# Modern Geometric Modeling

## "Modern Geometric Modeling: Theory and Applications" (Guest Editors: Prof. Rushan Ziatdinov, Prof. Kenjiro T. Miura)

## Special Issue Information

Dear Colleagues,

In recent decades, geometric modeling has evolved into an interesting and powerful branch of modern science and engineering. Its theories are mostly related to mathematics and computer science, and applications are commonly found in industrial design, graphics and animation, CAD/CAM, architecture, and other areas. Most of the popular approaches in geometric modeling include parametric spline curve and surfaces, and they are simple and intuitive for use for industrial designers. On the other hand, as is well known among researchers who study high-quality shapes, polynomial splines are not adequate for reaching highly-aesthetic requirements in industrial products. We believe that the field of geometric modeling needs breakthrough research which will result in a higher level of understanding of shape modeling and perception, the need of artificial intelligence in the CAD systems of the future, as well as the necessity of fundamentally new mathematical tools and paradigms which will revolutionize geometric modeling.

In view of the above, we invite you to submit your latest research in the area of geometric modeling to the Special Issue entitled “Modern Geometric Modeling: Theory and Applications”. The five most outstanding manuscripts will be accepted free of charge.

The scope of the Special Issue includes but is not limited to original research works within the subject of geometric modeling and its applications in engineering, physics, biology, medicine, computer graphics, architecture, etc., and also the theory of computational mathematics and geometry, which can be applied to problems of geometric modeling.

Guest Editors

## Accepted articles

"Modern Geometric Modeling: Theory and Applications"

**Guest Editor(s): Rushan Ziatdinov, Kenjiro T. M****iura****.**

**All articles can be accessed freely online. **

**Cheng, T.; Wu, Z.; Li, X.; Wang, C. Point Orthogonal Projection onto a Spatial Algebraic Curve. Mathematics 2020, 8(3), 317;**

Abstract: Point orthogonal projection onto a spatial algebraic curve plays an important role in computer graphics, computer-aided geometric design, etc. We propose an algorithm for point orthogonal projection onto a spatial algebraic curve based on Newton’s steepest gradient descent method and geometric correction method. The purpose of Algorithm 1 in the first step of Algorithm 4 is to let the initial iteration point fall on the spatial algebraic curve completely and successfully. On the basis of ensuring that the iteration point fallen on the spatial algebraic curve, the purpose of the intermediate for loop body including Step 2 and Step 3 is to let the iteration point gradually approach the orthogonal projection point (the closest point) such that the distance between them is very small. Algorithm 3 in the fourth step plays an important double acceleration and orthogonalization role. Numerical example shows that our algorithm is very robust and efficient which it achieves the expected and ideal result.

https://doi.org/10.3390/math8030317.

https://www.mdpi.com/2227-7390/8/3/317

**Li, C.; Zhu, C. Designing Developable C-Bézier Surface with Shape Parameters. Mathematics 2020, 8(3), 402;**

Abstract: Developable surface plays an important role in geometric design, architectural design, and manufacturing of material. Bézier curve and surface are the main tools in the modeling of curve and surface. Since polynomial representations can not express conics exactly and have few shape handles, one may want to use rational Bézier curves and surfaces whose weights control the shape. If we vary a weight of rational Bézier curve or surface, then all of the rational basis functions will be changed. The derivation and integration of the rational curve will yield a high degree curve, which means that the shape of rational Bézier curve and surface is not easy to control. To solve this problem of shape controlling for a developable surface, we construct C-Bézier developable surfaces with some parameters using a dual geometric method. This yields properties similar to Bézier surfaces so that it is easy to design. Since C-Bézier basis functions have only two parameters in every basis, we can control the shape of the surface locally. Moreover, we derive the conditions for C-Bézier developable surface interpolating a geodesic.

https://doi.org/10.3390/math8030402.

https://www.mdpi.com/2227-7390/8/3/402

**Chaidee, S.; Sugihara, K. The Existence of a Convex Polyhedron with Respect to the Constrained Vertex Norms. Mathematics 2020, 8(4), 645;**

Abstract: Given a set of constrained vertex norms, we proved the existence of a convex configuration with respect to the set of distinct constrained vertex norms in the two-dimensional case when the constrained vertex norms are distinct or repeated for, at most, four points. However, we proved that there always exists a convex configuration in the three-dimensional case. In the application, we can imply the existence of the non-empty spherical Laguerre Voronoi diagram.

https://doi.org/10.3390/math8040645.

