Point Orthogonal Projection onto a Spatial Algebraic Curve

Taixia Cheng, Zhinan Wu, Xiaowu Li and Chan Wang
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Taixia Cheng: Graduate School, Guizhou Minzu University, Guiyang 550025, China
Zhinan Wu: School of Mathematics and Computer Science, Yichun University, Yichun 336000, China
Xiaowu Li: College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China
Chan Wang: School of Finance, Central University of Finance and Economics, Beijing 100081, China

Mathematics, 2020, vol. 8, issue 3, 1-23

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.

Keywords: point orthogonal projection; the closest point; intersection; spatial algebraic curve; Newton’s steepest gradient descent method; foot-point; geometric correction method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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