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# Methods for Correcting Polyhedra Designs

IP.com Disclosure Number: IPCOM000076540D
Original Publication Date: 1972-Mar-01
Included in the Prior Art Database: 2005-Feb-24
Document File: 6 page(s) / 106K

IBM

## Related People

Appel, A: AUTHOR [+2]

## Abstract

In the interactive encoding of polyhedra, the description of a polyhedron is usually stored in a computer as two lists. One of these lists is a set of labeled points in space and their three-dimensional coordinates. The other of these lists is a table, setting forth as to how the labeled points of the first list are connected together to form the boundaries of surfaces. In Fig. 1, there is shown an example of a polyhedron wherein labeled points are interconnected to form such boundaries. The Vertex List and Topological Map of the polyhedron of Fig. 1 is as follows: VERTEX LIST: Point X Y Z 1 1.0 0.0 1.2 2 0.0 0.5 1.2 3 0.0 0.0 1.2 4 1.0 0.5 1.2 5 1.0 0.5 0.4 etc. TOPOLOGICAL MAP: Surface I 1 -> 2 -> 3 -> 4 Surface II 1 -> 4 -> 5 -> 6 -> 7 -> 12 etc.

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Methods for Correcting Polyhedra Designs

In the interactive encoding of polyhedra, the description of a polyhedron is usually stored in a computer as two lists. One of these lists is a set of labeled points in space and their three-dimensional coordinates. The other of these lists is a table, setting forth as to how the labeled points of the first list are connected together to form the boundaries of surfaces. In Fig. 1, there is shown an example of a polyhedron wherein labeled points are interconnected to form such boundaries. The Vertex List and Topological Map of the polyhedron of Fig. 1 is as follows: VERTEX LIST: Point X Y Z 1 1.0 0.0 1.2 2 0.0 0.5 1.2 3 0.0 0.0 1.2 4
1.0 0.5 1.2 5 1.0 0.5 0.4 etc. TOPOLOGICAL MAP: Surface I 1 -> 2 -> 3 -> 4 Surface II 1 -> 4 -> 5 -> 6 -> 7 -> 12 etc.

Generally, the Topological Map is easily interactively encoded. However, the coordinates of the vertex points are difficult to encode properly for several reasons as follows:

1) All of the points on a surface may not satisfy a unique surface equation. A plane has the following equation:. Ax + By + Cz + 1 = 0 (1). wherein x, y, and z is the vector of all points on the plane surface.

If more than three points are on a surface, equation (1) is overly determined. It is improbable that points encoded from an interactive graphics tablet or the screen of a cathode-ray tube in an interactive graphics unit will be coplanar. Thus, as shown in Fig. 2, surfaces 1, 2, 3 and 4 are overly determined. If all of the points on these surfaces are not coplanar, then the description of the object is invalid for purposes such as numerical control machining, or hidden-line elimination.

2) The point may not be the intersection of all of the surfaces that it lies on. The intersection of three or more planes is a point that satisfies the following equation: Ax + By + Cz + 1 = 0 (2). Wherein A, B, and C constitute the vector of planes which contain this point.

If the point is the intersection of more than three surfaces, it is not probable that by interactive encoding this point actually is the intersection of the planes involved. In Fig. 2, two points are shown which are overly determined, specifically points 1 and 2.

3) An interactive graphics tablet is nonlinear in the X and Y directions. Consequently, any attempt to draw an object that is symmetrical will result in inaccuracies.

4) Quite often, some planes are intended to be orthogonal, i.e., two of the parameters in the surface equation (1) set forth hereinabove have to be precisely
0. Also, some planes are intended to be inclined, i.e., one of the parameters in the surface equation (1) has to be exactly 0. These situations arise when it is intended to design symmetrical objects or objects that must fit together, or that are to be manufactured.

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5) In some situations, an attempt is made to draw an object such as a human being or an animal where only a rough idea is had as to how the polyhedron should app...