https://www.mdpi.com/2227-7390/8/4/645

**Inoguchi, J.-I.; Ziatdinov, R.; Miura, K.T. A Note on Superspirals of Confluent Type. Mathematics 2020, 8(5), 762;**

Abstract: Superspirals include a very broad family of monotonic curvature curves, whose radius of curvature is defined by a completely monotonic Gauss hypergeometric function. They are generalizations of log-aesthetic curves, and other curves whose radius of curvature is a particular case of a completely monotonic Gauss hypergeometric function. In this work, we study superspirals of confluent type via similarity geometry. Through a detailed investigation of the similarity curvatures of superspirals of confluent type, we find a new class of planar curves with monotone curvature in terms of Tricomi confluent hypergeometric function. Moreover, the proposed ideas will be our guide to expanding superspirals.

https://doi.org/10.3390/math8050762.

https://www.mdpi.com/2227-7390/8/5/762

**Li, F.; Hu, G.; Abbas, M.; Miura, K.T. The Generalized H-Bézier Model: Geometric Continuity Conditions and Applications to Curve and Surface Modeling. Mathematics 2020, 8(6), 924;**

Abstract: The local controlled generalized H-Bézier model is one of the most useful tools for shape designs and geometric representations in computer-aided geometric design (CAGD), which is owed to its good geometric properties, e.g., symmetry and shape adjustable property. In this paper, some geometric continuity conditions for the generalized cubic H-Bézier model are studied for the purpose of constructing shape-controlled complex curves and surfaces in engineering. Firstly, based on the linear independence of generalized H-Bézier basis functions (GHBF), the conditions of first-order and second-order geometric continuity (namely, G1 and G2 continuity) between two adjacent generalized cubic H-Bézier curves are proposed. Furthermore, following analysis of the terminal properties of GHBF, the conditions of G1 geometric continuity between two adjacent generalized H-Bézier surfaces are derived and then simplified by choosing appropriate shape parameters. Finally, two operable procedures of smooth continuity for the generalized H-Bézier model are devised. Modeling examples show that the smooth continuity technology of the generalized H-Bézier model can improve the efficiency of computer design for complex curve and surface models.

https://doi.org/10.3390/math8060924.

https://www.mdpi.com/2227-7390/8/6/924

**BiBi, S.; Abbas, M.; Miura, K.T.; Misro, M.Y. Geometric Modeling of Novel Generalized Hybrid Trigonometric Bézier-Like Curve with Shape Parameters and Its Applications. Mathematics 2020, 8(6), 967;**

Abstract: The main objective of this paper is to construct the various shapes and font designing of curves and to describe the curvature by using parametric and geometric continuity constraints of generalized hybrid trigonometric Bézier (GHT-Bézier) curves. The GHT-Bernstein basis functions and Bézier curve with shape parameters are presented. The parametric and geometric continuity constraints for GHT-Bézier curves are constructed. The curvature continuity provides a guarantee of smoothness geometrically between curve segments. Furthermore, we present the curvature junction of complex figures and also compare it with the curvature of the classical Bézier curve and some other applications by using the proposed GHT-Bézier curves. This approach is one of the pivotal parts of construction, which is basically due to the existence of continuity conditions and different shape parameters that permit the curve to change easily and be more flexible without altering its control points. Therefore, by adjusting the values of shape parameters, the curve still preserve its characteristics and geometrical configuration. These modeling examples illustrate that our method can be easily performed, and it can also provide us an alternative strong strategy for the modeling of complex figures.

https://doi.org/10.3390/math8060967.

https://www.mdpi.com/2227-7390/8/6/967

**Yan, X.; Kuang, M.; Zhu, J. A Geometry-Based Guidance Law to Control Impact Time and Angle under Variable Speeds. Mathematics 2020, 8(6), 1029;**

Abstract: To provide a feasible solution for a variable speed unmanned aerial vehicle (UAV) to home on a target with impact time and angle constraints, this paper presents a novel geometry-based guidance law composed of trajectory reshaping and tracking. A trajectory generation process using Bezier curves is introduced to satisfy the impact time and angle constraints under time-varying speed. The impact angle is satisfied by driving the UAV along a specified ending line. The impact time is satisfied by controlling the trajectory length, which is realized through adjusting one Bezier curve end point along the ending line. The adjustable range of this end point, along with the maximum trajectory curvature, is analyzed to ensure that the trajectory is flyable. Guidance command is generated using inverse dynamics. Numerical simulations under various scenarios are demonstrated to illustrate the performance and validate the effectiveness of the proposed method.

https://doi.org/10.3390/math8061029.

