Find matrix representation of transformation Given two lines l 1 </msub> : y =

Shea Stuart 2022-07-05 Answered
Find matrix representation of transformation
Given two lines l 1 : y = x 3 and l 2 : x = 1 find matrix representation of transformation f(in standard base) which switch lines each others and find all invariant lines of f.
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Answers (2)

eurgylchnj
Answered 2022-07-06 Author has 14 answers
Since the two straight lines intersect at the point P = ( 1 , 2 ) the transformation must have a fixed point in P. We can find such transformations in three steps.
1) translate the origin in P with the translation T P 1 ( x , y ) ( x 1 , y + 2 )
2) perform a simmetry S with axis the strignt line passing thorough the new origin ad such that bisect the angle between the two lines.
3) return to the old origin with the translation T P ( x , y ) ( x + 1 , y 2 ).
So the searched matrix has the form: M = T P S T P 1 .
This is not a linear transformation but an affine one, and, if you want, can be represented by a 3 × 3 matrix.
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icedagecs
Answered 2022-07-07 Author has 3 answers
I have an addition:
The translation matrices in omogeneous coordinate are:
T P = [ 1 0 1 0 0 2 0 0 1 ] T P 1 = [ 1 0 1 0 0 2 0 0 1 ]
and the reflection matrix can be found noting that the angle between the bisetrix and the x-axis is θ = 3 π 8 , Then the matrix is:
S = [ cos 2 θ sin 2 θ 0 sin 2 θ cos 2 θ 0 0 0 1 ]
Note that the invariant lines are the bisector and his orthogonal in P.
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