Character Motion Synthesis by topology coördinates

Character Motion Synthesis by topology coordinates

Edmond S. L. Ho and Tako Komura. Computer graphics forum (eurographics 2009), vol 28, issue 2, mar 2009.

This paper describes something called topology space. It describes how to convert general coördinates to topology coördinates and vica versa.

Topology coördinates consist of writhe, center and density. When considering two chain links, first we setup the writhe-matrix. The writhe matrix has the chain segemtns of the first chain length as a columns and the second chain segments as rows. The absolute values in the matrix determine how much each pair contributes to the total writhe. (see figure 6 in the paper)

The center is defined as roughly the center of the twist which can be extracted from the writhe matrix and the total writhe w.

When we look at the wirthe matrix, the highest values are usually conentrated around a single axis. horizontal if chain 1 twists around chain 2, vertically if chain 2 twists around chain 1 and diagonally if both chains twist around each other. Density is defined as the deviation from this principal axis in the writhe matrix when we scale down the writhe matrix to a square matrix.

Generating the general coördinates from the toplogy coordinates is a somewhat easier process so I won't describe it here.

If interpolating the general coördinates of joints would produce colliding joints, one could interpolate the topology coördinates instead.