https://www.mdpi.com/2227-7390/8/6/1029

**Majeed, A.; Abbas, M.; Miura, K.T.; Kamran, M.; Nazir, T. Surface Modeling from 2D Contours with an Application to Craniofacial Fracture Construction. Mathematics 2020, 8(8), 1246;**

Abstract: Treating trauma to the cranio-maxillofacial region is a great challenge and requires expert clinical skills and sophisticated radiological imaging. The aim of reconstruction of the facial fractures is to rehabilitate the patient both functionally and aesthetically. Bio-modeling is an important tool for constructing surfaces using 2D cross sections. The aim of this manuscript was to show 3D construction using 2D CT scan contours. The fractured part of the cranial vault were constructed using a Ball curve with two shape parameters, later the 2D contours were flipped into 3D with an equidistant z component. The surface created was represented by a bi-cubic rational Ball surface with C2 continuity. At the end of this article, we present two real cases, in which we had constructed the frontal and parietal bone fractures using a bi-cubic rational Ball surface. The proposed method was validated by constructing the non-fractured part.

https://doi.org/10.3390/math8081246.

https://www.mdpi.com/2227-7390/8/8/1246

**Pareja-Corcho, J.; Betancur-Acosta, O.; Posada, J.; Tammaro, A.; Ruiz-Salguero, O.; Cadavid, C. Reconfigurable 3D CAD Feature Recognition Supporting Confluent n-Dimensional Topologies and Geometric Filters for Prismatic and Curved Models. Mathematics 2020, 8(8), 1356;**

Abstract: Feature Recognition (FR) in Computer-aided Design (CAD) models is central for Design and Manufacturing. FR is a problem whose computational burden is intractable (NP-hard), given that its underlying task is the detection of graph isomorphism. Until now, compromises have been reached by only using FACE-based geometric information of prismatic CAD models to prune the search domain. Responding to such shortcomings, this manuscript presents an interactive FR method that more aggressively prunes the search space with reconfigurable geometric tests. Unlike previous approaches, our reconfigurable FR addresses curved EDGEs and FACEs. This reconfigurable approach allows enforcing arbitrary confluent topologic and geometric filters, thus handling an expanded scope. The test sequence is itself a graph (i.e., not a linear or total-order sequence). Unlike the existing methods that are FACE-based, the present one permits combinations of topologies whose dimensions are two (SHELL or FACE), one (LOOP or EDGE), or 0 (VERTEX). This system has been implemented in an industrial environment, using icon graphs for the interactive rule configuration. The industrial instancing allows industry based customization and itis faster when compared to topology-based feature recognition. Future work is required in improving the robustness of search conditions, treating the problem of interacting or nested features, and improving the graphic input interface.

https://doi.org/10.3390/math8081356.

https://www.mdpi.com/2227-7390/8/8/1356

**Hu, G.; Li, H.; Abbas, M.; Miura, K.T.; Wei, G. Explicit Continuity Conditions for G<sup>1</sup> Connection of S-<i>λ</i> Curves and Surfaces. Mathematics 2020, 8(8), 1359;**

Abstract: The S-λ model is one of the most useful tools for shape designs and geometric representations in computer-aided geometric design (CAGD), which is due to its good geometric properties such as symmetry, shape adjustable property. With the aim to solve the problem that complex S-λ curves and surfaces cannot be constructed by a single curve and surface, the explicit continuity conditions for G1 connection of S-λ curves and surfaces are investigated in this paper. On the basis of linear independence and terminal properties of S-λ basis functions, the conditions of G1 geometric continuity between two adjacent S-λ curves and surfaces are proposed, respectively. Modeling examples imply that the continuity conditions proposed in this paper are easy and effective, which indicate that the S-λ curves and surfaces can be used as a powerful supplement of complex curves and surfaces design in computer aided design/computer aided manufacturing (CAD/CAM) system.

https://doi.org/10.3390/math8081359.

https://www.mdpi.com/2227-7390/8/8/1359

**Majeed, A.; Abbas, M.; Qayyum, F.; Miura, K.T.; Misro, M.Y.; Nazir, T. Geometric Modeling Using New Cubic Trigonometric B-Spline Functions with Shape Parameter. Mathematics 2020, 8(12), 2102;**

Abstract: Trigonometric B-spline curves with shape parameters are equally important and useful for modeling in Computer-Aided Geometric Design (CAGD) like classical B-spline curves. This paper introduces the cubic polynomial and rational cubic B-spline curves using new cubic basis functions with shape parameter ξ∈[0,4]. All geometric characteristics of the proposed Trigonometric B-spline curves are similar to the classical B-spline, but the shape-adjustable is additional quality that the classical B-spline curves does not hold. The properties of these bases are similar to classical B-spline basis and have been delineated. Furthermore, uniform and non-uniform rational B-spline basis are also presented. C3 and C5 continuities for trigonometric B-spline basis and C3 continuities for rational basis are derived. In order to legitimize our proposed scheme for both basis, floating and periodic curves are constructed. 2D and 3D models are also constructed using proposed curves.

