Revisiting variable radius circles in constructive geometric constraint solving

Abstract

Variable-radius circles are common constructs in planar constraint solving and are usually not handled fully by algebraic constraint solvers. We give a complete treatment of variable-radius circles when such a circle must be determined simultaneously with placing two groups of geometric entities. The problem arises for instance in solvers using triangle decomposition to reduce the complexity of the constraint problem.This work offers a set of basic constructive methods that permits to determine variable radius circles simultaneously with placing two rigid geometric objects when geometric constraints are defined on both the circumference and center point of the constraint circle. The problem has been classified by the geometric entities in two groups, one is fixed and the other has translational and rotational movement, so that the variable radius circles satisfy the constraints on the geometric entities in these two groups. The number of solutions for each problem is also given.

Hardware Assistance for Constrained Circle Constructions II: Cluster Merging Problems

“…Likewise, the -map of a line L in the xy-plane, denoted , is a plane through L that is perpendicular to the xy-plane; e.g. [2]. The -map will be used to express constraints on the center of a variable-radius circle geometrically.…”

“…We also consider problems in which constraints are included on the center of the unknown circle. Analyzed algebraically in [2], we consider here the cases ', and ', .…”

“…For the oriented entity in the xy-plane, its cyclographic map is also denoted by in the literature; e.g., [2]. Consider the point , , on .…”

“…Note that elements in and can include circles of fixed radius, since they can be replaced with points after adjusting the constraints upon them. For this type of construction, the possible cases have been mapped out in detail in [2], [7], [8] and the algebraic systems formulated and solved. The most complicated case, involving only circles in the clusters and gives rise to a system with the algebraic degree 64.…”

“…The subsequent construction phase solves small systems of algebraic equations, and for 2D solvers with a focused scope these systems are easy to solve and numerically stable. However, the construction phase becomes demanding when one considers including variable-radius circles and extends the geometric vocabulary to include entities such as PH curves or conics; e.g., [2], [3], [6], [7], [8], [11]. For such constructions, the dominant algebraic approach delivers systems of equations that may require substantial computations as well as generating many solutions, not all of them of geometric significance.…”