A Rational Quadratic Trigonometric Spline (RQTS) as a Superior Surrogate to Rational Cubic Spline (RCS) with the Purpose of Designing

: A key technique for modeling 2D objects is built using a B é zier-like rational quadratic trigonometric function with two form parameters. Since they are generated employing weights, the suggested rational quadratic trigonometric spline curve schemes are helpful for shape modeling. The established method yields a curve with the best geometric properties, such as convex hull, partition of unity, afﬁne invariance, and diminishing variation. The parameters in the suggested method’s construction are helpful for several critical shape features such as local, global, and biased tension qualities, which give good control over the curve and allow for shape modiﬁcation as desired. In addition, the recommended approach is C 2 . Furthermore, the comparative study of both splines is discussed which revealed the proposed method as a superior alternative to rational cubic spline.


Introduction
Geometric modeling is the process of developing a mathematical description of an object's geometry using a CAD system. A geometric model is a mathematical description that is stored in computer memory. These processes include the creation of new geometric models from the system's basic building pieces. Geometric modeling is a subfield of applied mathematics and computational geometry that investigates methods and algorithms for mathematically describing shapes. In modern era for modeling, designing and analyzing various objects the role of classical Bézier curves cannot be overlooked when discussing the importance of free form curves in CAD/CAM, as the Bézier curve is a fundamental modeling tool in CAD/CAM. Because rational Bézier curves are produced using weights, they are superior to basic Bézier curves for form modeling. Additionally, rational Bézier curves may be used to represent conics, which have a range of technical uses. Regarding the importance of curves in various fields such as shape modeling specially in aerospace and automobiles, fingers print and face recognition, artificial intelligence, and even in media, a scheme is constructed using a Bézier-like rational quadratic trigonometric function with two form parameters. The suggested rational quadratic trigonometric spline curve schemes are useful for shape modeling because they are generated with weights. The established method produces a curve with the best geometric properties. The parameters in the suggested method's construction are beneficial for several critical shape features, providing good control over the curve and allowing for shape modification as desired.

1.
The method has good trigonometric splines properties.

2.
It possesses the best geometric properties of all splines. 3.
The scheme fulfills the criteria for geometric smoothness. 4.
The proposed technique incorporates key aspects of form design.

5.
Some conics can be represented using the proposed curve approach. 6. It contains two shape parameters that control shape effects such as interval tension and global tension. 7.
To establish that the suggested scheme is the better alternative, a quick comparative study of the proposed scheme and a rational cubic spline (RCS) is considered.
In Section 2, a Bézier-like rational quadratic trigonometric function is demonstrated in its interpolatory form, which is shaped using C 2 constraint equations at curve segment joints. The error analysis of a rational quadratic trigonometric spline (RQTS) is calculated in Section 3. Section 4 presents proofs of convex hull and affine invariance features for the RQTS curve, which are derived from Bernstein Bézier form. The form features of the suggested spline are addressed in Section 5. In Section 6, there is a demonstration of the proposed spline technique for various shape effects, including interval tension, biased Appl. Sci. 2022, 12, 3992 3 of 20 tension, and global tension. Section 7 comprises a comparison analysis of constructed RQTS with a rational cubic spline (RCS) regarding visual difference, time elapsed and error analysis. Section 8 is where the paper comes to a close.

Proposed Spline Method
Consider the data points {(t i , F i ), i = 0, 1, 2, . . . , n}, where domain t i , i = 0, 1, 2, . . . n, represents knots with t 0 < t 1 < · · · < t n and F i , i = 0, 1, 2, . . . n, be values at these knots. A rational quadratic trigonometric function R(t), with two shape parameters ρ i , σ i > 0, i = 0, 1,. . ., n − 1, is defined over the interval I i = [t i , t i+1 ], i = 0, 1, 2, . . . , n − 1, as: S i (t), i = 0, 1, 2, 3 are rational quadratic trigonometric basis functions defined as: Additionally, Furthermore, ∑ 3 i=0 S i (t) = 1 as demonstrated in Figure 1. By applying the following C 1 -continuity conditions The following tri-diagonal system of linear equations in 1 unknowns , 1, 2, … , 1, is achieved as follows: at the end points of the interval I i = [t i , t i+1 ], Bézier-like rational quadratic trigonometric Function (1) is transformed into a spline: M i 's are the computed values of 1st derivatives at the knots t i . ρ i , σ i > 0 provides positive denominator. For ρ i = σ i = 2, (2) turns into quadratic trigonometric interpolant R(t) ∈ C 1 [t 1 , t n ]. Now by putting the following C 2 continuity conditions The following tri-diagonal system of linear equations in n − 1 unknowns M i , i = 1, 2, . . . , n − 1, is achieved as follows: where With suitable end conditions, the above system is diagonally dominant and a unique solution for M i 's can be computed with the following limits on the shape parameters, The preceding discussion can be summarized as follows: RQTS has a solution that is unique due to the restriction on the shape parameters (5).