https://doi.org/10.3390/math8122102.

https://www.mdpi.com/2227-7390/8/12/2102

**Panchuk, K.; Myasoedova, T.; Lyubchinov, E. Spline Curves Formation Given Extreme Derivatives. Mathematics 2021, 9(1), 47;**

Abstract: This paper is dedicated to development of mathematical models for polynomial spline curve formation given extreme vector derivatives. This theoretical problem is raised in the view of a wide variety of theoretical and practical problems considering motion of physical objects along certain trajectories with predetermined laws of variation of speed, acceleration, jerk, etc. The analysis of the existing body of work on computational geometry performed by the authors did not reveal any systematic research in mathematical model development dedicated to solution of similar tasks. The established purpose of the research is therefore to develop mathematical models of formation of spline curves based on polynomials of various orders modeling the determined trajectories. The paper presents mathematical models of spline curve formation given extreme derivatives of the initial orders. The paper considers construction of Hermite and Bézier spline curves of various orders consisting of various segments. The acquired mathematical models are generalized for the cases of vector derivatives of higher orders. The presented models are of systematic nature and are universal, i.e., they can be applied in formation of any polynomial spline curves given extreme vector derivatives. The paper provides a number of examples validating the presented models.

https://doi.org/10.3390/math9010047.

https://www.mdpi.com/2227-7390/9/1/47

**Ammad, M.; Misro, M.Y.; Abbas, M.; Majeed, A. Generalized Developable Cubic Trigonometric Bézier Surfaces. Mathematics 2021, 9(3), 283;**

Abstract: This paper introduces a new approach for the fabrication of generalized developable cubic trigonometric Bézier (GDCT-Bézier) surfaces with shape parameters to address the fundamental issue of local surface shape adjustment. The GDCT-Bézier surfaces are made by means of GDCT-Bézier-basis-function-based control planes and alter their shape by modifying the shape parameter value. The GDCT-Bézier surfaces are designed by maintaining the classic Bézier surface characteristics when the shape parameters take on different values. In addition, the terms are defined for creating a geodesic interpolating surface for the GDCT-Bézier surface. The conditions appropriate and suitable for G1, Farin–Boehm G2, and G2 Beta continuity in two adjacent GDCT-Bézier surfaces are also created. Finally, a few important aspects of the newly formed surfaces and the influence of the shape parameters are discussed. The modeling example shows that the proposed approach succeeds and can also significantly improve the capability of solving problems in design engineering.

https://doi.org/10.3390/math9030283.

## Keywords

geometric modeling, CAGD, shape parameters, spline curve, curves and surfaces, computer aided geometric design, geometric continuity, basis functions, Bézier curve, Bézier surface, continuity conditions, generalized H Bézier, spatial algebraic curve, constrained vertex norm, generalized h Bézier model, mathematical model, trigonometric B-spline, developable surface, feature recognition, GDCT Bézier surface, iteration point, point orthogonal projection, spline basis, superspirals of confluent type, theory and applications, bicubic rational ball surface, CAD/CAM, classical B-spline curve, completely monotonic gauss hypergeometric function, computer graphics, convex configuration, dimensional case, end point, ending line, explicit continuity conditions, extreme vector derivatives, fractured part, generalized hybrid trigonometric Bézier, geometric continuity conditions, geometric filters, geometry based guidance law, GHT Bézier curve, linear independence, radius of curvature, shape adjustable property, shape designs and geometric representation, smooth continuity, terminal properties, tools for shape design, trigonometric B-spline curve, Bézier, geometric, curve, shape, modeling, surface, continuity, spline, mathematics, design, parameters, conditions, models, cubic, point, rational, applications, curvature, trigonometric, function, method, model, CAD, confluent, GDCT, algebraic, algorithm, approach, derivative, formation, geometry, industrial, polynomial, spatial, superspiral, theory, time, trajectory, vertex, adjacent, angle, configuration, constrained, constraint, construction, engineering, extreme, norms, orthogonal, projection, reconfigurable, computational, constructing, contour, convex, dimensional, face, feature, graphics, guidance, hypergeometric, information, iteration, linear, manufacturing, monotonic, parameter, parametric, recognition, representation, speed, vector, acceleration, aesthetic, analysis, application, applied, approach, architecture, article