is based on. There are generically two types of cusps in involutes. The first type is at the point where the involute touches the curve itself. This is a cusp of order 3/2. The second type is at the point where the curve has an inflection point. This is a cusp of order 5/2. This can be visually seen by constructing a map defined by where is the arclength parametrization of the curve, and is the slope-angle of the curve at the point . This maps the 2D plane into a surface in 3D space. For example, this maps the circle into the hyperboloid of one sheet. By this map, the involutes are obtained in a three-step process: map to , then to the surface in , then project it down to by removing the z-axis: where is any real constant. Since the mapping has nonzero derivative at all , cusps of the involute can only occur where the derivative of is vertical (parallel to the z-axis), which can only occur where the surface in has a vertical tangent plane. Generically, the surface has vertical tangent planes at only two cases: where the surface touches the curve, and where the curve has an inflection point. cusp of order 3/2 For the first type, one can start by the involute of a circle, with equationthen set , and expand for small , to obtainthus giving the order 3/2 curve , a semicubical parabola. cusp of order 5/2 For the second type, consider the curve . The arc from to is of length , and the tangent at has angle . Thus, the involute starting from at distance has parametric formulaExpand it up to order , we obtainwhich is a cusp of order 5/2. Explicitly, one may solve for the polynomial expansion satisfied by :or which clearly shows the cusp shape. Examples Involutes of a circle For a circle with parametric representation , one has . Hence , and the path length is . Evaluating the above given equation of the involute, one gets for the parametric equation of the involute of the circle. The term is optional; it serves to set the start location of the curve on the circle. The figure shows involutes for (green), (red), (purple) and (light blue). The involutes look like Archimedean spirals, but they are actually not. The arc length for and of the involute is