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

2004; Society for Industrial and Applied Mathematics; Volume: 18; Issue: 2 Linguagem: Inglês

10.1137/s0895480100374138

1095-7146

Franco Saliola, Walter Whiteley,

Innovations in Concrete and Construction Materials

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-and-joint 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 genericindependence 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.