Revisiting variable radius circles in constructive geometric constraint solving

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.