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-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 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.

[1]  E. M. Andreev.  ON CONVEX POLYHEDRA IN LOBAČEVSKIĬ SPACES , 1970 .

[2]  Robert Connelly,et al.  Generic Global Rigidity , 2005, Discret. Comput. Geom..

[3]  T. Tay.  On the Generic Rigidity of Bar-Frameworks , 1999 .

[4]  D. Pedoe,et al.  Geometry, a comprehensive course , 1988 .

[5]  D. Gans.  An introduction to non-Euclidean geometry , 1973 .

[6]  J. C. Owen,et al.  Algebraic solution for geometry from dimensional constraints , 1991, SMA '91.

[7]  Christoph M. Hoffmann,et al.  Variable-radius circles in cluster merging: II. Rotational clusters , 2002, Comput. Aided Des..

[8]  Robert Connelly,et al.  Second-Order Rigidity and Prestress Stability for Tensegrity Frameworks , 1996, SIAM J. Discret. Math..

[9]  W. Whiteley.  INFINITESIMAL MOTIONS OF A BIPARTITE FRAMEWORK , 1984 .

[10]  A. V. Pogorelov.  Extrinsic geometry of convex surfaces , 1973 .

[11]  Bruce Hendrickson,et al.  Conditions for Unique Graph Realizations , 1992, SIAM J. Comput..

[12]  Walter Whiteley,et al.  Constraining Plane Configurations in Computer-Aided Design: Combinatorics of Directions and Lengths , 1999, SIAM J. Discret. Math..

[13]  M HoffmanChristoph,et al.  Decomposition Plans for Geometric Constraint Systems, Part I , 2001 .

[14]  G. Laman.  On graphs and rigidity of plane skeletal structures , 1970 .

[15]  W. Whiteley.  Infinitesimally rigid polyhedra. I. Statics of frameworks , 1984 .

[16]  Christoph M. Hoffmann,et al.  Decomposition Plans for Geometric Constraint Systems, Part I: Performance Measures for CAD , 2001, J. Symb. Comput..

[17]  Glenn A. Kramer,et al.  Solving Geometric Constraint Systems , 1990, AAAI.

[18]  Lebrecht Henneberg,et al.  Die graphische Statik der Starren Systeme , 1911 .

[19]  Kokichi Sugihara,et al.  Detection of structural inconsistency in systems of equations with degrees of freedom and its applications , 1985, Discret. Appl. Math..

[20]  Walter Whiteley,et al.  Cones, infinity and one-story buildings , 1983 .

[21]  W. Whiteley,et al.  Statics of Frameworks and Motions of Panel Structures: A projective Geometric Introduction , 1982 .

[22]  C. Mann,et al.  Variable-Radius Circles in Cluster Merging Part II : Rotational Clusters , .

[23]  Glenn A. Kramer.  Solving geometric constraint systems a case study in kinematics , 1992, Comput. Aided Des..

[24]  Christoph M. Hoffmann,et al.  Decomposition Plans for Geometric Constraint Problems, Part II: New Algorithms , 2001, J. Symb. Comput..

[25]  H. Coxeter,et al.  The Geometric vein : the Coxeter Festschrift , 1981 .