic Terminology

In Building Information Modeling (BIM) environments, the term “automation” is used in a very broad sense; however, in practice, concepts referring to different technical layers are often conflated. In particular, automatic dimensioning, dimension export automation, and automatic quantity calculation formulas are frequently treated as if they were the same process in practice, although their objectives and operational mechanisms are fundamentally different. This study begins by clearly distinguishing these three concepts.

Automatic dimensioning is a process that analyzes the geometric structure of two-dimensional or three-dimensional objects to automatically identify meaningful reference points and, based on these, stably calculate and represent quantitative dimensions such as length, width, height, diameter, thickness, and distance. The primary purpose of automatic dimensioning is not direct quantity calculation but rather the numerical and quantitative interpretation of shape. Approaches to numerical interpretation of shape can be broadly divided into two categories. Typical BIM authoring environments operate primarily based on attribute-based dimensions derived from object parameters and properties, whereas line- and mesh-based environments such as 2D CAD, SketchUp, and Blender adopt geometry- and vertex-based automatic dimensioning methods that directly analyze the distribution of vertices, edges, and faces. Although both approaches pursue the same goal of dimension automation, the basis for determining reference points and the structure of reproducibility are fundamentally different.

Dimension export automation refers to the process of transferring dimensions generated within BIM to structured external formats such as Excel, IFC, JSON, CSV, or XML. In this stage, the function is limited to simply extracting already created dimension information, and it may become an object of reuse during platform conversion.

Automatic quantity calculation formulas represent rule-based automated computation techniques that execute predefined generalized equations or user-defined formulas using extracted dimensions as input variables. In reality, most BIM systems do not internally execute or manage complex user-defined formulas; calculations are typically performed in external engines such as Python, Dynamo, Grasshopper, Excel, Power BI, or other standalone tools. Therefore, this is closer to external rule-based computational automation using extracted dimensions as inputs rather than geometric analysis within BIM itself [13,14].

Generally, automatic dimensioning, dimension export automation, and automatic quantity calculation formulas are sequentially connected. First, automatic dimensioning quantifies shape; second, dimension export automation transfers this information in an external data format; and third, automatic quantity calculation formulas complete the final quantity. The focus of this study lies on the third stage—how to stabilize automatic quantity calculation formulas within BIM—while dimension export automation mainly functions as a reporting mechanism for outputting results.

Currently, most BIM software relies on object attribute information and proprietary black-box geometric engines to determine object shape and compute quantities [4,15]. Although this approach is convenient in practice, quantity results based on attribute information are provided in a black-box manner according to each platform vendor’s policies, making it difficult for users to directly verify them. This is not merely a convenience issue but a structural constraint that restricts the transparency and transferability of user-defined formulas. Clarifying this limitation is both the starting point and conclusion of this study.

3.2. Fundamental Concepts and Experiment

To verify the reproducibility and transferability limits of user-defined automatic quantity calculation, this study adopts two geometrically similar—but not identical—objects as basic experimental models.

Terminology Definition

• Similar objects: Objects that share the same structural dimensioning pattern (extreme-point search pattern) but may require different geometric interpretations for volume calculation.
• Geometrically identical objects: Objects whose vertices, edges, and faces are identical, allowing the same formula to be applied.

Experimental Objects

• Experimental Object 1: Cube with a circular cylindrical penetration.
• Experimental Object 2: Cube with an elliptical cylindrical penetration.

These represent regular and irregular geometries, respectively, and serve as benchmark models to compare conventional BIM attribute-based methods and geometry-based semantic vertex methods.

The circular penetration cube is symmetric and regular, making it relatively easy to recognize in conventional BIM using attribute information. Its main dimensions include cube width, depth, height, cylinder diameter, and penetration length and position. This case serves as a benchmark for existing methods.

The elliptical penetration cube exhibits asymmetry and directionality. Although it can be controlled by BIM object attributes, if such attributes are lost during IFC conversion, dimensional interpretation becomes difficult. Moreover, as geometric complexity increases, most BIM software employs internal algorithms (including numerical integration) to compute volume in a fully black-box manner, preventing users from directly inspecting the structure of quantity formulas.

Consequently, although automatic dimensioning visually produces the same dimensions for both experimental objects, fundamentally different formulas must be applied. However, these formulas are internally automated and not exposed to users. Thus, the comparative experiment aims to demonstrate this discrepancy.

The basic experiment is designed around two central questions:

• Question 1: How diversely can user-defined quantity formulas be defined?
• Question 2: What problems arise when a formula defined for one object is transferred to a geometrically non-identical similar object?

Based on this, the experiment proceeds in four stages.

Step 1—Semantic-Vertex-Based Automatic Dimensioning.

Semantic vertex detection is performed based on geometric vertices. Reference vertices of the cube are identified, boundary faces are detected, and essential dimensions are automatically generated. In particular, the major and minor axes of the elliptical penetration are extracted geometrically.

Figure 1 visualizes automatically detected semantic vertices for both experimental objects (circular and elliptical penetration cubes). Extreme points and face boundaries are represented as small triangular meshes whose origins exactly coincide with object vertices. The same geometric search algorithm is applied consistently to both objects, successfully identifying extreme points and generating small triangular meshes at those locations.

For the elliptical penetration, orthogonal reference lines corresponding to the major and minor axes are automatically generated and later used as input variables (A: major axis, B: minor axis) in quantity formulas. Thus, Figure 1 visually demonstrates that although automatic dimensioning is shape-dependent, the same detection principle can be applied to similar objects.

Step 2—Determination of User-Defined Quantity Formulas.

The most generalized approach subtracts the penetration volume from the total cube volume.

Circular penetration subtraction formula:

$$V_{cut,circle}=\pi r^{2}h,r=D/2$$

(1)

Elliptical penetration subtraction formula:

$$V_{cut,ellipse}=\pi abh$$

(2)

Since A and B are provided as full axes, they are converted to semi-axes:

$$a=A/2,b=B/2$$

(3)

Fixing the cube volume formula, the number of possible quantity formulas based on input dimensions is:

• Circular case: 4 diameter candidates × 4 length candidates → 16 possible formulas.
• Elliptical case: 2 candidates for a × 2 candidates for b × 4 length candidates → 16 possible formulas.

This combinatorial space does not represent computational burden but rather the extent of interpretive choices available to users. In practice, users select one formula based on convention and experience. In Figure 1, the selected input variables are marked as indices, emphasizing that the choice of representative inputs is user-driven rather than automatic.

Thus, the answer to Question 1 is that user-defined formulas are not merely mathematical combinations but reflect users’ decisions regarding which dimensions serve as representative inputs.

