Adjustable Bézier Curves with Simple Geometric Continuity Conditions
Abstract
:1. Introduction
2. Blending Functions
2.1. Construction of the Blending Functions
2.2. Properties of the Blending Functions
- (a)
- Degeneracy: When , the BL functions are the quartic Bernstein basis functions.
- (b)
- Non-negativity: When , for any and , we have , .
- (c)
- Normalization: For any , , and , we have .
- (d)
- Symmetry: For any , , , and , we have , and .
- (e)
- Endpoint property: For any , , and we have
- (f)
- Linear independence: For any , , and , the BL functions , are linearly independent.
- (e)
- The conclusions in (2) are obvious. We only prove (3a) and (3b). If
- (f)
- Let us consider the linear combination
3. Adjustable Bézier Curves
3.1. Construction of the Adjustable Bézier Curves
3.2. Properties of the Adjustable Bézier Curves
- (1)
- Convex hull property: The adjustable Bézier curves lie inside the convex hull of the control points. This is true, since the BL functions are nonnegative on and sum to 1.
- (2)
- Geometric invariance: From (13), we know that the adjustable Bézier curves are affine combinations of their control points. Thus, their shape is independent of the choice of the coordinate system.
- (3)
- Symmetry: The points , , and , , define two adjustable Bézier curves with the same shape but different parameterization.
- (4)
- Geometric property at the endpoints: From (2), (3) and (12) we get
- (5)
- Shape adjustability property: Even if the control points of an adjustable Bézier curve are fixed, its shape can still be adjusted by changing the values of the three parameters , , and .
4. Composite Adjustable Bézier Curves
5. Adjustable Bézier Curves with Tangent Polygon
6. Conclusions
Conflicts of Interest
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Yan, L. Adjustable Bézier Curves with Simple Geometric Continuity Conditions. Math. Comput. Appl. 2016, 21, 44. https://doi.org/10.3390/mca21040044
Yan L. Adjustable Bézier Curves with Simple Geometric Continuity Conditions. Mathematical and Computational Applications. 2016; 21(4):44. https://doi.org/10.3390/mca21040044
Chicago/Turabian StyleYan, Lanlan. 2016. "Adjustable Bézier Curves with Simple Geometric Continuity Conditions" Mathematical and Computational Applications 21, no. 4: 44. https://doi.org/10.3390/mca21040044
APA StyleYan, L. (2016). Adjustable Bézier Curves with Simple Geometric Continuity Conditions. Mathematical and Computational Applications, 21(4), 44. https://doi.org/10.3390/mca21040044