Hyperbolic Cords and Wheels

The cycloidal family of curves in R2, also known as the trochoid family, are equivalently generated by two classic methods: bungee cords and rolling wheels. What about their counterpart families in the hyperbolic unit disk? We review the two methods in Euclidean space, outline pertinent hyperbolic geometry tools, using both the Klein and Poincaré models, and show that the two methods give distinct, yet similar, results in hyperbolic space.