Step 3—Normal Quantity Calculation.

The cube dimensions are fixed at 2 × 2 × 2 m, so,

$$V_{cube}=8{ \mathrm{m}}^3$$

(4)

(i) Experimental Object 1—Circular penetration.

Diameter $D=0.782$ m → radius $r=0.391$ m, $h=2$ m.

$$V_{cut,circle}=\pi (0.391{)}^2\times 2\approx 0.96{ \mathrm{m}}^3$$

(5)

Final volume:

$$V_{total,1}=8-0.96=7.04{ \mathrm{m}}^3$$

(6)

(ii) Experimental Object 2—Elliptical penetration (normal case).

Major axis $A=1.197$ m; minor axis $B=0.782$ m:

$$a=0.5985,\hspace{0.25em}b=0.391,\hspace{0.25em}h=2$$

(7)

$$V_{cut,ellipse}=\pi \left(0.5985\right)\left(0.391\right)\times 2\approx 1.47{ \mathrm{m}}^3$$

(8)

Final volume:

$$V_{total,2}^{normal}=8-1.47\approx 6.53{ \mathrm{m}}^3$$

(9)

Step 4—Transfer Error Experiment (Question 2).

Assume the user mistakenly interprets 1.197 m as a circular diameter:

$$D_{misread}=1.197\Rightarrow r_{misread}=0.5985$$

(10)

$$V_{cut,error}=\pi (0.5985{)}^2\times 2\approx 2.25{ \mathrm{m}}^3$$

(11)

Erroneous total volume:

$$V_{total,2}^{error}=8-2.25=5.75{ \mathrm{m}}^3$$

(12)

Volume error:

$$\Delta V=6.53-5.75\approx 0.78{ \mathrm{m}}^3$$

(13)

This shows that transferring a circular formula to a non-identical object can cause significant errors.

3.3. Result Interpretation

• Semantic-vertex-based automatic dimensioning succeeds for both circular and elliptical cubes using the same algorithm. Thus, automatic dimensioning is reproducible across similar objects.
• Even with successful automatic dimensions, input variables for quantity formulas remain diverse and user-dependent.
• Although the two objects are geometrically similar, different formulas must be applied because their penetration volumes differ.
• Quantity formulas inherently depend on shape interpretation. While limited rule-based classification may distinguish circular from elliptical cases, it is insufficient for general automation across arbitrary objects.

Overall Conclusion

Whether user-defined dimensions can be stably transferred within BIM depends on two fundamental constraints:

• User-defined formulas are inherently based on subjective convention and experience.
• The lack of a robust criterion for geometric identity fundamentally limits automation.

Therefore, dimension automation is not merely a technical problem of generating dimensions but requires prior determination of object identity.

Table 1. Comparison of automatic dimensioning, dimension export automation, and automatic quantity calculation formulas based on the two experimental objects.

Criterion	Automatic Dimensioning	Dimension Export Automation	Automatic Quantity Calculation Formula
Repeatability across similar objects	Repeatable when the structural dimension pattern (extreme-point search pattern) is identical	Repeatable as long as the same dimension format is exported	High risk of failure (formulas themselves may differ)
Direct source of automation constraints	Success depends on geometric vertex/edge detection	Limited by standard format conversion and data transfer	Requires user selection + shape interpretation
Required level of shape identity	Operates at the level of geometric similarity	Operates at the level of geometric similarity	Requires conditions close to geometric identity
Dependence on user judgment	Relatively low	Not explicitly involved	Very high (core determinant of formula selection)
Dependence on shape interpretation	Depends on vertex and boundary detection	Not explicitly represented	Essential (circle vs. ellipse distinction must be reflected)
Conditions under which transfer fails	Fails when vertex topology changes significantly	Fails when exported formats are inconsistent	Fails when the same formula is applied to non-identical similar objects
Underlying cause of failure (summary)	Sensitive to mesh-level geometric differences	Limited by format standardization	Lack of a clear geometric identity criterion + non-uniqueness of user formulas

compares automatic dimensioning, dimension export automation, and automatic quantity calculation formulas for the two experimental objects.

4. Dimensioning Comparison

In this section, 2D CAD-based dimension representation and semantic-vertex-based dimension generation in a 3D BIM environment are compared and analyzed using the same isolated footing as a single test object. The purpose of this comparison is not to evaluate the level of automation or improvements in productivity but to examine the structural differences between 2D environments and semantic-vertex-based 3D environments from the perspective of dimension automation and clarify how user-dependent differences in dimension definition relate to the quantity takeoff process and its results. Under the assumption that the total quantity outcome remains comparable, the comparison focuses on the explicitness of dimension reference points, the reproducibility of the dimension generation process, and the scope of user-dependent intervention.

4.1. Quantity Takeoff and Dimensioning in 2D CAD and 3D BIM Environments

In this subsection, a 2D CAD-based dimensioning and quantity takeoff approach and a semantic-vertex-based 3D BIM modeling approach are applied to the same isolated footing object, and the equivalence of the final quantity takeoff results obtained from the two environments is compared. The purpose of this comparison is not to evaluate the accuracy or superiority of a specific method but rather to confirm that comparable total quantity outcomes within an acceptable range can be derived from two fundamentally different frameworks for dimension definition and interpretation.

This comparison also serves as an empirical bridge between the conceptual discussions in Section 3 and the practical quantity takeoff outcomes presented in this chapter. The analysis focuses on (1) the explicitness of dimension reference points, (2) the reproducibility of the dimension generation process, and (3) the scope of user-dependent intervention in defining dimensions and quantities.

Figure 2 illustrates an isolated footing modeled in a 2D CAD environment, where dimensions are represented as drawing annotations that are separated from the object geometry itself. In this environment, dimension reference points are largely implicit, relying on user interpretation rather than being explicitly embedded within the object. Consequently, the consistency of dimension-based quantity takeoff depends to a significant extent on how individual users interpret and apply the annotated dimensions.

• Dimension-Based Quantity Takeoff Formulations in a 2D CAD Environment.

The quantities of the isolated footing were calculated using dimension-based formulations that interpret the dimensional annotations in the 2D CAD drawing as input parameters. These formulations follow conventional engineering calculation practices and reflect how quantity takeoff is typically performed in 2D-based workflows.

Dimension-Based Quantity Takeoff Formulations (2D CAD Environment).

(i) Rubble layer.

$$V=2.3\times 2.3\times 0.2=1.058\approx 1.06 {\mathrm{m}}^3$$

(14)

(ii) Lean concrete layer.

$$V=2.3\times 2.3\times 0.06=0.317\approx 0.32 {\mathrm{m}}^3$$

(15)

(iii) Structural concrete.

