Nascent Papers

Trevor

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Çağatay

Coloring 3d line fields using Boy’s real projective plane immersion

It’s often useful to visualize a line field, a function that sends each point P of the plane or of space to a line through P; such fields arise in the study of tensor fields, where the principal eigendirection at each point determines a line (but not a vector, since if v is an eigenvector, so is −v). To visualize such a field, we often assign a color to each line; thus we consider the coloring of line fields as a mapping from the real projective plane (RP2) to color space. Ideally, such a coloring scheme should be smooth and one-to-one, so that the color uniquely identifies the line; unfortunately, there is not such mapping. We introduce Boy’s surface, an immersion of the projective plane in 3D, as a model for coloring line fields, and show results from its application in visualizing orientation in diffusion tensor fields. This coloring method is smooth and oneto- one except on a set of measure zero (the double curve of Boy’s surface).

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