N-Dimensional Polyhedron Modeling via Triangular Segmentation
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Solution Overview
Problem
Existing methods for modeling and truncating polyhedra are limited by complex formulas and require advanced computer processors and memory, unable to generalize to all general polyhedra, especially those with concave shapes and holes, and are computationally intensive.
Innovation Solution
A method using simple mathematical computations of angles and side lengths of triangles to model and truncate any n-dimensional General Polyhedron, allowing truncation along any edge by any proportion and angle, without relying on complex calculations of intersecting planes or half spaces.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If complex formulas and advanced computational methods are used to model and truncate polyhedra, then modeling precision and versatility are improved, but device complexity and computational resource requirements increase
Solution Approach 1:
The patent segments the polyhedron modeling process into discrete triangular faces defined by vertices and edges. Each triangular face is independently defined using simple vertex coordinates, and the entire polyhedron is constructed by assembling these triangular segments. This segmentation allows complex polyhedra to be modeled using basic geometric elements without requiring advanced computational formulas.
Solution Approach 2:
The patent uses triangular faces as reusable geometric primitives that can be copied and assembled to form any polyhedron. By defining polyhedra through collections of triangular faces with simple vertex coordinates, the method creates a universal template that can represent any general polyhedron type without requiring specialized computational routines for each shape.
2Adaptability or versatility
If traditional polyhedron definitions requiring two faces per edge are used, then geometric simplicity is maintained, but adaptability to model concave shapes and holes is lost
Solution Approach 1:
The patent creates a universal polyhedron definition where triangular faces with vertex coordinates can represent any polyhedron type - convex, concave, with holes, or without holes. The same simple data structure (vertices, edges, triangular faces) serves multiple functions and can model the entire spectrum of general polyhedra without requiring different definition systems for different shape categories.
Solution Approach 2:
The patent moves from traditional 3D polyhedron definitions to a 2D projection-based definition using triangular faces with vertex coordinates. By representing polyhedra through their 2D triangular face projections and vertex positions, the method gains the ability to easily represent complex topologies including holes and concavities that are difficult to define using traditional 3D geometric primitives.
3Adaptability or versatility
If midpoint truncation and symmetric plane constraints are used, then computational simplicity is maintained, but versatility to truncate by any proportion and angle is lost
Solution Approach 1:
The patent implements dynamic truncation where the truncation parameters (proportion and angle) can be freely adjusted without being constrained to fixed midpoints or symmetric planes. The triangular face definition allows vertices to be moved to any position along edges, enabling truncation at any proportion and angle while maintaining computational simplicity through the same basic vertex coordinate system.
Solution Approach 2:
The patent changes the truncation parameters from fixed values (midpoints, symmetric planes) to variable parameters (any proportion along edges, any angle). By modifying the vertex coordinates of triangular faces according to user-specified proportions and angles, the system achieves flexible truncation without requiring complex computational formulas, simply adjusting the positional parameters of existing vertices.
Data Source
AI summary
An apparatus and method for modeling all matter by modeling, truncating, and creating general polyhedra without requiring advanced computer processors and computer memory according to one embodiment includes at least one processor, means for inputting vertex data to the processor, a display in data communication with the processor, and computer memory coupled to the processor. The computer memory has recorded within it machine readable instructions for storing vertex data previously input to the processor, truncating a polyhedron created using the vertex data, and actuating the display. The instructions for creating and truncating a polyhedron utilize length data and angle data for triangles formed from the vertex data.