① Footing slab.

$$\begin{array}{cc}{V}_1& =(2\times 2\times 0.4)+{\displaystyle \frac{0.3}{6}}\left\{(2\times 2+0.4)\times 2+(2\times 0.4+2)\times 0.4\right\}\\ & =1.6+0.496=2.10 {\mathrm{m}}^3\end{array}$$

(16)

② Column.

$$V_2=0.4\times 0.4\times 1.0=0.16 {\mathrm{m}}^3$$

(17)

Total concrete volume.

$$V=V_1+V_2=2.10+0.16=2.26 {\mathrm{m}}^3$$

(18)

(iv) Formwork area.

① Footing slab.

$$A_1=2(2+2)\times 0.4=3.2 {\mathrm{m}}^2$$

(19)

② Inclined surface check.

$$\mathrm{t}\mathrm{a}\mathrm{n} \theta ={\displaystyle \frac{0.3}{0.8}}=0.375<0.577 (\mathrm{t}\mathrm{a}\mathrm{n} {30}^{\circ }=1/\sqrt{3})$$

(20)

Thus, formwork for the inclined surface is not required.

③ Column.

$$A_2=2(0.4+0.4)\times 1.0=1.6 {\mathrm{m}}^2$$

(21)

Total formwork area.

$$A=A_1+A_2=3.2+1.6=4.8 {\mathrm{m}}^2$$

(22)

(v) Reinforcement length.

① Footing reinforcement.

Main and secondary bars (D19):

$${\mathcal{l}}_1=2 \mathrm{m}\times 10\times 2=40 \mathrm{m}$$

(23)

Number of bars:

$$n={\displaystyle \frac{2}{0.2}}-10$$

(24)

Diagonal bars (D16):

$${\mathcal{l}}_2=\sqrt{2^2+2^2}\times 3\times 2=2.83\times 6=16.98 \mathrm{m}$$

(25)

② Column reinforcement.

Main bars (D22):

$${\mathcal{l}}_1=(1.7+0.4)\times 8=16.8 \mathrm{m}$$

(26)

Anchorage length: 0.4 m.

Number of hoops:

$$n={\displaystyle \frac{1.7}{0.3}}+1=7$$

(27)

Hoops (D10):

$${\mathcal{l}}_2=2(0.4+0.4)\times 7=11.2 \mathrm{m}$$

(28)

Diagonal hoops (D10):

$${\mathcal{l}}_3=2(0.4+0.4)\times 3=4.8 \mathrm{m}$$

(29)

2.

Dimension-Based Quantity Takeoff Formulations in a Semantic-Vertex-Based 3D BIM Environment

In this study, instead of relying on object property information typically used in conventional BIM environments, semantic vertices were assigned directly to selected object vertices, and dimensions between these semantic vertices were automatically generated. Quantities were then calculated by applying the dimension-based information extracted from the semantic-vertex-based 3D BIM model as input variables to the quantity takeoff formulations. Figure 3 presents an isolated footing modeled in a semantic-vertex-based 3D BIM environment with object-level dimension definitions. Figure 4 and Figure 5 illustrate examples of exporting automatically generated dimension data into structured external formats. This step does not perform new calculations but simply transfers existing information in an organized manner. Therefore, the analytical contribution of this paper lies not in the export mechanism itself but in how dimension inputs interact with quantity rules and shape transferability across similar objects.

Before presenting the detailed quantity takeoff formulations, it should be noted that all dimensional parameters denoted as D(1)–D(17) correspond to object-level dimensions defined in the semantic-vertex-based 3D BIM model and are illustrated in Figure 3. These parameters represent geometric dimensions directly measured from the model and are used consistently as input variables in the quantity takeoff formulations.

For reinforcement quantities, symbols such as SD19@200(i), SD16@_(i), SDM(i), and SD10@300(i) refer to individual reinforcement elements identified in the semantic-vertex-based reinforcement list extracted from the 3D BIM model, as shown in Figure 5. Unlike conventional BIM systems that often infer reinforcement lengths from predefined rules, the present approach derives reinforcement lengths from explicitly modeled geometry linked to semantic vertices.

In particular, when a representative reinforcement element is used (e.g., SD19@200(1)), this indicates that all corresponding reinforcement bars share identical geometric dimensions; therefore, the length of a single representative bar is multiplied by the total number of bars.

The number of hoops (n = 6) for column reinforcement is determined based on direct geometric measurement of the modeled object, rather than on a predefined spacing rule, reflecting the actual modeled configuration in the 3D BIM environment. With these definitions, the following quantity takeoff formulations are derived directly from dimension-based information extracted from the semantic-vertex-based 3D BIM model.

Dimension-Based Quantity Takeoff Formulations (Semantic-Vertex-Based 3D BIM Environment)

(i) Rubble layer.

$$V=D\left(1\right)\times D\left(2\right)\times D\left(3\right)=1.058\hspace{0.17em}{\mathrm{m}}^3$$

(30)

(ii) Lean concrete layer.

$$V=D\left(4\right)\times D\left(5\right)\times D\left(6\right)=0.317\hspace{0.17em}{\mathrm{m}}^3$$

(31)

(iii) Structural concrete.

① Footing slab.

$$\begin{array}{cc}{V}_1& =\left(D\right(7)\times D(8)\times D(9\left)\right)\\ & +{\displaystyle \frac{D\left(10\right)}{6}}\{(2\times D(11)+D(12))\times D(13)\\ & \left.+(2\times D(12)+D(11\left)\right)\times D\left(14\right)\right\}\\ & =1.6+0.496=2.096\hspace{0.17em}{\mathrm{m}}^3\end{array}$$

(32)

② Column.

$$V_2=D\left(12\right)\times D\left(14\right)\times D\left(15\right)=0.16\hspace{0.17em}{\mathrm{m}}^3$$

(33)

Total concrete volume.

$$V=V_1+V_2=2.096+0.16=2.256\hspace{0.17em}{\mathrm{m}}^3$$

(34)

(iv) Formwork area.

① Footing slab.

$$A_1=2\left(D\right(7)+D(8\left)\right)\times D\left(9\right)=3.2\hspace{0.17em}{\mathrm{m}}^2$$

(35)

② Inclined surface check.

$$\mathrm{t}\mathrm{a}\mathrm{n}\theta ={\displaystyle \frac{D\left(10\right)}{D\left(16\right)}}=0.375<0.577\hspace{0.25em}(\mathrm{t}\mathrm{a}\mathrm{n}{30}^{\circ }=1/\sqrt{3})$$

(36)

Thus, formwork for the inclined surface is not required.

