Classification and Generation of 3D Discrete Curves

We present a quantitative approach to the representation, classification, and generation of families of 3D (three-dimensional) particular discrete closed curves. Any 3D continuous curve can be digitalized and represented as a 3D discrete curve. Thus, a 3D discrete curve is composed of constant orthogonal straight-line segments. In order to represent 3D discrete curves, we use the orthogonal direction change chain code. The chain elements represent the orthogonal direction changes of the contiguous straight-line segments of the discrete curve. This chain code only considers relative direction changes, which allows us to have a curve descriptor invariant under translation and rotation. Also, this curve descriptor may be starting point normalized, invariant under coding direction, and mirroring curves may be obtained with ease. Thus, using the above-mentioned chain code it is possible to have a unique 3D-curve descriptor. By evaluating all possible combinations of chain elements of curves and considering some restrictions, we obtain interesting families of particular curves at different orders (different number of chain elements), such as: open, closed, planar, angular, binary, mirror-symmetric, and random curves. Finally, we present some examples of curve classifications, such as: multiple occurrences of any curve within another curve. Mathematics Subject Classification: 65D17