Interpolation with PH Quintic Spirals

This paper finds a spiral segment matching G² Hermite conditions for a single Pythagorean hodograph quintic polynomial. We have significantly simplified the existing methods with computationally stable spiral conditions and discovers more spiral regions for the existence of unsolved problems in the past. A spiral is free of local curvature extrema, making spiral design an interesting mathematical problem with importance for both physical and aesthetic applications. In the construction of highways or railway routes and in the path planning of non-holonomic mobile robots, it is often desirable to have a spiral segment with given positions, tangents, and curvatures at the end points.