Constraining Plane Configurations in CAD: Circles, Lines, and Angles in the Plane

This paper investigates the local uniqueness of designs of m-circles (lines and circles) in the plane up to inversion under a set of angles of intersection as constraints. This local behavior is studied through the Jacobian of the angle measurements in a form analogous to the rigidity matrix for a framework of points with distance constraints. After showing directly that the complete set of angle constraints on v distinct m-circles gives a matrix of rank 3v − 6, we show that the Jacobian is column equivalent by a geometric correspondence to the rigidity matrix for a bar-andjoint framework in Euclidean 3-space. As a corollary, the complexity of the independence of angle constraints on generic plane circles is the complexity of the old unsolved combinatorial problem of generic rigidity in 3-space. This theory is not known to have a polynomial time algorithm for generic independence that offers a warning about the complexity of general systems of geometric constraints even in the plane. Our correspondence extends to all dimensions. Angle constraints on spheres in 3-space then match the even more complex first-order theory of frameworks in 4-space. This theory is not predicted to have a polynomial time algorithm for generic points.