Error Analysis of RQTS
The error of RQTS Function (1) is calculated in this segment by the following theorem.

Theorem 2.
For G(t) ∈ C 3 [t 0 , t n ], let R(t) be the rational quadratic trigonometric spline (1) interpolates G(t) in [t 0 , t 1 ], then for ζ i > 0, ξ i > 0, the following holds: Proof. The error is determined without compromising generality since the quadratic trigonometric function is interpolated locally in the subinterval [t i , t i+1 ]. Let R (t) be the quadratic trigonometric function of G(t) ∈ [t 0 , t n ], interpolated in [t i , t i+1 ] as defined in (1) then by applying the Peano-kernel theorem, where E t (t − τ) 2 + is the kernel of integral defined for the quadratic trigonometric function as follows: with Two main steps are used in the evidence of the error evaluation; discussing the properties of function a 1 (τ, t) and b 1 (τ, t) and in the next step, compute the absolute values To compute the roots of a 1 (τ, t) and b 1 (τ, t), put τ = t in Equation (6) as It implies It can easily be observed that Then the two roots of a 1 (τ, t) are in terms of real value. Similarly, it can be noted that for σ ≤ σ * , b 1 (t, t) ≤ 0 and for So, the following result holds for σ ∈ 0, π 2 σ * / ∈ 0, π 2 , ξ i > 1, Thus prove the above result.

Geometric Properties
The perfect geometric properties of the RQTS curve are defined by the subsequent propositions: Proof . Consider an affine transformation T, given by . , 3, be the control points then The illustration of the Proposition 1, for affine invariance property, is shown in Figure 2.

□
The illustration of the Proposition 1, for affine invariance property, is shown in Figure 2.

Proposition 3 (Variation Diminishing (VD) Property). The intersection of RQTS curve
∑ for ∈ , with any 1 dimensional hyper plane will be equal to or less than the point at which that plane intersects with the control polygon P as determined by control points , , , ∈ ℝ .
The VD property for an open and a close curve is displayed in Figure 4a and Figure  4b respectively.

Proposition 3 (Variation Diminishing (VD) Property). The intersection of RQTS curve
] with any N − 1 dimensional hyper plane will be equal to or less than the point at which that plane intersects with the control polygon P as determined by control points P The VD property for an open and a close curve is displayed in Figure 4a and

Remark 2.
The VD property also upsets for ρ k < 2 or σ k < 2, for any k, as revealed in Figure 6a

Shape Properties
The proposed spline method is advantageous for different purposes where the changes needed locally or globally. By changing and , the desire changes can be made in the specific region or in the whole shape of the object.

Consider an interval
, for a fixed 1, … , , if , → ∞, then the curve converges to control polygon in , .

Biased Interval Tension Property
If or → ∞, for any , then curve inclines towards a control vertex in , .

Biased Global Tension Property
If or → ∞, ∀ , then curve inclines towards control vertices for all intervals , .

Demonstration
The RQTS curve method is used to interpolate data points ∈ ℝ , 1, … , , of various existing objects. The local and global tension for RQTS is designated with the examples. This illustration shows the shape parameter's default value 2, ought to be selected, except it is specified.

Shape Properties
The proposed spline method is advantageous for different purposes where the changes needed locally or globally. By changing ρ i and σ i , the desire changes can be made in the specific region or in the whole shape of the object.

Global Tension Property
Consider the interval [t i , t i+1 ], ∀i, if ρ i , σ i → ∞ , ∀i, then the curve converges to control polygon in [t i , t i+1 ], ∀i.

Biased Interval Tension Property
If ρ k or σ k → ∞, for any k, then curve inclines towards a control vertex in [t k , t k+1 ].

Biased Global Tension Property
If ρ i or σ i → ∞, ∀i, then curve inclines towards control vertices for all intervals [t i , t i+1 ].

