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Geometric Photo Manipulation - Hidden Points: The z-buffer

Tom Farmer

The particular examples of Figure 3 avoid the problem of having parts of the object that should be hidden from view (unless the line-of-sight is chosen carelessly). If we had painted all six faces of the cube, though, there would often be two distinct points and of the object that are projected to the same point in the viewing plane. But only the point that is nearest to the viewer's eye should be visible. Fortunately, this problem has a well-known solution in computer graphics called a z-buffer, an array having the same row and column dimensions as the output array. For each pixel of the output image, the buffer stores a value representing the distance from a point of the three-dimensional object to the viewer's eye (or to an appropriate plane if the eye is at infinity). The z-buffer can be initialized with values that are larger than any distance that arises from the object. Then the color of an output pixel is changed only if the distance from the given point to the eye is smaller than the value held in the z-buffer. If this is the case, the value in the buffer is updated to be the distance just computed. This device is further described in the next section, and code to implement it is included in the programs perspective.m and orthographic.m.

Tom Farmer, "Geometric Photo Manipulation - Hidden Points: The z-buffer," Loci (October 2005)


Journal of Online Mathematics and its Applications