On Parametrizing Polygons, 2
Dec. 9th, 2004 08:38 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowContinuing a previous discussion:
The method of parametrizing polygons I discussed earlier works very well on triangles, but not quite as well on polygons with more sides. However, it can also be applied to subsets of polygons, and "class 1 polygons", as I've dubbed them, work particularly well - almost as well as triangles.
So, what are they? If you remember how coordinate geometry works, you can do the following. Draw a regular polygon with n sides - all sides equal, and all angles equal - and note the coordinates of the vertices. Now pick four numbers a,b,c,d (but don't make all of them zero), and if the first vertex of your polygon has coordinates (x,y), find the point with coordinates (ax+by,cx+dy). That's the first vertex of a new polygon, and you can find the other vertices similarly from the other vertices of the original polygon. The new polygon is class 1.
If you don't like coordinate geometry, think of it this way. Start with your regular polygon, and do any combination of the following.
1) Translate it - move it bodily a certain distance in a certain direction.
2) Rotate it by a certain angle around a certain center. (The center may be anywhere.)
3) Reflect it in some specified line.
4) Pick a direction, and stretch or compress it by a specified factor in that direction. (So, for example, you might double or halve its dimensions horizontally, without changing it vertically.)
5) Shear it. That is, pick a line, and shift every point parallel to the line by an amount proportional to its distance from the line. (For example, you might pick a vertical line. Then, if a point lies to the right of the line, you might shift it upwards, say, two inches for every inch it is away from the line. Points to the left of the line get shifted downwards, by the same formula.)
The result of all this is a class 1 polygon.
Now, if I apply my methods to n-sided class 1 polygons, I find that I can parametrize them up to similarity with just two parameters. The first, e, is the eccentricity. e lies between 0 and 1, and in some sense measures how flat the polygon is: if e=0, you still have a regular polygon, and if e=1, the polygon is flat - all of its vertices lie in a straight line. The second, s, is the symmetry. If a polygon has eccentricity e, then the symmetry is restricted by the following condition: s^2 is at most e^m, where m=n if n is odd and m=n/2 if n isodd even. The meaning of s is harder to describe, but it turns out that if s is as far from 0 as possible - i.e., if s^2=e^m - then the polygon has an axis of symmetry. Depending on the value of n, one of three things can happen.
If n is odd, an axis of symmetry has to run from one vertex to the midpoint of the opposite side. For example, think of an isosceles triangle. Two of the sides have equal length, and they meet at the apex; the third side is the base. The line from the apex to the midpoint of the base is the axis of symmetry; if you reflect the triangle in that line, you'll get the same triangle back. If n is odd but greater than three, a symmetric class 1 polygon will still have an apex, and the sides to either side of it will have the same length and make the same angle with the axis; so will the next two sides, and so on around until you hit the opposite side, which will be perpendicular to the axis.
A polygon with an even number of sides can have an axis of symmetry in two different ways. The axis might run from a vertex to the opposite vertex (think of a rhombus - a diamond shape - for instance) - that's a diagonal axis - or it might run from the midpoint of a side to the midpoint of the opposite side (think of a rectangle), and that's a lateral axis.
If n is even, it turns out that a class 1 polygon with an axis of symmetry actually has two, and they're perpendicular. (Again, think of rhombuses or rectangles.) If n is even but not a multiple of 4, then the two axes are of different kinds - one is diagonal, and the other is lateral. (Think of hexagons.) But if n is a multiple of 4, the axes are of the same kind, either both diagonal or both lateral. It turns out that, in this case, the symmetry parameter has something to say: if s is positive, the axes are both diagonal, and if s is negative, they're both lateral. (Squares and other regular polygons with an even number of sides have several axes of both kinds, but s is zero for those.)
So that's how parameters work in this situation. Pairs (e,s) corresponding to n-sided class 1 polygons lie in the region of the coordinate plane bounded by the straight line e=1 and the curve s^2=e^m. In the case n=4, the polygons are parallelograms, and the region is the triangle with vertices (0,0), (1,1), and (1,-1). The square is at (0,0); rectangles lie along the lower edge, rhombuses along the upper, and flat parallelograms along the right side.
