Figure 6

A coordinate frame is calculated using points A and B and the average of their normals as shown in (a). The change of coordinates from (X, Y, Z)-space to (U, V, N)-space is given by the rotation matrix (U, V, N) (the 3 × 3 matrix represented in the left upper corner in the 4 × 4 matrix in (b)) and the translation vector to the origin (assuming the point A is taken as the origin, (-AU, -AV, -AN) shown at the bottom of the 4 × 4 matrix is the translation vector). Coordinates of point C can be expressed in the new coordinate system by using a matrix vector multiplication.