Example: Circle
In polar coordinates, the family of circles centered about the origin is the level curves of
where is the radius of the circle. Then the orthogonal trajectories are the level curves of defined by:
The lack of complete boundary data prevents determining . However, we want our orthogonal trajectories to span every point on every circle, which means that must have a range which at least include one period of rotation. Thus, the level curves of, with freedom to choose any, are all of the curves that intersect circles, which are (all of the) straight lines passing through the origin. Note that the dot product takes nearly the familiar form since polar coordinates are orthogonal.
The absence of boundary data is a good thing, as it makes solving the PDE simple as one doesn't need to contort the solution to any boundary. In general, though, it must be ensured that all of the trajectories are found.
Read more about this topic: Orthogonal Trajectory
Famous quotes containing the word circle:
“A man should not go where he cannot carry his whole sphere or society with him,Mnot bodily, the whole circle of his friends, but atmospherically. He should preserve in a new company the same attitude of mind and reality of relation, which his daily associates draw him to, else he is shorn of his best beams, and will be an orphan in the merriest club.”
—Ralph Waldo Emerson (1803–1882)