Torus-Line Segment Intersection via Inscribed Sphere Iteration
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Solution Overview
Problem
Existing methods are impractical for determining the intersection point of a torus and a line segment due to the complexity of solving high-order equations involved.
Innovation Solution
An image processing device and method that acquires an inscribed sphere within the torus, calculates its intersection with the line segment, and iteratively refines the inscribed sphere's position to converge on the torus's intersection with the line segment, allowing for output of the torus's intersection point without directly solving the high-order equation.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If the intersection of the torus and line segment is calculated by directly solving the high-order equation, then the exact intersection point can be obtained, but the calculation becomes difficult and unrealistic
Solution Approach 1:
The patent introduces an inscribed sphere as an intermediary object between the torus and the line segment. Instead of directly calculating the complex torus-line segment intersection, the method calculates the simpler sphere-line segment intersection and uses the inscribed sphere's properties (contact circle with torus) to derive the torus intersection point, thereby simplifying the calculation while maintaining accuracy
Solution Approach 2:
The patent replaces the complex algebraic system (solving high-order equations for torus intersection) with a geometric system (using inscribed sphere properties and contact circles). This substitution transforms an analytically difficult problem into a geometrically tractable one by leveraging the simpler geometry of spheres and their relationship with tori
2Productivity
If the inscribed sphere method is used to calculate the torus intersection, then the calculation becomes realistic and efficient, but iterative refinement is required to achieve sufficient precision
Solution Approach 1:
The patent implements a feedback mechanism where the inscribed sphere is iteratively repositioned based on the calculated intersection point. The contact circle of the inscribed sphere with the torus is used to determine a new sphere position that brings the contact circle closer to the actual intersection, creating a convergent iterative process that refines the solution
Solution Approach 2:
The method performs preliminary calculations using the inscribed sphere to obtain an approximate intersection point, which then serves as the basis for subsequent iterative refinements. This preliminary action provides a good starting point that reduces the number of iterations needed to achieve the desired precision
Data Source
AI summary
An image processing device includes an inscribed sphere acquiring section that acquires an inscribed sphere which intersects line segment and is inscribed in torus in calculation of an intersection of the torus and a line segment, a calculating section that calculates an intersection of the inscribed sphere and the line segment, a reacquiring section that acquires the position of a new inscribed sphere in such a manner that a contact circle between the new inscribed sphere and the torus gets closer to the intersection, a recalculating section that calculates an intersection of the new inscribed sphere and the line segment, and an intersection output section that outputs the intersection of the torus and the line segment on the basis of an intersection of the inscribed sphere and the line segment obtained by at least one time of calculation of the intersection of the new inscribed sphere and the line segment.