In the last post “Projection method – geometry and maths” I forgot to show what the parameters s_i are doing.
To get the center of perfect 2n-fold rotational symmetry at the origin you have to put
s_i=0.5 for all i.
The distances from the lines to the origin are then 0.5, 1.5, 2.5 and so on. But often one wants to see a different part of the design. Then you simply have to shift it around. You add to each vector (x,y) in space a constant vector (x_trans,y_trans) and use the new vector (x+x_trans,y+y_trans). Putting this into the second equation of the last post you get:
(a_i,b_i) * (x+x_trans,y+y_trans)=a_i*x+b_i*y+a_i*x_trans+b_i*y_trans=s_i + k_i .
Rearranging terms gives
a_i*x+b_i*y=s_i+a_i*x_trans+b_i*y_trans + k_i .
Thus we can move the design around by setting
Note that other changes in the s_i destroy the rotational symmetry of the structure.