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# Picture Plane Distortion Process

IP.com Disclosure Number: IPCOM000078461D
Original Publication Date: 1973-Jan-01
Included in the Prior Art Database: 2005-Feb-26
Document File: 3 page(s) / 63K

IBM

## Abstract

There is described herein a general process for data presentation which is effected by the deforming of the picture plane.

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Picture Plane Distortion Process

There is described herein a general process for data presentation which is effected by the deforming of the picture plane.

The salient component of the process is illustrated in Fig. 1. In this figure, a surface of revolution 10, with a cross-section curve of F = f(P) = Z, is set over the picture plane 12 and touches the picture plane at a point 14. Point 14 is generally the origin of the surface of revolution, but can be anywhere. To obtain the spatial transformation of any point on picture plane 12, i.e., point,(P(X,Y)), there has to be found the intersection of a line connecting P and the point (0, 0,
1) with the surface of revolution. The X and Y coordinates of this point of intersection are the coordinates on the picture plane of P'(X,Y), which is the spatial transformation of P(X,Y). Fig. 2 illustrates the cross section of the arrangement shown in Fig. 1, which indicates that the line is the true length. The phenomenological model shown in Fig. 1 need not be analyzed in detail because, as is shown in Fig. 3, several mathematical short cuts can be availed of.

As shown in Fig. 3, R = square root of X/2/ +Y/2/. Taking X' = (X/R) . R' and Y' = (Y/R) . R' bypasses the problem of the calculating of direction cosines of three-dimensional lines. Utilizing Fig. 3, the following equations can be employed to produce the structure shown in Fig. 1. F = P/2/ F = (-1 over R) P + 1 P/2/ = -P over R + 1 P/2/ + P over R - 1 = 0 P = 1 over 2 ( - 1 over R + 1 over R/2/ +
4)/1/2/) = R' X' = X over R . R', Y' = Y over R . R' The R' is the transformation of
R. The foregoing equations illustrate parabolic transformation.

It is readily apparent that for picture plane distortions based upon surfaces of revolution, only the cross section of the surface need be dealt with.

A distortion based upon a sphere is illustrated in Fig. 4, wherein the left circle is the plan view of the sphere and the right circle is the cross section. This figure illustrates spherical transformation and therein, with regard to the left circle, rho = square root X/2/+Y/2/, theta = Tan/-1/(Y over X). With regard to the right circle, Tan alpha = 1 over rho, beta = 90 degrees - alpha, Sin (90 degrees - alpha) = Sin (90 degrees - Tan/-1/ (1 over rho )) for the transformation (rho, theta) -> (rho',theta). The parabolic distortion illustrated in Fig. 3 is preferred to the spherical transformation illustrated in Fig. 4 because of calculation efficiency.

Fig. 5 illustrates the use of a polynomial function to minimize or eliminate distortion near the origin. By the technique shown in Fig. 5, the close-up region will be a realisti...