# Geometric properties of indirect Pythagorean hodograph curves

An offset curve is a curve of fixed distance away from a given planar curve along its normal direction. They arise in a variety of applications, which include CNC machining, railway design and shape blending. However, for polynomial curves that are widely used in computer-aided design and manufacturing, their offset curves may not be polynomial or rational. Only a special set of polynomials curves has rational offsets, and they are called "Pythagorean hodograph curves". For these curves, there exists a polynomial such that this polynomial, together with the x- and the y-component of the derivative (or hodograph) of the given curve form a "Pythagorean triple", hence the name. An extension of this concept are those curves that do not have a polynomial Pythagorean hodograph, but can have a rational Pythagorean hodograph, after a properly chosen reparameterization, and hence they are called indirect Pythagorean hodograph curves. While the geometric properties of Pythagorean hodograph curves are well understood by now, little is known so far about indirect Pythagorean hodograph curves, and the aim of this project is to change that. We are therefore looking for a highly motivated student with a thorough background in Mathematics (esp. Algebra, Analysis, Geometry, and Numerics) and basic programming skills.