⌈1 0⌉⌈x⌉=⌈ x⌉
⌊0 -1⌋⌊y⌋=⌊-y⌋
The above 2x2 matrix reflects a point over the x-axis. (x,y)→(x,-y).
Derive a matrix that will reflect a point over the line y=kx.
(In reply to
re: researched solution by Jer)
The following eliminates the trig functions, but still starts out with the Wikipedia formula involving theta, rather than from scratch:
atan(k) has a sine = k/sqrt(1+k^2) and cosine = 1/sqrt(1+k^k).
cos(2*atan(k)) = 1/(1+k^2) - k^2/(1+k^2) = (1-k^2)/(1+k^2)
sin(2*atan(k)) = 2*k/(1+k^2)
[cos(2*atan(k)) sin(2*atan(k))]
[sin(2*atan(k)) -cos(2*atan(k))]
then becomes
[(1-k^2)/(1+k^2) 2*k/(1+k^2)]
[2*k/(1+k^2) (k^2-1)/(1+k^2)]
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Posted by Charlie
on 2015-04-02 08:50:32 |
completing the derivation
