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Reflection Matrix (Posted on 2015-04-01) Difficulty: 3 of 5
⌈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.

No Solution Yet Submitted by Jer    
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Solution completing the derivation Comment 3 of 3 |
(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)]
 

  Posted by Charlie on 2015-04-02 08:50:32
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