The tangency problem of variable radius circle to lines, circles and ellipses

Variable-radius circles are common constructs in planar constraint solving and are usually not handled completely by algebraic constraint solvers, especially when the ellipse is the geometric entities. Consider the cluster merge problems, there are two constraints between the variable radius circle and each of the rigid geometric objects. It can be handled when the rigid geometric objects concerning point, ray (oriented line) and cycle (oriented circle). The cluster merge problem become more complex when the ellipse added into the geometric elements. This paper proposed two methods to find the variable radius circle tangent to three geometric objects, including line, circle and ellipse. These methods construct a system of equation from the tangency properties between geometric entities and variable radius circle, consider the geometric entities is oriented (the 2nd method) or not oriented (the 1st method). For all cases, an upper bound of the number of variable radius circle is calculated.