line of symmetry for shapes

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2 total messages Started by Jim Streicher Mon, 05 Oct 1992 00:16

#756

Author: Jim Streicher
Date: Mon, 05 Oct 1992 00:16

57 lines
2369 bytes

I have a question concerning the determination of
planar symmetry from two irregular "concentric" shapes.

An example of the two shapes might be ...


                         -------
                       /         \
                      /    ____   \_
                      |   /    \     \
                      /  |    _/     |
                     /   |   |       |
                    /    |___/   x    \ 
                    |           ______/
                     \____     /
                          \___/


In general, the shapes will not have a line of symmetry
such that one half of the shape is the mirror image of the other.

I need to find the "best" line of symmetry and quantify the error
based on some numerical algorithm.

The symmetry must be based on the mass between the concentric
shapes, (ie the area indicated with an 'x').


By eyeballing these shapes, I guess the line of sym should be drawn
like this:



                         -------
                       /         \
                      /    ____   \_
                      |   /    \     \
          ------------/--|----_/-----|------------ 
                     /   |   |       |
                    /    |___/   x    \ 
                    |           ______/
                     \____     /
                          \___/


Obviously, this may be way off, and I don't know how far off it is.


Does anyone know of a algorithm to determine the best line of symmetry?
Any thoughts on this problem?

---
Jim Streicher                                         (215) 898-8099
jim@heart.seas.upenn.edu (best)                       Dept. of Bioengineering
jim@thrombus.seas.upenn.edu (forwarded to heart)      U. of Penn.
streiche@eniac.seas.upenn.edu (forwarded to heart)    Philadelphia, PA. 19104

Re: line of symmetry for shapes

#779

Author: Mattias Bryntesson
Date: Mon, 05 Oct 1992 19:36

37 lines
1779 bytes

In article <91604@netnews.upenn.edu> jim@heart.seas.upenn.edu (Jim Streicher) writes:
>
>I have a question concerning the determination of
>planar symmetry from two irregular "concentric" shapes.
>
...

Wouldn't some kind of Monte Carlo approach be nice?

(i)   Pick a number of random points in the plane with some
      distribution (for example uniform). Store the points
      that lies between the two shapes. (The cytoplasm? ;-)

(ii)  Pick a random line orientation and find its location
      so that it has half the points on each of its sides.

If the distribution is uniform the line should divide the area
into two equally big areas. If this is not enough you could also
use some kind of momentum criterion to find a good line in (ii).
For example you could specify the variance to be equal:

              sum  x(i)^2      =          sum  x(i)^2
       i: x(i) in area 1           i: x(i) in area 2

where x(i) is the distance from the line to point i, and area 1
and 2 are the areas on the two sides of the line.

(The exact criterions depends on what you're looking for, but
Monte Carlo should be a fast method, probably faster than doing
some other kind of area estimation, especially if you want to do
some weighted integration in your criterion.)

Good Luck!

Mattias

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