③ Column.

$$A_2=2\left(D\right(12)+D(14\left)\right)\times D\left(15\right)=1.6\hspace{0.17em}{\mathrm{m}}^2$$

(37)

Total formwork area.

$$A=A_1+A_2=4.8\hspace{0.17em}{\mathrm{m}}^2$$

(38)

(v) Reinforcement length.

① Footing reinforcement.

Main and secondary bars (D19).

$$l_1=\mathrm{SD}19@200\left(1\right)\times 10\times 2=37\hspace{0.17em}\mathrm{m}$$

(39)

SD19@200(1) represents a single representative bar length, as all main and secondary bars have identical geometric dimensions in the model.

Number of bars.

$$n={\displaystyle \frac{2}{0.2}}=10$$

(40)

Diagonal bars (D16).

$$l_2=\mathrm{SD}16@\left(1\right)+\mathrm{SD}16@\left(2\right)+\mathrm{SD}16@\left(3\right)+\mathrm{SD}16@\left(4\right)+\mathrm{SD}16@\left(5\right)+\mathrm{SD}16@\left(6\right)=15.12\hspace{0.17em}\mathrm{m}$$

(41)

② Column reinforcement.

Main bars (D22).

$$l_1=\mathrm{SDM}\left(1\right)+\mathrm{SDM}\left(2\right)+\mathrm{SDM}\left(3\right)+\mathrm{SDM}\left(4\right)+\mathrm{SDM}\left(5\right)+\mathrm{SDM}\left(6\right)+\mathrm{SDM}\left(7\right)+\mathrm{SDM}\left(8\right)=15.57\hspace{0.17em}\mathrm{m}$$

(42)

Anchorage length: 0.4 m.

Number of hoops $n=6$.

Hoops (D10).

$$l_2=\mathrm{SD}10@300\left(1\right)+\mathrm{SD}10@300\left(2\right)+\mathrm{SD}10@300\left(3\right)+\mathrm{SD}10@300\left(4\right)+\mathrm{SD}10@300\left(5\right)+\mathrm{SD}10@300\left(6\right)=6\hspace{0.17em}\mathrm{m}$$

(43)

Diagonal hoops (D10).

$$l_3=\mathrm{SD}10@900\left(1\right)+\mathrm{SD}10@900\left(2\right)=1.41\hspace{0.17em}\mathrm{m}$$

(44)

Summary of Key Findings

The total structural concrete volume obtained from the 3D BIM model was slightly smaller than that derived from the 2D CAD drawings (by 0.004 m³, or approximately 0.18%). This difference is negligible in practice and is mainly attributable to rounding effects and internal dimensional precision within the 3D dimension parameters. The formwork areas were identical in both environments, including the consistent exclusion of formwork for inclined surfaces based on the same inclination criterion (tan θ = 0.375 < 0.577).

In contrast, reinforcement quantities exhibited noticeable differences. The 2D CAD-based approach generally estimates reinforcement lengths using simplified external member dimensions, whereas the 3D BIM-based method calculates reinforcement lengths based on explicit modeling rules that account for concrete cover, actual bar placement, and realistic reinforcement layouts. Consequently, the 3D BIM-based quantities reflect measured reinforcement lengths rather than assumed values, including a reduced number of diagonal reinforcement sets within the footing.

These findings imply that quantity takeoff is not merely a computational problem but fundamentally a modeling and interpretation problem. The observed differences therefore originate from methodological definitions in modeling rather than from computational inconsistencies.

Table 2. Comparison of dimension-based quantity takeoff results between 2D CAD and 3D BIM environments for an isolated footing.

Category	Item	2D CAD	3D	Difference (3D–2D)	Difference (%)
Volume	Rubble layer (m3)	1.058	1.058	0	0
Volume	Lean concrete layer (m3)	0.317	0.317	0	0
Volume	Structural concrete—footing slab (m3)	2.1	2.096	−0.004	−0.19
Volume	Structural concrete—column (m3)	0.16	0.16	0	0
Volume	Total structural concrete (m3)	2.26	2.256	−0.004	−0.18
Area	Formwork—footing slab (m2)	3.2	3.2	0	0
Area	Formwork—column (m2)	1.6	1.6	0	0
Area	Total formwork area (m2)	4.8	4.8	0	0
Reinforcement	Footing main & secondary bars (D19), length (m)	40	37	−3.00	−7.50
Reinforcement	Footing diagonal bars (D16), length (m)	16.98	15.12	−1.86	−10.95
Reinforcement	Column main bars (D22), length (m)	16.8	15.57	−1.23	−7.32
Reinforcement	Column hoops (D10), length (m)	11.2	6	−5.20	−46.43
Reinforcement	Column diagonal hoops (D10), length (m)	4.8	1.41	−3.39	−70.63

summarizes the dimension-based quantity takeoff results for a single isolated footing obtained from the 2D CAD and 3D BIM environments.

The maximum reinforcement quantity difference (–70.63%) in Table 2 cannot be adequately explained by simply stating that ‘3D BIM is more realistic.’ Instead, this study interprets the discrepancy as a structural difference between rule-based 2D CAD estimation and geometry-constrained 3D BIM modeling. 2D CAD-based takeoff relies on representative outer dimensions and often simplifies cover, hooks, and bends, while 3D BIM explicitly models reinforcement within the concrete solid and constrains bar paths geometrically. Consequently, reinforcement lengths in 3D are calculated based on actual placement rather than approximated representative dimensions. Particularly for hoops and repetitive bars, small path differences accumulate across multiple bars, leading to substantial total-length deviations. Therefore, the difference in Table 2 reflects a methodological discrepancy rather than a calculation error in either system.

4.2. User-Dependent Quantity Takeoff in 3D Semantic-Vertex-Based Modeling

This section presents a comparative analysis of two user-defined cases applied to a 3D semantic-vertex-based isolated footing concrete model. In each case, users intentionally adopt different quantity takeoff formulas and dimension interpretation strategies. The comparison is limited to concrete quantities, while reinforcement quantities are excluded because a single, invariant quantity takeoff formula is applied that leaves no room for variation.

For concrete quantities, the analysis is conducted under the assumption that the final total volume remains identical despite the use of different quantity takeoff formulas. Through this comparison, it is confirmed that even when the same object and identical dimension reference points (semantic vertices) are provided, users may adopt different dimension interpretation approaches and quantity takeoff rules. This result indirectly demonstrates the presence of user dependency in the quantity takeoff process, while also indicating that dimension-based quantity takeoff results can converge to an equivalent outcome despite differences in calculation procedure.

