Lecture 2 - Transformations for animation (slides)

Look along the z z axis and the the axis is graphics

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Unformatted text preview: ical form for Perspective Projection WeWe look along the-axis and the the yy axis is ‘up’! look along the z z -axis and the the -axis is Graphics Lecture 2: Slide 6 ! 5 / 45 Transformation of viewpoint Transformation of viewpoint! Coordinate system for for definition Coordinate system definition Coordinate system for viewing! Coordinate system for viewing ! Graphics Lecture 2: Slide 7 ! 6 / 45 Flying Sequences! •  The required transformation is in three parts:! 1.  Translation of the origin! 2.  Rotate about y-axis! 3.  Rotate about x-axis ! •  The two rotations are to line up the z-axis with the view direction ! Graphics Lecture 2: Slide 8 ! 1. Translation of the Origin 1. Translation of the Origin ! 0 1 B0 A=B @0 0 0 1 0 0 0 0 1 0 1 Cx Cy C C Cz A 1 Graphics Lecture 2: Slide 9 ! 8 / 45 2. Rotate about y until until in thein -z plane1plane ! 2. Rotate about y d is d is y the y-z 0 cos ✓ B0 B=B @ sin ✓ 0 0 1 0 0 sin ✓ 0 cos ✓ 0 1 0 0 dz /v 0C B 0 C=B 0A @dx /v 1 0 = dx /v 0 1 0 0 Graphics Lecture 2: Slide 10 ! 1 = sin ✓ = cos ✓ kvk = v q correct this if you have an earlier version of the slides. dx /v 0 dz /v 0 1 0 0C C 0A 1 d2 + d2 x z dz /v 3. Rotate about x until d points along the z -axis 3. Rotate about x until d points along the z-axis! v cos 0 1 0 B0 cos C=B @0 sin 0 0 0 sin cos 0 = sin = = q d2 + d2 x z v/|d| dy /|d| 10 0 1 0 0C B0 v/|d| C=B 0A @0 dy /|d| 1 0 0 1 0 0 dy /|d| 0C C v/|d| 0A 0 1 Graphics Lecture 2: Slide 11 ! 10 / 45 Combining the matrices ! •  A single matrix that transforms the scene can be obtained from the matrices A, B and C by multiplication ! ! T = CBA •  And for every point P of the scene, we calculate ! Pt = T P •  The view is now in ‘canonical’ form and we can apply the standard perspective or orthographic projection. ! Graphics Lecture 2: Slide 12 ! Verticals! •  So far we have not looked at verticals ! •  Usually, the y direction is treated as vertical, and by doing the Ry transformation...
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