Possibilities and Advantages of Rational Envelope and Minkowski Pythagorean Hodograph Curves for Circle Skinning
Abstract
:1. Introduction
2. Motivation and Previous Work
2.1. Rational Envelope (RE) Curves
2.2. Applying RE/MPH Curves for Skinning
3. Using RE/MPH Curves for Skinning with More Flexibility
3.1. Determining the Tangent Vectors
3.2. Constructing the Envelope Curves
Algorithm 1 A flexible approach for using RE/MPH curves for skinning purposes |
Require: An admissible sequence of circles |
Ensure: The skinning curves as envelope curves with continuity |
1: |
2: |
3: for to do |
4: |
5: for to n do |
6: |
7: ▹ see (11) |
8: |
9: for to do |
10: |
11: ▹ see (12) |
12: |
13: ▹ see (13) and (14) |
14: ▹ see (15) |
15: ▹ see (16) |
16: |
17: return |
4. Intersections of the Envelope Curves
4.1. Intersection Detection
- , or
- , that is, the tangent vector of at the parameter value is light-like.
- If , then .
- If , then the equation can be written asSince , rearranging the equation we get
- If the third coordinate is zero for a point on the MAT, then it means that the radius of the corresponding cycle is zero. A cycle of zero radius is called null-cycle, and it is interpreted as a point. Moreover, the tangent lines drawn to this null-cycle touch it at the same point, thus this point becomes the point of the envelopes, i.e., .
- If the tangent vector of the MAT at parameter value is light-like, then the tangent line at is a generator line of the representing cone of . In this case, the piercing point of the tangent line and the plane is a point on the cycle. As it is not possible to draw tangent lines from this point to the cycle, then the point of the envelope cannot be anything else than the point itself, so .
- is irregular and is a singular point;
- is regular, but holds; or
- , i.e., the tangent vector of the RE curve at the parameter value is light-like.
4.2. Eliminating the Intersections
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Kruppa, K.; Kunkli, R.; Hoffmann, M. Possibilities and Advantages of Rational Envelope and Minkowski Pythagorean Hodograph Curves for Circle Skinning. Mathematics 2021, 9, 843. https://doi.org/10.3390/math9080843
Kruppa K, Kunkli R, Hoffmann M. Possibilities and Advantages of Rational Envelope and Minkowski Pythagorean Hodograph Curves for Circle Skinning. Mathematics. 2021; 9(8):843. https://doi.org/10.3390/math9080843
Chicago/Turabian StyleKruppa, Kinga, Roland Kunkli, and Miklós Hoffmann. 2021. "Possibilities and Advantages of Rational Envelope and Minkowski Pythagorean Hodograph Curves for Circle Skinning" Mathematics 9, no. 8: 843. https://doi.org/10.3390/math9080843
APA StyleKruppa, K., Kunkli, R., & Hoffmann, M. (2021). Possibilities and Advantages of Rational Envelope and Minkowski Pythagorean Hodograph Curves for Circle Skinning. Mathematics, 9(8), 843. https://doi.org/10.3390/math9080843