Demonstration
The RQTS curve method is used to interpolate data points F i ∈ R 2 , i = 1, . . . , n, of various existing objects. The local and global tension for RQTS is designated with the examples. This illustration shows the shape parameter's default value ρ i = σ i = 2, ought to be selected, except it is specified.
In Figure 7a-e, different objects are interpolated using developed RQTS with periodic end conditions. Figures 8a, 9a, 10a, 11a, 12a and 13a are the default RQTS with periodic end conditions and Figure 8b-d demonstrate the global tension property with the increasing values of shape parameters ρ i = σ i = 3, 5 and 100. Figure 9b-d demonstrate that rising values of shape constraints ρ k = σ k = 3, 5 and 100, for the base interval to stretch the curve in the base interval and consequently reveal interval tension. Figure 10b In Figure 14(a1,b1) default shape of objects is shown using RQTS whereas Figure 14(a2,b2) demonstrates the RQTS with local tension, using ρ i = σ i = 100 at the mandatory portion of the shapes.        3, 5 and 10, respectively, for any interval . In Figure 14(a1,b1) default shape of objects is shown using RQTS whereas Figure  14(a2,b2) demonstrates the RQTS with local tension, using 100 at the mandatory portion of the shapes.

Comparative Study of a RCS and RQTS
A concise comparative study of proposed RQTS and a rational cubic spline (RCS) is discussed here. For this purpose, visual differences and time pass by two splines are kept in mind.

Visual Difference of Two Splines
The RQTS behaves similar to a RCS. To notice the visual differences of two splines, some objects are interpolated by both splines; RCS and RQTS.

Remark 3.
It is worthy of note that the suggested RQTS allows for both local and global modifications in the geometry of the objects.

Comparative Study of a RCS and RQTS
A concise comparative study of proposed RQTS and a rational cubic spline (RCS) is discussed here. For this purpose, visual differences and time pass by two splines are kept in mind.

Visual Difference of Two Splines
The RQTS behaves similar to a RCS. To notice the visual differences of two splines, some objects are interpolated by both splines; RCS and RQTS.

Comparative Study of a RCS and RQTS
A concise comparative study of proposed RQTS and a rational cubic spline (RCS) is discussed here. For this purpose, visual differences and time pass by two splines are kept in mind.

Visual Difference of Two Splines
The RQTS behaves similar to a RCS. To notice the visual differences of two splines, some objects are interpolated by both splines; RCS and RQTS.

Time Elapsed by Two Splines
The comparative study is also justifiable by comparing the time required to complete the tasks by both splines; RCS and RQTS. Table 1 shows time passed by two spline; RCS and RQTS, for different objects has Figure 15. Different objects (a1,b1,c1,d1,e1) are interpolated using RCS and (a2,b2,c2,d2,e2) using RQTS.

Time Elapsed by Two Splines
The comparative study is also justifiable by comparing the time required to complete the tasks by both splines; RCS and RQTS. Table 1 shows time passed by two spline; RCS and RQTS, for different objects has been considered. Moreover, it is easy to see that RCS takes longer to execute than RQTS, presenting RQTS as a preferable option for RCS.

Error Analysis for RCS and RQTS
For comparing these splines, the error calculated by RCS and RQTS is defensible. Table 2 shows it for seven different trigonometric, logarithmic, exponential, and polynomial functions: sin(t), cos(t), tan(t), sec(t), log(t), e t , and √ t + 6 + (t + 2) 2 revealed in column 3. Column 2 specifies the domain of the functions. Columns 4 and 5 illustrate the errors of these functions using RCS and RQTS, respectively. The difference in errors determined by RCS and RQTS is shown in Column 6 of Table 2. The difference between the errors of both splines is determined in column 6 to determine the accuracy of the splines.

Conclusions
In light of the importance of curve modeling in various fields, a C 2 interpolation system is built utilizing a rational quadratic trigonometric function for the purpose of object form design. Convex hull, partition of unity, affine invariance, and variation decreasing are geometric properties of the curves achieved through the developed method. The suggested method is accommodating for numerous shape effects to change the object as desired locally or globally. The parameters in the recommended method are beneficial for several critical shape features, providing good control over the curve and allowing for shape modification as desired. Furthermore, the comparative study revealed that the proposed method is a superior alternative to RCS.