Direct Orthogonal Distance to Quadratic Surfaces in 3D

Area of Science:

• Computer Vision
• Geometric Modeling
• Robotics

Background:

• Calculating orthogonal distances to quadratic surfaces is a fundamental geometric problem.
• Existing methods may lack efficiency, stability, or broad applicability across surface types.

Purpose of the Study:

• To develop a simple, efficient, and stable direct solution for the orthogonal distance to arbitrary quadratic surfaces.
• To provide a method that handles all real quadratic surface classes and detects special geometric cases.

Main Methods:

• The problem is reformulated as the intersection of three quadratic surfaces.
• Orthogonality conditions are used to derive two of these surfaces.
• A sixth-order polynomial in one coordinate is directly generated for a closed-form solution.

Main Results:

• A direct, stable, and efficient algorithm for computing orthogonal distances to quadratic surfaces is presented.
• The method successfully detects intersection points at infinity and handles all real quadratic surface types.
• Geometric detection of continuums of orthogonal points, such as for spheres, is achieved.

Conclusions:

• The proposed direct method offers a robust and versatile solution for orthogonal distance computation.
• This algorithm has significant implications for computer vision, geometric modeling, and robotics applications.
• The method's performance and extensions to higher dimensions are discussed.