Accordingly, this comparative analysis suggests that the reliability of dimension-based quantity takeoff is not derived from procedural uniformity but from its ability to accommodate user-specific methodological differences while preserving consistent results.

• User-Defined Quantity Takeoff Case 1 (Conventional Standard Formulation)

Figure 6 and Figure 7 present the semantic-vertex-based dimension generation of an isolated footing model and the associated Excel data extraction required for applying Equation (45).

The volume of the sloped portion of an isolated footing can be represented by a prismoidal geometry in which the upper and lower sections are parallel rectangles and the side slopes vary linearly with height. Under this common engineering assumption, the volume V of the sloped faces is calculated using the prismoidal formula as

$$V={\displaystyle \frac{h}{6}}\left\{(2a+a^{\prime })b+(2a^{\prime }+a)b^{\prime }\right\}$$

(45)

where $h$ is the height of the sloped region and $a,b$ and $a^{\prime },b^{\prime }$ denote the plan dimensions of the lower and upper sections, respectively.

By mapping the dimensional parameters defined in this study, the sloped height corresponds to $d(8{)}^{\prime }$, the lower section dimensions to $d(5{)}^{\prime }$ and $d(4{)}^{\prime }$, and the upper section dimensions to $d(6{)}^{\prime }$ and $d(7{)}^{\prime }$. Accordingly, the sloped volume of the isolated footing is expressed as

$$V={\displaystyle \frac{d(8{)}^{\prime }}{6}}\left\{\left(2d\right(5{)}^{\prime }+d\left(6{)}^{\prime }\right)d(4{)}^{\prime }+(2d(6{)}^{\prime }+d(5{)}^{\prime }\left)d\right(7{)}^{\prime }\right\}=0.496\hspace{0.25em}({\mathrm{m}}^3)$$

(46)

This formulation provides an exact analytical volume of the sloped faces under the assumption of linear side slopes and parallel upper and lower sections.

2.

User-Defined Quantity Takeoff Case 2 (Segmentation-Based Summation Method)

The volume of the sloped region of an isolated footing is calculated via user-defined geometric decomposition. In this study, the sloped region is decomposed into three distinct components: one central cuboid, four right triangular prisms, and eight tri-rectangular tetrahedra. The volume of each component is evaluated independently using closed-form geometric expressions.

It should be noted that the decomposition strategy is not unique; alternative segmentation schemes are possible. However, the selected configuration ensures orthogonal subcomponents that admit exact closed-form volume solutions.

First, the volume of the central cuboid $V_{1 }$ is defined as the product of its three orthogonal dimensions. The general expression is given by

$$V_1=i\cdot j\cdot h$$

(47)

Applying the dimensional parameters defined in this study, the volume of the central cuboid is expressed as

$$V_1=d{6}^{″}\cdot d{7}^{″}\cdot d{13}^{″}=0.048\hspace{0.25em}\left({\mathrm{m}}^3\right)$$

(48)

Second, four right triangular prisms are formed along the sloped faces. Each prism has a right triangular cross-section, and its volume is calculated as the product of the triangular area and the extrusion length. Accordingly, the total volume of the four triangular prisms $V_2$ is expressed as

$$V_2={\displaystyle \frac{1}{2}}\cdot k\cdot h\cdot l\times 4$$

(49)

Using the dimensional notation adopted in this study, this term is written as

$$V_2={\displaystyle \frac{1}{2}}\cdot d{9}^{″}\cdot d{13}^{″}\cdot d{15}^{″}\times 4=0.192\hspace{0.25em}({\mathrm{m}}^3)$$

(50)

Finally, four tri-rectangular tetrahedra are formed at the corners of the sloped region. A tri-rectangular tetrahedron has three mutually orthogonal edges and admits an exact closed-form volume. The volume of a single tetrahedron is given by

$$V_3={\displaystyle \frac{1}{6}}\cdot k\cdot m\cdot h$$

(51)

Thus, the total volume contribution of the eight tetrahedral components is expressed as

$$V_3={\displaystyle \frac{1}{6}}\cdot k\cdot m\cdot h\times 8$$

(52)

and, when mapped to the dimensional parameters used in this study, as

$$V_3={\displaystyle \frac{1}{6}}\cdot d{9}^{″}\cdot d{11}^{″}\cdot d{13}^{″}\times 8=0.256\hspace{0.25em}({\mathrm{m}}^3)$$

(53)

Thus, the total volume of the sloped region of the isolated footing is obtained by summing the volumes of the three components as

$$V_{\mathrm{total}}=V_1+V_2+V_3$$

(54)

Based on Equation (54), the total volume is calculated as shown below

$$V_{\mathrm{total}}=0.048\hspace{0.25em}+0.192\hspace{0.25em}+0.256\hspace{0.25em}=0.496 ({\mathrm{m}}^3)$$

(55)

By explicitly pairing the general geometric formulas with the corresponding study-defined dimensional parameters, the proposed formulation preserves a clear and transparent relationship between the geometric decomposition principle and the applied quantity takeoff rules for the sloped region of isolated footings.

Figure 8 shows the essential dimensional information required for applying Equations (47), (49), (52) and (54) to calculate the concrete volume of the sloped portion of an isolated footing based on a user-defined quantity takeoff rule.

Figure 9 and Figure 10 present the user-defined modeling configuration and the corresponding Excel export required for applying Equations (47), (49) and (52) in the concrete quantity takeoff of an isolated footing.

4.3. Structural Limitations in the Transferability of User-Defined Quantity Takeoff Formula

• Purpose of the Comparison

The purpose of this section is to verify whether the semantic-vertex-based dimensioning approach—applied to geometries originating from both 2D CAD and 3D BIM environments—can structurally support the reproducibility and transferability of user-defined quantity takeoff rules among similar objects. Figure 11 and Figure 12 show, respectively, the semantic vertex detection and the dimension variables selected as input parameters for the semantic-vertex-based user-defined quantity takeoff equation in the 2D CAD and 3D BIM environments.

Because reproducibility and transferability are examined among similar objects, the geometric relationships between such objects must first be clarified. In this study, all objects that do not satisfy complete geometric identity are broadly regarded as similar objects. However, the degree of difference among similar objects is not further classified in this section. For analytical clarity in the subsequent discussion, the following conceptual distinctions are used:

• Identical objects: Objects that are geometrically completely identical in shape, topology, and spatial configuration.
• Quasi-identical objects: Objects that share the same geometric structure but differ in modeling history or platform (Figure 12A,B).
• Modified similar objects: Objects whose shapes are partially modified or reconstructed, resulting in altered geometric structure (Figure 13).

