The gray triangle ABC has circumcircle and incircle as shown. As the point t goes around the incircle, the tan-colored triangle "rotates" through the set of all triangles that have this same circumcircle and incircle, thus solving a famous problem called the "Poncelet porism". During this "rotation", points such as the orthocenter, labeled H, produce orbits. Shown here are the orbits of H and of the midpoint of a side of the tan-colored triangle.
This configuration (and others) show that H (and other special points) are really functions, not merely points. For example, the domain of H is the set of all triangles.
For more, see GEOMETRY IN ACTION, Chapter 7.