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In this paper, we present the analytical expressions for computing the minimum distance between a point and a torus, which is called the orthogonal projection point problem. If the test point is on the outside of the torus and the test point is at the center axis of the torus, we present that the orthogonal projection point set is a circle perpendicular to the center axis of the torus; if not, the analytical expression for the orthogonal projection point problem is also given. Furthermore, if the test point is in the inside of the torus, we also give the corresponding analytical expression for orthogonal projection point for two cases.

In this paper, we discuss how to compute the minimum distance between a point and a spatial parametric surface and to return the nearest point on the surface as well as its corresponding parameter, which is also called the point projection problem (the point inversion problem) of a spatial parametric surface. It is very interesting for this problem due to its importance in geometric modeling, computer graphics and computer vision [

In the various methods mentioned above, all the iterative processes can produce one iterative solution. Different from the above methods, we consider the special situation which the test point have countless corresponding solutions for the orthogonal projection problem. We present the analytical expression for computing the minimum distance between a point and a torus. If the test point is on the outside of the torus and the test point is at the center axis of the the torus, we know that the orthogonal projection point set is a circle which is perpendicular to the center axis of the torus; If not, the analytical expression for the orthogonal projection point problem is also given. In addition, if the test point is in the inside of the torus and is on the major planar circle, then the corresponding analytical expression for orthogonal projection point set is minor planar circle. Moreover, if the test point is in the inside of the torus and is not on the major planar circle, we also present the corresponding analytical expression for orthogonal projection point of the test point.

The torus

in

Firstly, we deal with the first kind of circumstance which the test point is on the center axis of the torus

Assume that the coordinates of arbitrary point of major planar circle is

It is not difficult to find that line segment

From (1) and (4), we get that the corresponding parameter value of intersection of parametric equation for the line segment

or another form

By (3) and (5), we obtain

In the case of the test point being at the center axis of the torus, Formula (6) indicates that the corresponding orthogonal projection point set of the test point is a circle which parallels to major planar circle (see

In the following content, we try to discuss the second orthogonal projection case which test point

From (2) and (7), we obtain that the corresponding intersection of the plane

the torus is

And because the intersections of the torus

by (8) and (9), the corresponding parameter value of intersection of the line segment

Because the intersections of the torus

from (10) and (11), the intersection parameter of the line segment

In the following, we explain that the distance between the intersection

And because of

so it exists inequality relationship

This demonstrates that the distance between the second intersection and the test point is longer than the distance between the first intersection and the test point. Thus the distance between the intersection

Remark 1. If the test point

In this subsection, we suppose that test point

and

the corresponding minor planer circle. We directly present the corresponding analytical expression according to the test points being at different positions for major planar circle. Since Formula (9) denotes two minor planer circles, in fact, orthogonal projection point set of arbitrary test point being on the major planar circle just only has one minor planar circle. According to this reason, we try to present a unified and concise analytical expression of the only one minor planar circle for arbitrary test point being on the major planar circle. If

For more special cases, if test point

If test point

If test point

If test point

Remark 2. In this subsection, we fully present the corresponding orthogonal projection point or point set of arbitrary test point which is in the inside of the torus, namely, the corresponding analytical expression of orthogonal projection point for the minimum distance between the test point and the torus. If the test point is not on the major planar circle, then the corresponding analytical expression of orthogonal projection point is only one point. If the test point is on the major planar circle, then the corresponding analytical expression of

orthogonal projection point is minor planar circle. Besides that, if the test point

This paper investigates the problem related to a point projection on the torus surface. We present the analytical expression for the orthogonal projection of computing the minimum distance between a point and a torus for all kind s of positions. An area for future research is to develop a method for computing the minimum distance between a point and a general completely center symmetrical surface.

We would like to take the opportunity to thank the reviewers for their thoughtful and meaningful comments. This work is supported by the National Natural Science Foundation of China (Grant No. 61263034), the Scientific and Technology Foundation Funded Project of Guizhou Province (Grant No. [

Xiaowu Li,Linke Hou,Juan Liang,Zhinan Wu,Lin Wang,Chunguang Yue, (2016) The Analytical Expressions for Computing the Minimum Distance between a Point and a Torus. Journal of Computer and Communications,04,125-133. doi: 10.4236/jcc.2016.44011