Iterating skew evolutes and skew involutes: a linear analog of bicycle kinematics

The evolute of a plane curve is the envelope of its normals. Replacing the normals by the lines that make a fixed angle with the curve yields its skew evolute. We study the geometry and dynamics of the skew evolute maps and of their inverses, the skew involute maps. Among the motivations for this study are relations of this subject with tire track geometry and with mathematical billiards. We prove a version of the 4-vertex theorem where the role of circles is played by logarithmic spirals.