Point Orthogonal Projection onto a Spatial Algebraic Curve

Taixia Cheng,Zhinan Wu,Xiaowu Li,Chan Wang

doi:10.3390/math8030317

Taixia Cheng, Zhinan Wu + Show 2 more

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https://doi.org/10.3390/math8030317

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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.

Highlights

Orthogonal projection is widely used and plays an important role in geometric modeling, computer graphics and computer aided geometric design
There is a close relationship between the orthogonal projection problem and distance projection problem [1]; the study of the distance projection problem is helpful for the study of the orthogonal projection problem to some extent
The second advantage of the Newton’s steepest gradient descent method (9) is that the iteration point fallen on the spatial algebraic curve is closer to the orthogonal projection point PΓ, it is very convenient for implementation of the subsequent sub-algorithms

Summary

Introduction

Orthogonal projection is widely used and plays an important role in geometric modeling, computer graphics and computer aided geometric design. Pegna et al [1] first proposed the concept of orthogonal projection, and discussed the calculation projecting problem of spatial parametric curve orthogonal projection onto a parametric surface and algebraic surface. The so-called orthogonal projection is to find a point on the curve such that the line segment connected to this point and the given point is perpendicular to the tangent line of the curve at this point. There is a close relationship between the orthogonal projection problem and distance projection problem [1]; the study of the distance projection problem is helpful for the study of the orthogonal projection problem to some extent. Hartmann [2] proposed a first-order tangent line method to calculate point orthogonal projection onto parametric curve and surface

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