3 ms·

Bridges, String art, and Bezier Curves

30 points by akg 14y ago

mistercow 14y ago
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.
- tylerneylon 14y ago
  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 14y ago
    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)
akg 14y ago
The book "Architectural Geometry" http://www.amazon.com/gp/product/193449304X/103-2668819-9693439 http://www.amazon.com/gp/product/193449304X/103-2668819-9693... is a fantastic read with plenty of nice geometric insights.
leeoniya 14y ago
reminds me of the exhibits at the Saint Louis Arch http://www.dzre.com/alex/slarch/ http://www.dzre.com/alex/slarch/ and http://www.sciencefriday.com/video/04/23/2009/how-the-arch-got-its-shape.html http://www.sciencefriday.com/video/04/23/2009/how-the-arch-g...
tylerneylon 14y ago
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 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 http://fridaypuzzl.es/?p=180

30 points by akg 14y ago

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.

tylerneylon 14y ago
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 14y ago
  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)

akg 14y ago

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

leeoniya 14y ago

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

tylerneylon 14y ago

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