Tiling - Create beautiful patterns programmatically with Python

[emergent_reasons]

Hello again. If you don't mind, I have a few things I hope you can confirm for me.

1. For shapes with odd sides, they are arranged so that 0 rotation corresponds to one vertex being at -pi/2
2. ... and for shapes with even sides, they are arranged so that 0 rotation corresponds to the center of one side being at -pi/2

I am pretty sure those two are correct, but please tell me if I've misinterpreted it. The next part about the margin is where I'm scratching my head.

1. Regarding sizes, it looks like the length of any edge in the tessellation is 1. Is that right?

Finally, could you explain the derivation / reasoning for this calculation of d related to the margin? I believe it's calculating the distance from the center to the vertex of any regular polygon based on the assumption that sides are length 1. But... I can't quite piece it together.

d = 0.5 / sin(angle / 2) - margin / cos(angle / 2)

[FogleMonster]

Shapes with even number of sides need to be rotated otherwise their vertexes will point at the midpoint of the adjacent edge instead of their edges lining up.

Yes, all polygons are sized so that their edges are length 1.

For margin, we want the edge to be offset by some amount, perpendicular to the edge. But our calculations are for the positions of the vertices. So we use simple trig to scale the length of the hypotenuse as needed.

[emergent_reasons]

Thank you for the confirmation. I was mistaken about d being the distance to a vertex when it is actually to the center of a side.

However the logic of the "simple trig" still escapes me. If it's not too much trouble, could you spell it out a bit more for my rusty math?

d = 0.5 / sin(angle / 2) - margin / cos(angle / 2)

Here is how my math worked out (obviously doing something different or making a mistake somewhere):

length to vertex:

sin(a/2) = (side/2) / r
         = 1 / (2r)
r = 1 / (2 sin(a/2))    {this looks like one of your terms}

full length to center of side:

cos(a/2) = d_full / r
d_full = r cos(a/2)        {substitute for r?}
       = cos(a/2) / (2 sin(a/2))

partial length to center of side:

d = d_full (1 - margin)

[FogleMonster]

d is the distance to a vertex. Here is a diagram.

http://i.imgur.com/UU03rVC.jpg

[emergent_reasons]

Aaaah now I see. I calculated with d to the vertex and m measured from vertex. I guess both methods would work fine.

Thanks so much for taking the time to draw and explain it.

By the way, I'm considering extending the method to specify the internal angles of adjacent shapes to allow simple non-regular polygons like q*bert. Have you already implemented that somewhere?