This distinction is introduced for analytical convenience rather than as a formal classification scheme.

Accordingly, this comparison does not aim simply to confirm whether identical quantities are calculated in both environments. Instead, it focuses on examining how dimensional information—derived based on each environment’s own procedures and input definitions—is connected to user-defined quantity takeoff formulas.

In particular, assuming a common representative formula, this comparison seeks to clarify:

• How the required dimensional input variables are derived in 2D CAD-based and 3D BIM-based modeling frameworks;
• How these variables may differ between the two environments despite geometric similarity;
• What structural limitations and constraints emerge when the established procedural framework is applied to similar shape-modified objects.

Importantly, transferability in this study is evaluated primarily within each modeling environment, rather than as a direct cross-platform transfer between 2D-derived and native BIM models. This distinction underlies the interpretation of results in Section 2 and Section 3.

2.

Comparison Environment and Method

In this study, a semantic-vertex-based analysis framework was applied to geometries originating from both 2D CAD and 3D BIM environments. Importantly, the 2D CAD drawing itself was not treated as a purely planar artifact; instead, its core geometric information was reinterpreted under a 3D spatial reference system and reconstructed in Blender as a three-dimensional object suitable for semantic-vertex detection.

This reconstructed 3D representation of the 2D CAD drawing served as the basis for semantic-vertex identification, automatic dimension generation, and index mapping. Figure 11 illustrates the result of importing and reconstructing the 2D CAD geometry in Blender, including semantic-vertex-based dimension generation and corresponding vertex index mapping.

Figure 12A presents the 3D object reconstructed from the 2D CAD drawing, while Figure 12B shows a native 3D BIM-modeled object. Although these two objects share functionally identical geometric structures, they are not strictly identical because they originate from different modeling platforms and histories. Consequently, each follows its own semantic-vertex detection procedure.

Nevertheless, within each modeling environment, semantic-vertex detection remains internally consistent, enabling reliable reproduction of dimensional inputs and quantity rules. Therefore, Figure 12A,B are treated as quasi-identical objects that are structurally comparable but distinguished from the similar shape-modified object in Figure 13.

To conduct the final quantity takeoff for both objects, a representative calculation formula was required; therefore, Equation (45) in Section 4.2 was adopted as the reference formula. This served two purposes:

(i) To demonstrate that the same representative formula cannot be directly applied to shape-modified similar objects.

(ii) To show that even under the same representative formula, the required dimensional inputs may be defined differently depending on the modeling environment.

This setup enabled a systematic examination of whether the user-defined quantity takeoff rules applied to Figure 12 could be reproduced or transferred to the similar modified object in Figure 13, how users define input variables, and under what conditions such transfer becomes structurally impossible.

3.

Transfer Results

The comparative analysis confirmed that, as shown in Figure 11 and Figure 12, both 2D CAD-derived and native 3D BIM objects can automatically generate and record the necessary dimensional information through semantic-vertex detection.

Although the geometry in Figure 12A originates from a reinterpretation of 2D geometry into 3D space, while Figure 12B is a native 3D model, the dimensional variables required for the predefined quantity takeoff formula were consistently derived and mapped through semantic-vertex detection in both cases.

Although their semantic-vertex representations originate from different modeling environments, each environment maintains internal consistency, enabling strong transferability of user-defined quantity rules within that environment.

Figure 12A,B demonstrate that semantic-vertex detection combined with triangular mesh representation can systematically generate the required dimensional information.

The following Equation (56), corresponding to the representative quantity takeoff formula in Equation (45) of Section 4.2, was adopted as the final calculation formula for the sloped part of the isolated footing:

$$V\prime ={\displaystyle \frac{h}{6}}\left[(2a+a^{\prime })b+(2a^{\prime }+a)b^{\prime }\right]$$

(56)

As seen in Figure 12A,B, the selection of dimensional input variables ultimately belongs to the user’s interpretation domain, regardless of whether they originate from 2D or 3D modeling. However, with respect to the core objective of this experiment, it is evident that Equation (56) cannot be directly applied to the modified similar object in Figure 13. This limitation can be explained through the following two-step causal structure.

(i) structural cause: change in semantic-vertex set.

As shown in Figure 13, the sloped part of the isolated footing undergoes geometric modification such that its reference surfaces, edges, extreme points, and reference planes no longer share the same topology as in Figure 12. Consequently, the semantic-vertex set is no longer equivalent.

Because Equation (56) presupposes a specific cross-sectional continuity and a particular pair of reference sections, the geometric prerequisites for defining variables $a,a^{\prime },b,b^{\prime }$ in the same manner are no longer satisfied in Figure 13. This represents a definition-level limitation, which subsequently leads to an automation-level limitation, as the same variables cannot be automatically generated via semantic-vertex detection.

(ii) change in quantity takeoff structure: B → C.

Since the upper column part in Figure 12 and Figure 13 is geometrically identical, transfer remains possible for that portion. However, because the lower sloped part differs in Figure 13, a new quantity takeoff formulation becomes necessary.

Volume Structure in Figure 12.

The total volume of the isolated footing in Figure 12 consists of two parts:

$$V_{Fig.12}=A\hspace{0.25em}\left(\mathrm{column} \mathrm{part}\right)+B\hspace{0.25em}\left(\mathrm{lower} \mathrm{sloped} \mathrm{part}\right)$$

(57)

The upper rectangular column part is a simple rectangular prism and can be calculated using three semantic dimensions $d\left(5\right),d\left(6\right),d\left(7\right)$, which remain unchanged in Figure 13:

$$A=d\left(5\right)\times d\left(6\right)\times d\left(7\right)$$

(58)

The lower sloped part exhibits a continuously varying cross-section and therefore requires an integral-type formulation:

$$B={\displaystyle \frac{d\left(4\right)}{6}}\left[(2a+a^{\prime })b+(2a^{\prime }+a)b^{\prime }\right]$$

(59)

Volume Structure in Figure 13.

Similarly, the total volume in Figure 13 consists of two parts:

$$V_{Fig.13}=A\hspace{0.25em}\left(\mathrm{column} \mathrm{part}\right)+C\hspace{0.25em}\left(\mathrm{lower} \mathrm{sloped} \mathrm{part}\right)$$

(60)

Since the upper column part is identical, the same formula for $A$ is transferable:

$$A=d\left(5\right)\times d\left(6\right)\times d\left(7\right)$$

(61)

However, the reconstructed lower part in Figure 13 can no longer be represented by Equation (59). Instead, it can be decomposed into two triangular cross-section prisms and one central rectangular prism. Their sum defines part $C$:

$$C={\displaystyle \frac{1}{2}}d(1)d(2)d(3)\times 2+d(5)d(4)d(3)$$

(62)

4.4. Descriptive Interpretation of Transferability

Because Figure 12 and Figure 13 share the same upper rectangular column part (A), the quantity takeoff formula for this part is transferable among similar objects. This is due to the preservation of the same semantic-vertex set and input dimensions.

