

Bridges, String art, and Bezier Curves - akg
http://plus.maths.org/content/bridges-string-art-and-bezier-curves

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mistercow
Incidentally, the "rotated parabola" representation of quadratic bezier curves
can be extremely useful when writing highly optimized code, as it lends itself
to closed form solutions in some situations where the parametric
representation does not.

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tylerneylon
Is there a place where I can read more about this? Like how to write code
along these lines, and maybe how to understand why it works?

~~~
mistercow
My own experience with it is largely gleaned from personally exploring and
optimizing algorithms. At some point I'll write a blog post about it.

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)

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tylerneylon
There's a generalization of this problem where we consider the set of all
lines between (a, 0) and (0, b) with

||(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:

<http://fridaypuzzl.es/?p=187>

and if you love puzzles, try to forget what you just read, and here is the
puzzle:

<http://fridaypuzzl.es/?p=180>

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akg
The book "Architectural Geometry"
[http://www.amazon.com/gp/product/193449304X/103-2668819-9693...](http://www.amazon.com/gp/product/193449304X/103-2668819-9693439)
is a fantastic read with plenty of nice geometric insights.

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leeoniya
reminds me of the exhibits at the Saint Louis Arch
<http://www.dzre.com/alex/slarch/> and
[http://www.sciencefriday.com/video/04/23/2009/how-the-
arch-g...](http://www.sciencefriday.com/video/04/23/2009/how-the-arch-got-its-
shape.html)

