This may sound paradoxical. Intuitively, the longer path the particle needs to glide, the longer time it will have to take. However, consider that acceleration rate is high when the slope is sharp, therefore it is possible for a curve whose curvature balances out so that no matter where the particle starts, it always takes the same time to reach the end. The unique solution is the cycloid . The latter meaning will be used here.
Anallagmatic curve is any curve that has inversion that's itself.
Circle and Cassinian oval are examples. P such that P is the intersection of tangents of C that meets in angle alpha. For example, the orthoptic of a parabola is its directrix, and the orthoptic of an ellipse , hyperbola , or deltoid is a circle. The phrases in parenthesis are reminders. Not all curves in this project appears below.