Construction of Local-Shape-Controlled Quartic Generalized Said-Ball Model
Abstract
:1. Introduction
- The QGS-Ball basis functions are proposed by introducing three shape parameters.
- The QGS-Ball curves and surfaces are proposed, and the impact of shape parameters on the curves and surfaces is discussed.
- The CQGS-Ball curves are defined based on the novel QGS-Ball curves, and the continuity conditions of G1 and G2 smooth joining of QGS-Bal curves are derived.
2. Quartic Generalized Said-Ball Basis Functions
2.1. Definition of QGS-Ball Basis Functions
2.2. Properties of QGS-Ball Basis Functions
- (1)
- Non-negativity: For , there are , where .
- (2)
- Normality: .
- (3)
- Symmetry under the particular case: When , the QGS-Ball basis functions are symmetric, that is, and .
- (4)
- Endpoint properties:
- (5)
- Unimodal property: The QGS-Ball basis functions have only one maximum value on .
- (6)
- Monotonicity of parameters: Consider as a constant, is a decreasing function about , is an increasing function about and a decreasing function about , is an increasing function about , is a decreasing function about and an increasing function about , and is a decreasing function about .
- (7)
- Degeneracy: The QGS-Ball basis functions reduce into the traditional quartic Said-Ball basis functions when . It reduces into the quartic Bernstein basis functions when . It reduces into the cubic Bernstein basis functions when .
- (1)
- Because of , and , there are
- (2)
- According to Equation (1), there are
- (3)
- When , Equation (1) can be written as
- (4)
- The endpoint properties can be obtained by simple calculation of .
- (5)
- The unimodality of QGS-Ball basis functions can be verified by derivation. and have unimodality according to the property (3), so it is only necessary to prove that and have unimodality.
- (6)
- If is regarded as a constant, is a decreasing function about , is an increasing function about and a decreasing function about , is an increasing function about , is a decreasing function about and an increasing function about , and is a decreasing function about , property (6) is proved.
- (7)
- When , then the QGS-Ball basis functions can be written asWhen , then the QGS-Ball basis functions can be written asWhen , then the QGS-Ball basis functions can be written as
3. Quartic Generalized Said-Ball Curve
3.1. Definition and Properties of QGS-Ball Curve
- (1)
- Endpoint properties: For and , the QGS-Ball curve at the endpoints satisfyThe first and second derivatives of the curve at the endpoints satisfy
- (2)
- Symmetry under the particular case: When , the shape of the QGS-Ball curve with as the control polygon and the shape of the QGS-Ball curve with as the control polygon are the same, but the direction is opposite, i.e.,
- (3)
- Convexity: The QGS-Ball curve is involved in the convex hull of the control polygon.
- (4)
- Geometric invariability and affine invariability: Because the QGS-Ball basis functions satisfy the normalization, the affine transformation is performed on the QGS-Ball curve , the new curve is obtained by using the linear transformation and the translation , that is,It is the QGS-Ball curve corresponding to the new control points , which is obtained by the same affine transformation for .
- (5)
- Shape adjustability: The global and local shape of the QGS-Ball curve can be modified via the parameters.
- When five control points are given, only one unique quartic Bézier curve can be generated, while the QGS-Ball curve containing multiple shape parameters defines a family of curves.
- Because the QGS-Ball curve contains multiple shape parameters, the curves can be modified flexibly via the parameters while keeping the control points unchanged.
- Since the QGS-Ball curve contains shape parameters, shape optimization can be performed on the curves.
3.2. Impact of Shape Parameters on the QGS-Ball Curve
3.3. Performance Comparison of QGS-Ball Curves and Other Ball Curves
4. Smooth Joining of Combined Quartic Generalized Said-Ball Curves
4.1. Continuity Conditions of G1 Smooth Joining of QGS-Ball Curves
4.2. Continuity Conditions of G2 Smooth Joining of QGS-Ball Curves
4.3. Examples of CQGS-Ball Curves
5. Quartic Generalized Said-Ball Surface
5.1. Definition and Properties of the QGS-Ball Surface
- (1)
- Corner interpolation: The four corners of the QGS-Ball surfaces are interpolated to the four corners of the surface’s control mesh, that are
- (2)
- Boundary property: For the QGS-Ball surfaces , four boundary curves are the QGS-Ball curves generated by their corresponding outermost control points, respectively, that are
- (3)
- Tangent planarity of corners: For the QGS-Ball surfaces , the tangent planes at the four corners are determined by , , , and , respectively.
- (4)
- Symmetry: If the given control mesh vertices are symmetric, the QGS-Ball surfaces are also symmetric.
- (5)
- Convexity: The QGS-Ball surface is located in the convex hull of its control mesh.
- (6)
- Geometric invariability and affine invariability: Given the shape parameters and , the QGS-Ball surfaces are only related to the control vertices .
- (7)
- Shape adjustability: Given the control vertices , the global and local shape of QGS-Ball surfaces can be modified via the parameters and .
5.2. Impact of Shape Parameters on the QGS-Ball Surfaces
- (1)
- Given the control vertices and the shape parameters , the QGS-Ball surfaces move in the same direction as the control vertices by altering the shape parameters , that is, the shape parameters affect the local surface shape around the control vertices . In addition, the shape of borderline curves and is changed, while the shape of borderline curves and is not changed (see Figure 10).
- (2)
- Given the control vertices and the shape parameters , the QGS-Ball surfaces move in the same direction as the control vertices by altering the shape parameters , that is, the shape parameters affect the local surface shape around the control vertices . In addition, the shape of borderline curves and is changed, while the shape of borderline curves and is not changed (see Figure 11).
- (3)
- Given the control vertices , if the shape parameters and are increased (or decreased) at the same time, the QGS-Ball surfaces will approach (or move far from) its control mesh.
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Property | QGS-Ball Curves | Rational Cubic/Quartic Said-Ball Conics [9] | Generalized Ball Curves [10] | Said-Ball Curves [15] | Generalized Said-Ball Curves [20] | |
---|---|---|---|---|---|---|
Same | End-point properties | √ | √ | √ | √ | √ |
Convex hull property | √ | √ | √ | √ | √ | |
Symmetry | √ | √ | √ | √ | √ | |
Affine invariability | √ | √ | √ | √ | √ | |
Different | Computational complexity | Low | High | Low | Low | Low |
Number of shape parameters | 3 | * | 0 | 2 | 2 | |
Shape adjustability | Global and local | Global | × | Global | Global | |
Extra degree of freedom | √ | √ | × | √ | √ |
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Zheng, J.; Ji, X.; Ma, Z.; Hu, G. Construction of Local-Shape-Controlled Quartic Generalized Said-Ball Model. Mathematics 2023, 11, 2369. https://doi.org/10.3390/math11102369
Zheng J, Ji X, Ma Z, Hu G. Construction of Local-Shape-Controlled Quartic Generalized Said-Ball Model. Mathematics. 2023; 11(10):2369. https://doi.org/10.3390/math11102369
Chicago/Turabian StyleZheng, Jiaoyue, Xiaomin Ji, Zhaozhao Ma, and Gang Hu. 2023. "Construction of Local-Shape-Controlled Quartic Generalized Said-Ball Model" Mathematics 11, no. 10: 2369. https://doi.org/10.3390/math11102369
APA StyleZheng, J., Ji, X., Ma, Z., & Hu, G. (2023). Construction of Local-Shape-Controlled Quartic Generalized Said-Ball Model. Mathematics, 11(10), 2369. https://doi.org/10.3390/math11102369