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#ActualParadigm Shifter

Posted 12 June 2013 - 05:01 PM

You divide the x, y, z values by the w component to get the position in 3D space which is embedded in the 4D projective homogeneous space.

 

So (x, y, z, w) = (1, 1, 1, 1) and (x, y, z, w) = (2, 2, 2, 2) both refer to the same point in 3D space, (1, 1, 1).

 

For direction vectors, w is 0, which would cause division by zero. These correspond to points at infinity in 3D space in the direction of the x, y, z components.

 

That's basically it in a nutshell.

 

4D coordinates are used because you can represent any affine transform in 3D space with a (4x4) matrix. You can't do that in 3D space because the origin of 3D space (0, 0, 0) always maps to the origin after being transformed by a matrix (so translations are impossible in 3D space using just a matrix).

 

That doesn't happen in homogeneous 4D space since the origin of 3D space is represented by (0, 0, 0, 1) [or (0, 0, 0, w) for w != 0].

 

(0, 0, 0, 0) doesn't correspond to to any point or direction in 3D space, and is not a valid value for 4D homogeneous projective space.

 

So 4D vectors (x, y, z, w) with w != 0 corresponds to all points in 3D space and all directions ("points at infinity") in 3D space as well.

 

Hope that helps.


#3Paradigm Shifter

Posted 12 June 2013 - 05:00 PM

You divide the x, y, z values by the w component to get the position in 3D space which is embedded in the 4D projective homogeneous space.

 

So (x, y, z, w) = (1, 1, 1, 1) and (x, y, z, w) = (2, 2, 2, 2) both refer to the same point in 3D space, (1, 1, 1).

 

For direction vectors, w is 0, which would cause division by zero. These correspond to points at infinity in 3D space in the direction of the x, y, z components.

 

That's basically it in a nutshell.

 

4D coordinates are used because you can represent any affine transform in 3D space with a (4x4) matrix. You can't do that in 3D space because the origin of 3D space (0, 0, 0) always maps to the origin after being transformed by a matrix (so translations are impossible in 3D space using just a matrix).

 

That doesn't happen in homogeneous 4D space since the origin is (0, 0, 0, 1) [or (0, 0, 0, w) for w != 0].

 

(0, 0, 0, 0) doesn't correspond to to any point or direction in 3D space, and is not a valid value for 4D homogeneous projective space.

 

So 4D vectors (x, y, z, w) with w != 0 corresponds to all points in 3D space and all directions (points at infinity) in 3D space as well.

 

Hope that helps.


#2Paradigm Shifter

Posted 12 June 2013 - 05:00 PM

You divide the x, y, z values by the w component to get the position in 3D space which is embedded in the 4D projective homogeneous space.

 

So (x, y, z, w) = (1, 1, 1, 1) and (x, y, z, w) = (2, 2, 2, 2) both refer to the same point in 3D space, (1, 1, 1).

 

For direction vectors, w is 0, which would cause division by zero. These correspond to points at infinity in 3D space in the direction of the x, y, z components.

 

That's basically it in a nutshell.

 

4D coordinates are used because you can represent any affine transform in 3D space with a (4x4) matrix. You can't do that in 3D space because the origin of 3D space (0, 0, 0) always maps to the origin after being transformed by a matrix.

 

That doesn't happen in homogeneous 4D space since the origin is (0, 0, 0, 1) [or (0, 0, 0, w) for w != 0].

 

(0, 0, 0, 0) doesn't correspond to to any point or direction in 3D space, and is not a valid value for 4D homogeneous projective space.

 

So 4D vectors (x, y, z, w) with w != 0 corresponds to all points in 3D space and all directions (points at infinity) in 3D space as well.

 

Hope that helps.


#1Paradigm Shifter

Posted 12 June 2013 - 05:00 PM

You divide the x, y, z values by the w component to get the position in 3D space which is imbedded in the 4D projective homogeneous space.

 

So (x, y, z, w) = (1, 1, 1, 1) and (x, y, z, w) = (2, 2, 2, 2) both refer to the same point in 3D space, (1, 1, 1).

 

For direction vectors, w is 0, which would cause division by zero. These correspond to points at infinity in 3D space in the direction of the x, y, z components.

 

That's basically it in a nutshell.

 

4D coordinates are used because you can represent any affine transform in 3D space with a (4x4) matrix. You can't do that in 3D space because the origin of 3D space (0, 0, 0) always maps to the origin after being transformed by a matrix.

 

That doesn't happen in homogeneous 4D space since the origin is (0, 0, 0, 1) [or (0, 0, 0, w) for w != 0].

 

(0, 0, 0, 0) doesn't correspond to to any point or direction in 3D space, and is not a valid value for 4D homogeneous projective space.

 

So 4D vectors (x, y, z, w) with w != 0 corresponds to all points in 3D space and all directions (points at infinity) in 3D space as well.

 

Hope that helps.


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