However, although the lower sloped parts appear visually similar, they are not geometrically identical. Their reference surfaces, extreme points, and cross-sectional structures differ, meaning that variables $a,a^{\prime },b,b^{\prime }$ used in Equation (56) cannot be generated or matched in the same way in Figure 13. Instead, new dimensions $d\left(1\right),d\left(2\right),d\left(3\right),d\left(4\right),d\left(5\right)$ are required, implying that the structure of the quantity takeoff formula itself must change from $B$ to $C$.

Therefore, transfer of quantity takeoff formulas is robust within a stable modeling environment when the semantic-vertex set is preserved (Figure 12A,B), but becomes structurally constrained when the geometry itself is modified (Figure 13), even if the objects remain visually similar.

This result suggests that, within the tested scope of this study, quantity takeoff transfer is most reliable for completely identical or quasi-identical objects within the same environment, whereas for shape-modified similar objects, dimension automation and formula transfer are substantially limited.

Table 3. Transferability of quantity takeoff formulas for similar isolated footing objects in a vertical comparison format. Transferability is evaluated within each modeling environment for Figure 12A,B, where strong internal consistency is maintained. In Figure 13, although the foundation upper part (A) remains transferable, the lower sloped part requires reformulation from B to C due to a change in the semantic-vertex set.

Attribute (Row-Based Comparison)	Figure 12A 2D → 3D Reconstructed	Figure 12B Native 3D BIM	Figure 13 Shape-Modified Similar Object
Modeling environment	2D-to-3D reconstructed modeling environment	Native 3D BIM modeling environment	Same platform as Figure 12 with modified geometry
Overall volume structure	A + B	A + B	A + C
Upper part (A)	Rectangular prism	Rectangular prism	Rectangular prism (identical)
Formula for A	(A = d(5),d(6),d(7))	(A = d(5),d(6),d(7))	(A = d(5),d(6),d(7))
Lower part type	Continuously sloped	Continuously sloped	Reconstructed composite geometry
Formula for lower part	Integral form (B)	Integral form (B)	Decomposed form (C)
Lower part equation	(B=\frac{d(4)}{6}[(2a+a′)b+(2a′+a)b′])	Same as Figure 12A	(C=\frac{1}{2}d(1)d(2)d(3)\times2+d(5)d(4)d(3))
Semantic-vertex set	Internally consistent within the reconstructed environment	Internally consistent within native BIM	Changed from Figure 12
Required input dimensions	(d(4), a, a′, b, b′)	(d(4), a, a′, b, b′)	(d(1), d(2), d(3), d(4), d(5))
Transfer criterion	Transfer evaluated within the reconstructed environment	Transfer evaluated within native BIM	Transfer across shape change
Transferability (A)	Strong	Strong	Strong
Transferability (lower part)	Strong	Strong	Not transferable (B→C required)
Overall transferability	Strong	Strong	Partial
Structural reason	Stable semantic-vertex mapping within the same environment	Stable native semantic system	Semantic set changed; formula must change

summarizes the transferability of quantity takeoff formulas for similar isolated footing objects based on Figure 12 and Figure 13.

5. Comprehensive Discussion of Experimental Results

5.1. Discussion on Dimension Generation and Quantity Takeoff in 2D CAD and 3D BIM Environments (Based on Section 4.1)

This study compared the quantity takeoff process by applying a 2D CAD-based dimension definition method and a semantic-vertex-based dimension generation method in a 3D BIM environment to the same isolated footing object. The purpose of this comparison was not to judge the superiority of a particular environment but to elucidate the structural differences that different dimensioning systems impose on the quantity takeoff process. The analysis focused on the explicitness of dimension reference points, the reproducibility of dimension generation, and the scope of user intervention.

The experiment in Section 4.1 first demonstrated that the two environments are fundamentally different in terms of the explicitness of dimension reference points. In the 2D CAD environment, dimensions are represented as drawing annotations, and reference points are not structurally embedded within the object; therefore, they depend largely on user interpretation. In contrast, in the semantic-vertex-based 3D BIM environment, reference points are directly assigned to object vertices and explicitly recorded within the model, and dimension generation is performed in a reproducible manner through an automated procedure. This confirmed that even when the same quantity takeoff formula is applied, the manner in which input dimensions are defined is fundamentally different.

However, despite these differences in input dimension definitions, the total concrete volume calculated in the two environments was practically equivalent. The 2D CAD-based result was 2.26 m³, whereas the 3D BIM-based result was 2.256 m³, showing only a negligible difference of approximately 0.004 m³ (0.18%). This difference was mainly attributable to rounding effects and internal dimensional precision within the 3D model. This indicates that semantic-vertex-based dimension automation can reproduce quantity results that are structurally equivalent to those obtained from 2D-based calculations.

In contrast, significant differences appeared in reinforcement quantities. The 2D CAD approach relied on estimates based on external dimensions and nationally varying quantity takeoff standards, whereas the 3D BIM approach directly measured bar lengths based on the modeled geometry, considering concrete cover and actual bar placement. As a result, the 3D BIM approach yielded values closer to real reinforcement layouts, including a reduction in diagonal reinforcement quantities. These differences did not originate from computational errors but from differences in dimension definition methods and modeling interpretations.

Consequently, the experiment in Section 4.1 clearly confirms that quantity takeoff is not merely a computational issue but fundamentally a matter of how dimensions are defined and how objects are modeled. In particular, when mediated by the concept of semantic vertices, dimension definition and quantity takeoff automation for 2D and 3D objects can be interpreted within a structurally equivalent logical framework.

5.2. Structural Meaning of User Dependency and Semantic-Vertex-Based Dimension Automation (Based on Section 4.2)

Section 4.2 compared cases in which different user-defined quantity takeoff formulas were applied to the same 3D object. The results showed that even when the same object and the same semantic vertices were used, users could adopt different dimension interpretation strategies and calculation formulas; nevertheless, the final total quantity converged to the same value. This empirically demonstrates that user dependency in quantity takeoff is very high, while also suggesting that semantic vertices can structurally manage this dependency.

