An Improved Curvature Circle Algorithm for Orthogonal Projection onto a Planar Algebraic Curve

Zhinan Wu,Xiaowu Li

doi:10.3390/math7100912

Abstract

Point orthogonal projection onto planar algebraic curve plays an important role in computer graphics, computer aided design, computer aided geometric design and other fields. For the case where the test point p is very far from the planar algebraic curve, we propose an improved curvature circle algorithm to find the footpoint. Concretely, the first step is to repeatedly iterate algorithm (the Newton’s steepest gradient descent method) until the iterated point could fall on the planar algebraic curve. Then seek footpoint by using the algorithm (computing footpoint q ) where the core technology is the curvature circle method. And the next step is to orthogonally project the footpoint q onto the planar algebraic curve by using the algorithm (the hybrid tangent vertical foot algorithm). Repeatedly run the algorithm (computing footpoint q ) and the algorithm (the hybrid tangent vertical foot algorithm) until the distance between the current footpoint and the previous footpoint is near 0. Furthermore, we propose Second Remedial Algorithm based on Comprehensive Algorithm B. In particular, its robustness is greatly improved than that of Comprehensive Algorithm B and it achieves our expected result. Numerical examples demonstrate that Second Remedial Algorithm could converge accurately and efficiently no matter how far the test point is from the plane algebraic curve and where the initial iteration point is.

Highlights

Reconstructing curve/surface is an important work in the field of computer aided geometric design, especially in geometric modeling and processing where it is crucial to fit curve/surface in high accuracy and reduce the error of representation curve/surface
After a lot of testing and observation, when the point on the curve is close to the orthogonal projection point, we find that Comprehensive Algorithm A presents two characteristics: (1) difference between the first distance and the second distance decreases slower and slower, where the first distance and the second distance are the one between the previous iterative point pc on the planar algebraic curve and the orthogonal projection point pΓ, and the one between the current iterative point pc on the planar algebraic curve and the orthogonal projection point pΓ, respectively; (2) the rate goes even slower at which the absolute value of the inner product gradually approaches zero
No matter how far the test point is from the planar algebraic curve, Second Remedial Algorithm converges very robustly and efficiently

Summary

Introduction

Reconstructing curve/surface is an important work in the field of computer aided geometric design, especially in geometric modeling and processing where it is crucial to fit curve/surface in high accuracy and reduce the error of representation curve/surface. The representation of the four curve types are the explicit-type, implicit-type, parametric-type and subdivision-type. Because implicit representation has unique advantage in the process of computer aided geometric design, it has wide and far-reaching applications. From scattered and unorganized three-dimensional data, Bajaj et al [1]. They [2,3] have constructed the algebraic B-spline surfaces with least-squares fitting feature using tensor product technique. Schulz et al [4] constructed an enveloping algebraic surface using gradually approximate algebraization method. Kanatani et al [5]

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Results

Conclusion