FIGURE 10 The geometry underlying the embedding of a one-dimensional perspective transformation into a two-by-two homogeneous coordinate matrix. Real points are defined to lie in along the line y = 1. The upper left shows a one-dimensional object with nine equally spaced subdivisions. Shearing along the x-dimension does not move the object off of the line y = 1. The coordinates of all of the intervals of the object are simply translated as shown in the upper right. Shearing along the y-dimension moves points off of the line y = 1. A point off this line is remapped back onto y = 1 by projecting a line from the origin through the point. The intersection of the projection line with the line y = 1 is the remapped coordinate. This is equivalent to rescaling the transformed vector to make its final coordinate equal to unity. Projection lines are shown as dashed lines, and the resulting coordinates along y = 1 are shown as small circles. Note that the distances between projected points become progressively smaller from right to left. The gray line parallel to the skewed object intersects the line y = 1 at the far left. This point is the vanishing point of the transformation. A point infinitely far to the left before transformation will map to this location. In this case, the skew is not sufficiently severe to move any part of the object below the origin. Points below the origin will project to the left of the vanishing point with a reversed order, and a point infinitely far to the right before transformation will map to the vanishing point. Consequently, at the vanishing point, there is a singularity where positive and negative infinities meet and spatial directions become inverted. Two-dimensional perspective transformations can be envisioned by extending the second real dimension out of the page. Three-dimensional perspective transformations require a four-dimensional space.
As x goes to positive infinity, x' will go to 5// This corresponds to the vanishing point in the transformed image. The same vanishing point applies as x goes to negative infinity, Note that a point with the original coordinate —1// causes the denominator to become zero. This corresponds to the intersection of the skewed one-dimensional image with the y-axis.
Points to one side of the value — 1// are mapped to positive infinity, while those on the other side are mapped to negative infinity. The geometry underlying these relationships can be seen in Fig. 10. From a practical standpoint, the singularities involving division by zero or the projection of the extremes in either direction to the same point are generally irrelevant since they pertain to the physical impossible situation where the direction of light or radiation is reversed.
In two dimensions, there are two parameters that control perspective. In the matrix formulation here, they are / and g
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