If there are two links, and , then the C-space can be nicely visualized as a square with opposite faces identified. Each coordinate, and , ranges from 0 to , for which . Suppose that each link has length . This yields . A point is transformed as

To obtain polynomials, the technique from Section 4.2.2 is applied to replace the trigonometric functions using and , subject to the constraint . This results in

for which the constraints for must be satisfied. This preserves the torus topology of , but now the C-space is embedded in . The coordinates of each point are ; however, there are only two degrees of freedom because each pair must lie on a unit circle.

Multiplying the matrices in (4.59) yields the polynomials, ,

and

for the and coordinates, respectively. Note that the polynomial variables are configuration parameters; and are not polynomial variables. For a given point , all coefficients are determined.

Steven M LaValle 2012-04-20