Next up, we'll start doing things with the parameters.
The method of parametrizing polygons I discussed earlier works very well on triangles, but not quite as well on polygons with more sides. However, it can also be applied to subsets of polygons, and "class 1 polygons", as I've dubbed them, work particularly well - almost as well as triangles.
So, what are they? If you remember how coordinate geometry works, you can do the following. Draw a regular polygon with n sides - all sides equal, and all angles equal - and note the coordinates of the vertices. Now pick four numbers a,b,c,d (but don't make all of them zero), and if the first vertex of your polygon has coordinates (x,y), find the point with coordinates (ax+by,cx+dy). That's the first vertex of a new polygon, and you can find the other vertices similarly from the other vertices of the original polygon. The new polygon is class 1.
If you don't like coordinate geometry, think of it this way. Start with your regular polygon, and do any combination of the following.
1) Translate it - move it bodily a certain distance in a certain direction.
2) Rotate it by a certain angle around a certain center. (The center may be anywhere.)
3) Reflect it in some specified line.
4) Pick a direction, and stretch or compress it by a specified factor in that direction. (So, for example, you might double or halve its dimensions horizontally, without changing it vertically.)
5) Shear it. That is, pick a line, and shift every point parallel to the line by an amount proportional to its distance from the line. (For example, you might pick a vertical line. Then, if a point lies to the right of the line, you might shift it upwards, say, two inches for every inch it is away from the line. Points to the left of the line get shifted downwards, by the same formula.)
The result of all this is a class 1 polygon.
Now, if I apply my methods to n-sided class 1 polygons, I find that I can parametrize them up to similarity with just two parameters. The first, e, is the eccentricity. e lies between 0 and 1, and in some sense measures how flat the polygon is: if e=0, you still have a regular polygon, and if e=1, the polygon is flat - all of its vertices lie in a straight line. The second, s, is the symmetry. If a polygon has eccentricity e, then the symmetry is restricted by the following condition: s^2 is at most e^m, where m=n if n is odd and m=n/2 if n is
If n is odd, an axis of symmetry has to run from one vertex to the midpoint of the opposite side. For example, think of an isosceles triangle. Two of the sides have equal length, and they meet at the apex; the third side is the base. The line from the apex to the midpoint of the base is the axis of symmetry; if you reflect the triangle in that line, you'll get the same triangle back. If n is odd but greater than three, a symmetric class 1 polygon will still have an apex, and the sides to either side of it will have the same length and make the same angle with the axis; so will the next two sides, and so on around until you hit the opposite side, which will be perpendicular to the axis.
A polygon with an even number of sides can have an axis of symmetry in two different ways. The axis might run from a vertex to the opposite vertex (think of a rhombus - a diamond shape - for instance) - that's a diagonal axis - or it might run from the midpoint of a side to the midpoint of the opposite side (think of a rectangle), and that's a lateral axis.
If n is even, it turns out that a class 1 polygon with an axis of symmetry actually has two, and they're perpendicular. (Again, think of rhombuses or rectangles.) If n is even but not a multiple of 4, then the two axes are of different kinds - one is diagonal, and the other is lateral. (Think of hexagons.) But if n is a multiple of 4, the axes are of the same kind, either both diagonal or both lateral. It turns out that, in this case, the symmetry parameter has something to say: if s is positive, the axes are both diagonal, and if s is negative, they're both lateral. (Squares and other regular polygons with an even number of sides have several axes of both kinds, but s is zero for those.)
So that's how parameters work in this situation. Pairs (e,s) corresponding to n-sided class 1 polygons lie in the region of the coordinate plane bounded by the straight line e=1 and the curve s^2=e^m. In the case n=4, the polygons are parallelograms, and the region is the triangle with vertices (0,0), (1,1), and (1,-1). The square is at (0,0); rectangles lie along the lower edge, rhombuses along the upper, and flat parallelograms along the right side.
Next up, we'll start doing things with the parameters.
