$$\epsilon \kappa $$-Curves: controlled local curvature extrema

κ-Curves: Interpolation at Local Maximum Curvature

We present a method for constructing almost-everywhere curvature-continuous, piecewise-quadratic curves that interpolate a list of control points and have local maxima of curvature only at the control points. Our premise is that salient features of the curve should occur only at control points to avoid the creation of features unintended by the artist. While many artists prefer to use interpola...

Magnetic Curves: Curvature-Controlled Aesthetic Curves Using Magnetic Fields

We describe “magnetic curves”, a particle-tracing method that creates curves with constantly changing curvature. It is well known that charged particles in a constant magnetic field trace out circular or helical trajectories. Motivated by John Ruskin’s advice to use variation in curvature to achieve aesthetic curves, we propose to continuously change the charge on a simulated particle so that i...

Local extrema in random trees

The extension of permutation statistics to labelled trees is the subject of a number of articles. Generating functions for the number of labelled trees of several types according to the number of ascents and descents are given in [4]. A functional equation satisfied by the generating function for the number of labelled trees according to the number of descents and leaves is given in [5]. Centra...

Curvature-controlled curve editing using piecewise clothoid curves

Two-dimensional curves are conventionally designed using splines or Bézier curves. Although formally they are C2 or higher, the variation of the curvature of (piecewise) polynomial curves is difficult to control; in some cases it is practically impossible to obtain the desired curvature. As an alternative we propose piecewise clothoid curves (PCCs). We show that from the design point of view th...

Extrema of Curvature Functionals on the Space of Metrics