Symbolic Computation of the Orthogonal Projection of Rational Curves onto Rational Parameterized Surfaces

This paper focuses on the orthogonal projection of rational curves onto rational parameterized surface.Three symbolic algorithms are developed and studied. One of them, based on regular systems, is able to compute the exact parametric loci of projection. The one based on Gröbner basis can compute the minimal variety that contains the parametric loci. The remaining one computes a variety that contains the parametric loci via resultant. Examples show that our algorithms are efficient and valuable.


Introduction
Computing the projection of a point onto a surface is to find a closest point on the surface, and projection of a curve onto a surface is the locus of all points on the curve project onto the surface.The orthogonal projection problem attracted great interest in minimal distance computation [1,2], calculating the intersection of curves and surfaces [3], surface curve design [4,5], curve or surface selecting [6], and shape registration [7].And many algorithms have been developed.The work in [8] proposed a second-order tracing method for calculating the orthogonal projection of parametric curves onto B-spline surfaces.The work in [9] focused on projecting points onto conics.The work in [10] developed a secondorder algorithm for orthogonal projection onto curves and surfaces.The work in [11] used a torus patch to approach the surface in projection computation.In [12], an efficient algorithm is presented for projecting a point to its closest point.Among these methods, the common steps are to find the approach projective point in normed space by iteration techniques which rely on good initial values and then determine the approximate parameters in parametric space, which is called a point inversion problem.
Numerical methods above are efficient and stable in computing orthogonal projection and are easy applied.However, there exist common drawbacks as follows: the computation relies on samplings and the step size determines the accuracy of the result.The projective locus might be invisible while the locus is smaller than the step size.And the curve is always assumed to keep close enough to the surface so that a single solution is guaranteed.Symbolic methods would be necessary to overcome the shortcomings.Previous applications of symbolic methods in CAGD could be seen in [13][14][15].In order to apply symbolic methods, we only are concerned about curves and surfaces that have rational parametric representations.As known to all, common representations of surface and curves are NURBS [16], which is formed by rational patches.And since the parametric locus could uniquely determine the projection in 3D space, we focus on the parametric locus of orthogonal projection.Moreover, the range of surfaces and curves is restricted in R 3 .
Classical symbolic tools applied in this paper are regular systems [17] (triangular decomposition), Gröbner basis [18], and resultant (see [19,20]).Parametrization of curves and surfaces is a hard task in the area [21].But, for convenience, we only consider parametric curves and surfaces.With the rational assumptions of curves and surfaces, the orthogonal condition would be transformed into a simple polynomial system.Then the orthogonal projection problem equals determining the real solution of the polynomial system, which can be solved by symbolic or mix symbolic-numeric techniques.
In this paper, three algorithms are presented to compute the orthogonal projection of a rational parameterized curve onto a rational parameterized surface.The algorithm based 2 Mathematical Problems in Engineering on regular systems is able to compute the exact loci of orthogonal projection, and the false points will be detected.By means of Gröbner bases, we can get the minimal variety that contains the projective loci.And the resultant method efficiently computes a variety that contains the projective loci.The former two algorithms can particularly be used to compute point projections.
Compared with numerical algorithms, our algorithms have distinct advantages: (1) We generate the exact results without numerical errors.
(2) Both point projection and curve projection are included.
(3) There is no point inversion problem involved since we directly are concerned about the parametric loci.
In addition, the decomposition method in [22] would generate duplicate zeros between different regular systems and Huang and Wang [15] proposed a method to simplify the result.We improve Huang's method and directly consider the symbolic representation of zeros.Once the redundancy of zeros is judged, the corresponding regular system could be deleted without changing the zeros.
An early version of this paper has been reported on the 4th International Congress on Mathematical Software [23], in which the main algorithms and proofs are missing.The rest of the paper is organized as follows.In Section 2, some concepts and properties of regular systems, Gröbner basis, and resultant are introduced.Section 3 presents the main theorems.And Section 4 describes the algorithms based on the theorems in Section 3. In Section 5, we demonstrate nontrivial examples and experiment results.This paper is summarized in a brief conclusion in Section 6.

