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# Computer Generation of Warped Surface Models

IP.com Disclosure Number: IPCOM000079491D
Original Publication Date: 1973-Jul-01
Included in the Prior Art Database: 2005-Feb-26
Document File: 4 page(s) / 82K

IBM

## Related People

Appel, A: AUTHOR [+1]

## Abstract

There is described herein a technique for making models of warped surfaces. These models are advantageously employed in the description and presentation of complex mathematical functions.

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Computer Generation of Warped Surface Models

There is described herein a technique for making models of warped surfaces. These models are advantageously employed in the description and presentation of complex mathematical functions.

An example of such complex mathematical functions wherein the use of such a warped surface model is beneficial are those of "Catastrophe Theory". Catastrophe Theory describes the nature of general systems in terms of the behavior of the stationary (maximum and minimum) values of an equation, of the form set forth in equation (1) hereinbelow. Heretofore, it has been quite difficult to describe the geometry of the surface generated as the loci of the maximum and the minimum of the values of equation (1). V = x(n) over n + A x/n-2/ over n-2 + B x/n-3/ over n-3 + .... + R x. (1). Catastrophe Theory may be used to model the behavior of many different systems, typical examples of such systems being the study of embryo growth, cell differentiation, animal behavior, and other problems.

Heretofore, the most successful presentations of these generated surfaces has been by means of computer generated moving pictures or, in some instances, with the use of stereoscopic pictures. However, the latter type pictures have proven to be very difficult to comprehend. For some functions where u = f(x,w) (2)

v = f'(x,w) it is possible to consider a particular three-dimensional section of the surface (u,v,x) as a single-ruled warped surface. A single-rules surface has the property such that, through each point of the surface, a straight line can be drawn that will lie entirely on the surface. Thus, for those instances where conditions, as set forth in equation (2) hereinabove, are true, a string model of the surface (generated by the stationary values of equation (1)) can be made.

In accordance with the technique described herein, the shape of the string model is calculated and the model is constructed as follows: The intersection of the warped surface rulings and a tube of usually but not necessarily square cross section is calculated, as shown in Fig. 1, wherein there is depicted for each straight line ruling, two piercing points. For example, a typical ruling has the piercing points PP1 and PP2; another ruling has the piercing points PP3, and PP4. The rulings are selected such that they are uniformly spaced in the vertical direction. The next step in the technique, is to draw these piercing points on each side of the tube. Fig. 2 shows such points drawn on the tube. Fig. 3 shows the unfolding of the sides of the tube. The drawings of the unfolded sides of the tube are suitably made using a conventional digital plotter; and each side is cut out and pasted on a suitable substrate such as plexiglass. The piercing points are drilled through and the sides of the substrate are assembled into boxes. With such assembling the model is threaded as shown in Fig. 4.

In Fig. 5, there is shown a flow chart for making the necessary calcu...