The quantity takeoff method proposed in this study does not aim to exclude or standardize user-defined formulas. Rather, it seeks to clarify the limitations of a structure in which the dimensional variables required by users are explicitly generated and recorded based on semantic vertices so that the same rules can be repeatedly applied and reused. In this sense, semantic-vertex-based dimension automation should be understood not merely as an automation tool but as a medium that structurally records and enables the transfer of users’ judgments and interpretations.

From this perspective, a user-defined quantity takeoff formula is not a simple equation but a structural information system that integrates semantic vertices with dimension interpretation rules. The semantic-vertex-based approach provides a minimal technical foundation that can consistently maintain and reproduce such a system, thereby opening the possibility of systematically accumulating and reusing user experience in future BIM environments.

5.3. Structural Limitations of Transferability Among Similar Objects (Based on Section 4.3)

Section 4.3 compared three cases: a 2D-to-3D reconstructed object (Figure 12A), a native 3D BIM object (Figure 12B), and a partially modified similar object (Figure 13).

First, despite having different modeling histories and platforms, Figure 12A,B exhibited stable semantic-vertex-based dimension generation within each respective environment. In both cases, the same A+B volume structure could be applied, and the required input variables $a,a^{\prime },b,b^{\prime },d\left(4\right)$ were consistently defined through semantic vertices. This demonstrates that semantic-vertex-based dimension automation ensures strong transferability within each specific modeling environment, indicating that platform differences themselves do not constitute a barrier to transfer.

However, in Figure 13, the situation changed fundamentally. While the upper column part A remained identical and thus transferable, the lower sloped part underwent partial geometric modification, leading to changes in reference surfaces, extreme points, and cross-sectional structure. As a result, the original variables $a,a^{\prime },b,b^{\prime }$ could no longer be defined in the same manner. Consequently, the original integral-based formula B used in Figure 12 could not be directly applied; instead, a new formula C—combining triangular cross-section prisms and a central rectangular prism—became necessary.

This clearly shows that although semantic-vertex-based dimension automation can reliably generate dimensions, it does not guarantee the transferability of quantity takeoff formulas to similar objects with modified geometry.

Therefore, to transfer user-defined quantity takeoff formulas reliably, it is insufficient to judge objects merely as “similar.” A clear criterion for identifying completely identical objects (geometric identity criteria) must be established in advance. Otherwise, changes in the semantic-vertex set lead to changes in the structure of the quantity takeoff formula itself, fundamentally constraining automatic transfer.

This suggests that future BIM automation research should clearly distinguish between “automatic dimension generation” and “transferability of quantity takeoff formulas for identical objects.”

5.4. Significance, Limitations, and Future Research Directions

The academic contributions of this study can be summarized in three points. First, it demonstrated that dimension definitions for 2D-to-3D reconstructed objects and native 3D objects can be interpreted within a structurally equivalent framework through the concept of semantic vertices. Second, it empirically showed that strong transferability of quantity takeoff formulas is possible for identical objects within the same environment, but that structural limitations arise when geometry is modified. Third, it highlighted the industrial need for a structure in which user-defined quantity takeoff formulas can be explicitly recorded and reused based on semantic vertices.

At the same time, this study has limitations: the experiments were restricted to a single isolated footing object, automatic semantic-vertex detection was not addressed, and generalization to various object types was not pursued. This was an intentional choice to clarify the structural principles and limits of transferability.

Future research should proceed in the following directions:

(i) Integrate AI-based semantic-vertex detection techniques with the user-defined quantity takeoff recording structure proposed in this study to establish a semi-automated or fully automated framework that integrates semantic-vertex identification, dimension generation, and formula application.

(ii) Expand the experimental scope to various BIM object types, including footings, columns, beams, walls, and slabs, to verify whether the transferability logic proposed in this study holds generally.

(iii) Define quantitative geometric identity criteria to specify under what conditions quantity takeoff formulas can be transferred.

(iv) Develop a robust formula transfer framework that can adapt to partial geometric modifications, enabling automatic adjustment or rational reconstruction of quantity takeoff formulas.

If these directions are integrated, they could lead to the development of an intelligent BIM-based quantity takeoff system capable of systematically accumulating and reusing user experience and knowledge.

6. Conclusions

This study systematically analyzed the conditions and limits under which user-defined, dimension-based quantity takeoff criteria can be reproduced and transferred between 2D CAD and 3D BIM environments, under the premise that the total quantity for a single object remains reliable. Rather than comparing computational accuracy, this study experimentally clarified that how dimensions are defined, structured, and recorded fundamentally determines the reliability and transferability of quantity takeoff results.

By introducing the concept of semantic vertices, this study showed that 2D and 3D objects can be interpreted in a structurally equivalent manner with respect to dimension definition and quantity takeoff. Even when dimension generation procedures were applied differently across environments for the same isolated footing object, the final total quantity remained consistent within a practically acceptable range when the essential dimensional information required by users was properly encoded through the semantic-vertex structure. This confirms that quantity reliability depends not on the modeling environment itself but on the structural definition and management of dimensional reference criteria.

Furthermore, even when the same object and identical semantic vertices were provided, different users could adopt different dimension interpretations and quantity takeoff formulas. However, when the required dimensional variables were clearly mapped through a semantic-vertex-based structure, the final total quantity consistently converged to the same value. Accordingly, this study reinterprets user dependency not as an error to be eliminated but as an inherent characteristic of practical workflows, which can be explicitly managed, recorded, and reproduced through semantic vertices.

In the expanded experiment on an isolated footing, geometric identity was confirmed as the decisive condition for transferability. For geometrically identical objects, user-defined formulas and dimensional variables were stably reproduced and transferred across repeated tasks. In contrast, for shape-modified similar objects, the semantic-vertex set itself changed, altering the structure of the required dimensional variables and preventing the direct transfer of the original quantity takeoff formula. This empirically demonstrates that the limitation of transferability arises not from similarity but from the loss of geometric identity.

In summary, semantic-vertex-based automated dimensioning guarantees the automation of dimension generation but does not guarantee the automatic transfer of quantity takeoff formulas when object identity is not preserved. Therefore, a clear, quantitative criterion for geometric identity is a prerequisite for reliable formula transfer.

Future research should integrate automated or AI-assisted semantic vertex detection with the proposed semantic-vertex-based dimensional recording structure, extend this framework to various BIM object types, and develop mechanisms that can adapt or reconstruct quantity takeoff formulas under partial geometric modification. These advances would support an empirically grounded structural quantity takeoff framework in which user knowledge and experience can be systematically accumulated and reused within intelligent BIM environments.

7. Patents

A patent application related to this research has been filed.