Circle to Circle Transition with a Single Cubic Spiral

Z. Habib and M. Sakai (Japan)


B´ezier cubic, curvature, spiral, Mathematica.


This paper derives a spiral condition for a single B´ezier cu bic transition curve of G2 contact, between two circles with one circle inside the other. In high way, railway route, or satellite path design it is often desirable to have a transition curve from circle to circle. A spiral is free of local cur vature extrema, making spiral design an interesting math ematical problem with importance for both physical and aesthetic applications. Since B´ezier cubics are commonly used in modern design systems because they are of low de gree, are easily evaluated, and allow inflection points, it would be convenient to employ cubic spirals so that spirals may be used in a variety of CAD systems. We simplify and complete the analysis on earlier results on planar cu bic B´ezier spiral segments which are proposed as transition curve elements, examine techniques for curve design using the new results, and derive lower and upper bounds for the distance between two circles.

Important Links:

Go Back