On PH quintic spirals joining two circles with one circle inside the other

The paper derives a spiral condition, for a single Pythagorean hodograph quintic transition curve of G² contact, between two circles with one circle inside the other. A spiral is free of local curvature extrema, making spiral designs an interesting mathematical problem with importance for both physical and aesthetic applications. In the construction of highways or railway routes in particular, it is often desirable to have a transition curve from circle to circle. Here, we treat an open problem on planar quintic spiral segments, called transition curve elements, examine techniques for curve design using the new results, and derive lower and upper bounds for the distance between the two circles. The proposed method is applicable for non-tangent and non-concentric circles.