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Geometric Photo Manipulation - Examples

Tom Farmer

Generating pictures such as the ones in this article may require a little mathematics as well as a considerable amount of experimentation. For example, Figure 1 involves, among other things, expressing an arc of a helix in parametric form with arc length as the parameter. Parameterizations using some other parameter would be apt to deform the pictures or leave gaps. Figure 4 below requires that same type of parameterization for the circles that generate a torus; it is also an application of the tangent, normal, and binormal vectors of a parametric curve.

Figure 4

A picture like the one in Figure 5 can be a fun way to clarify a three-dimensional object. In this case we are able to visualize the shape of the solid obtained as the intersection of two cylinders arising from two input images.

Figure 5

There is no end to the possible examples, and many are apt to lead to challenges that can pique students' interest in mathematical topics.

Tom Farmer, "Geometric Photo Manipulation - Examples," Convergence (October 2005)