Lecture 2 - Transformations for animation (slides)

By matrix multiplication 0 1 10 0 0 b0 1 0 0c c mp b

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Unformatted text preview: dinates can Perspective projection of homogenous coordinates can also be done also be done by matrix multiplication:! by matrix multiplication: 0 1 10 0 0 B0 1 0 0C C Mp = B @0 0 1 0A 0 0 1/f 0 01 0 1 x x By C B y C C Mp B C = B @z A @ z A 1 z/f Graphics Lecture 2: Slide 20 ! 19 / 45 Perspective Projection Matrix: Normalisation Perspective Projection Matrix: Normalisation Perspective projection matrix: Normalisation ! •  Remember we can normalise homogeneous coordinates, so ! Remember we can normalise homogeneous coordinates, so Remember we can normalise homogeneous coordinates, so 01 0 1 1 01 0 x x x x By C B y C By C B y C C hich M B C=B the same as Mpp B zC = B z C wwhich is the same as ! @ A @ z A which isis the same as A@ A @z ! 1 z/f 1 z/f as required. as required. ! 0 1 0 1 xf /z xf /z Byf /z C Byf /z C B C B @ ff C A @ A 11 •  as required. ! ! Graphics Lecture 2: Slide 21 ! 20 45 20 / / 45 Projection matrices are singular Projection matrices are singular ! •  Notice that both projection matrices are singular (i.e. ‘non-invertible’, zero-determinant, …)! Notice that both projection matrices are singular 2 0 1 B0 Mp = B @0 0 00 10 01 0 1/f 1 0 0C C 0A 0 0 1 B0 Mo = B @0 0 0 1 0 0 0 0 0 0 1 0 0C C 0A 1 •  This is because a projection transformation cannot be This is because a projection transformation cannot be inverted. inverted. ! •  Given 2D image, we in general reconstruct the original Given a 2Daimage, we cannotcannot in general reconstruct the 3D original 3D scene. ! scene. Graphics Lecture 2: Slide 22 ! 2 A.K.A ‘non-invertible’, zero-determinant, . . . Homogenous Coordinates as Vectors Homogenous coordinates as vectors ! We now take a second look at homogeneous coordinates, and their •  We now take a second look at homogeneous relation to vectors. coordinates, and their relation to vectors. ! •  In the previous lecture we described the fourth ordinate In the previous lecture we described the fourth ordinate as a scale as factor.a scale factor. ! ! Homogeneous Cartesian Homogeneous Cartesian ! 01 x By C BC @z A s ! 0 1 x/s @y/s...
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This document was uploaded on 03/26/2014.

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