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#ActualParadigm Shifter

Posted 17 October 2013 - 02:02 PM

You can just write down the matrix if you know what it does to (1, 0, 0), (0, 1, 0) and (0, 0, 1), it is just the images of those vectors after the transform as the rows of the matrix.

 

So to map (1, 0, 0) to (1, 0, 0); (0, 1, 0) to (0, 1, 0) and (0, 0, 1) to (0.5, 0, -1) you would use the matrix

 

(1, 0, 0)

(0, 1, 0)

(0.5, 0, -1)

 

and to map back you use the inverse of that matrix.

 

The matrix has determinant -1 (see http://www.wolframalpha.com/input/?i=det{{1%2C+0%2C+0}%2C+{0%2C+1%2C+0}%2C+{0.5%2C+0%2C+-1}} ), so it corresponds to a reflection and the orientation is flipped. You can't represent that transform with a unit quaternion since unit quaternions correspond to rotations and they all have determinant +1.


#2Paradigm Shifter

Posted 15 October 2013 - 01:59 PM

You can just write down the matrix if you know what it does to (1, 0, 0), (0, 1, 0) and (0, 0, 1), it is just the images of those vectors after the transform as the rows of the matrix.

 

So to map (0, 0, 1) to (0, 0, 1); (0, 1, 0) to (0, 1, 0) and (0, 0, 1) to (0.5, 0, -1) you would use the matrix

 

(1, 0, 0)

(0, 1, 0)

(0.5, 0, -1)

 

and to map back you use the inverse of that matrix.

 

The matrix has determinant -1 (see http://www.wolframalpha.com/input/?i=det{{1%2C+0%2C+0}%2C+{0%2C+1%2C+0}%2C+{0.5%2C+0%2C+-1}} ), so it corresponds to a reflection and the orientation is flipped. You can't represent that transform with a unit quaternion since unit quaternions correspond to rotations and they all have determinant +1.


#1Paradigm Shifter

Posted 15 October 2013 - 01:58 PM

You can just write down the matrix if you know what it does to (1, 0, 0), (0, 1, 0) and (0, 0, 1), it is just the images of those vectors after the transform as the rows of the matrix.

 

So to map (0, 0, 1) to (0, 0, 1); (0, 1, 0) to (0, 1, 0) and (0, 0, 1) to (0.5, 0, -1) you would use the matrix

 

(1, 0, 0)

(0, 1, 0)

(0.5, 0, -1)

 

and to map back you use the inverse of that matrix.

 

The matrix has determinant -1 (see http://www.wolframalpha.com/input/?i=det{{1%2C+0%2C+0}%2C+{0%2C+1%2C+0}%2C+{0.5%2C+0%2C+-1}} ), so it corresponds to a reflection and the orientation is flipped. You can't represent that transform with a unit quaternion since unit quaternions correspond to rotations and they all have determinant +1.


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