A Hybrid Algorithm for Point Orthogonal Projecting onto a Planar Algebraic Curve

Algorithm of point orthogonal projection onto a planar algebraic curve plays an important role in computer graphics, computer aided geometric design, especially interactive design, etc. A hybrid algorithm is proposed for calculating the orthogonal projection problem of point to plane algebraic curves based on the Newton gradient descent method, the midpoint foot point method formed by tangent and curvature circles, and the hybrid geometric acceleration orthogonal method. Firstly, Newton’s gradient descent method is used to make the initial iteration point fall on the plane algebraic curve. Secondly, based on the foot-point which is the middle point formed by the tangent and the curvature circle, Newton’s gradient descent method is used to make the iterative point falling on the plane algebraic curve gradually move to the position where the orthogonal projection point is very close. Finally, hybrid geometry accelerating orthogonal method is used to make the iterative point converge to the orthogonal projection point quickly till the orthogonal projection point is achieved. Three examples of closed plane algebraic curves were used for experiments. Validated by the convergence calculations, the results show that the algorithm is robust and efficient when the test points are far away or the degree of algebraic curves is high.