Bridges, String art, and Bezier Curves
plus.maths.org
plus.maths.org
In short, though, there are two main properties that are difficult to calculate in parametric form, but relatively easy in rotated parabola segment form. The first is arc length, which is needed for dividing a curve into uniform line segments, and the second is point-to-curve distance.
One other thing that is sometimes needed is to be able to ensure that a curve does not change direction, and split it into two segments if it does. This is trivial with the parabolic representation, as a change in direction can only happen at the parabola's vertex.
(Note: it's actually been quite a while since I worked with all of this, so I may have misremembered some details here)
||(a, b)||_p = 1
as in the Lp norm; and ask, what curve is produced (as their boundary) ?
And the answer has the nice form: The set of points (x,y) which satisfy
||(x, y)||_q = 1
where q = p / (p+1) [which can be written as 1/q - 1/p = 1.]
This is written up as the solution to a puzzle here:
and if you love puzzles, try to forget what you just read, and here is the puzzle: