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# Determining Intersection Lines of Intersecting Polyhedra

IP.com Disclosure Number: IPCOM000074743D
Original Publication Date: 1971-Jun-01
Included in the Prior Art Database: 2005-Feb-23
Document File: 4 page(s) / 57K

IBM

## Related People

Appel, A: AUTHOR [+2]

## Abstract

This technique relates to a method for determining the intersection curve and the essential bounding polygons of two intersecting polyhedra. The polyhedra may both be solid or real whereby the resulting solid is the spatial sum of the two original solids. Alternatively, one of the two intersecting polyhedra can be imaginary (empty space) and the resulting solid is effectively carved from the solid (real) polyhedra. In this manner, complex shapes can be composed from simple shapes which have previously been stored in a computer. The technique is particularly useful in the interactive design of solids and graphic display consoles where the designer can visually manipulate the solids in the computer storage until he produces the desired shape.

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Determining Intersection Lines of Intersecting Polyhedra

This technique relates to a method for determining the intersection curve and the essential bounding polygons of two intersecting polyhedra. The polyhedra may both be solid or real whereby the resulting solid is the spatial sum of the two original solids. Alternatively, one of the two intersecting polyhedra can be imaginary (empty space) and the resulting solid is effectively carved from the solid (real) polyhedra. In this manner, complex shapes can be composed from simple shapes which have previously been stored in a computer. The technique is particularly useful in the interactive design of solids and graphic display consoles where the designer can visually manipulate the solids in the computer storage until he produces the desired shape. With the use of this technique, there is avoided the need to calculate coordinates in space except in the situation where the knowledge of these coordinates is essential. In addition, there is not needed the tedious and time-consuming drawing in three-dimensional space of all of the lines on a solid. Furthermore, with this technique, there is eliminated the necessity of having to organize graphic data as a picture is generated and a mathematical description of a solid can be generated to enable the rendering of the solid from the computer more rapidly, and with more accuracy as compared to the speed and accuracy of a draftsman. The mathematical description of a solid is so organized that it can be employed to generate numerically controlled machine tool instructions.

Heretofore, the classic solution of the problem solved by this technique has been by the use of descriptive geometry techniques, the solution comprising the determining of all piercing points of edges of one polyhedron within the faces of the other polyhedra. The draftsman then has to use his spatial imagination or undertake a careful analysis of the shapes of the polyhedra in order to form the curve of intersection. The intersection curve is generally nonplanar and consists of many line segments. In addition, there may be more than one complete spatial loop and the curve may cross itself. The time required for elegant geometric configurations can be of the order of an hour.

In accordance with this technique, the strategy for determining the intersection curve and the essential bounding polygons of two intersecting polyhedra comprises the following:

1) The storing in secondary storage of a selection of conventional three- dimensional geometric shapes such as, cubes, triangular prisms, many-sided polyhedra, L-shaped polyhedra and polyhedra representing standard extruded metal shapes or other common engineering shapes. These three-dimensional shapes and their enclosing flat surfaces are processed by known program techniques to determine: (a) equations of lines (b) equations of surfaces (c) inward pointing vectors for each surface. These vectors point toward the side of the surf...