Preliminaries
Assume that K is a field with characteristic 0 and K[ 1 , . . .,   ] denotes the polynomial ring on K with ordered indeterminates  1 <  2 < ⋅ ⋅ ⋅ <   .For a polynomial  ∈ K[ 1 , . . .,   ],  K() = { ∈ K | () = 0} is called the zero set of , where K is a field extension of K.And  K() is simply denoted as () in this paper when there is no ambiguity.Definition 1.For  ⊆ K , one defines where 1 ≤  ≤ .
It is obvious that √  :  ∞ = √( :  ∞ ).Furthermore, Let R be a commutative ring with identity.Consider (), () ∈ R[]: The Sylvester Matrix of () and () with respect to  is defined to be where the former  rows are only related to the coefficients of  and the last  rows are only involved with the coefficients of .
We denote (, , ) to be the determinant of (, , ).And (, , ) is called the resultant of  and  with respect to .

The Main Results
In this section, we consider the orthogonal projection of a rational parameterized curve onto a rational parameterized surface.
Rational parameterized curves are defined as the images of mappings form where And rational parameterized surface is defined as the images of mappings form where Given a rational parameterized curve  with parametric equation Φ() and a rational parameterized surface  with parametric equation Ψ(, V), the orthogonal projection of  onto  is defined to be the set Γ  of points (, V, ) satisfying the following condition: where (, V) stands for the normal vector of , the above condition can be written as The problem of orthogonal projection is to find the solution of system (13).And note that (13) can be treated as polynomial systems, where Φ, Ψ are rational mappings.
Proof.Equation ( 13) could be simplified as the following form by substituting Φ() and Ψ(, V): That implies  1 = 0,  2 = 0 and Ψ 0 ̸ = 0, Φ 0 ̸ = 0 as 0 will not be denominators. Then and Proof.Since it is directly that And the second statement of the theorem holds according to Proposition 3.

Remark 12. For the polynomial system
) had been established [17], where [, V, ] means the variable order is  < V < .

Theorem 13. G is a Gröbner basis of
under a variable order  < V <  < .Then Proof.According to the properties of radical ideal and saturation of ideal, we have The last two equations hold under the statement of Proposition 6.And apparently Γ  (, V) = (G ∩ R[, V]).

Algorithms
For a polynomial set P and a set  ⊆ K2 , we denote ([P, ]) = (P) − .Then, for a polynomial system we define (Ω) = ∪  =1 ([T  ,   ]).Theorem 11 induces that the exact loci of projection could be decomposed into the union of zeros of regular systems, which could be in a complex form.In order to analyze the result easier, we developed an algorithm, which is improved from SIM [15], to simplify regular systems.Proposition 16.Algorithm 1 is correct.
Proof.Steps 1 and 2 are similar to SIM; we only need to prove Step 3.
So T can be deleted from Ω2 and U would be substituted by U \ (T).
Given a rational curve  and a rational surface , Algorithm 2 computes the exact parametric loci of the orthogonal projection of  onto .

Examples and Comparison
Example 1 (point projection).Consider the algebraic surface : where Let  = (−10, 0, 30) be a point in 3D space. could be treated as a constant function with variable .Algorithm 2 yields {[{ 1 ,  2 }, 0]}, where Algorithm 3 returns the same loci as above.Since { 1 ,  2 } is a triangular system, it is easy to check that ( 1 ,  2 ) contains only finitely isolated points.The point projections are shown in Figure 1.
Output A variety that contains Γ  (, V).

Figure 1 :
Figure 1: Projection of  onto  in Example 1.

Figure 3 :
Figure 3: Parametric loci and 3D curve of orthogonal projection for Example 3.

Figure 4 :
Figure 4: Parametric loci and 3D curve of orthogonal projection for Example 4.