How to shift the design

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

s_i=0.5+a_i*x_trans+b_i*y_trans.

Note that other changes in the s_i destroy the rotational symmetry of the structure.

 

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