Lecture 2 - Transformations for animation (slides)

However it is possible to invert the vertical

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Unformatted text preview: rst, things work out correctly! •  However it is possible to invert the vertical ! Graphics Lecture 2: Slide 13 ! Transformations verticals Transformations and and verticals ! Graphics Lecture 2: Slide 14 ! 13 / 45 Rotation about a general line ! •  Special effects, such as rotating a scene about a general line can be achieved by multiple transformations ! •  The transformation is formed by: ! –  Making the line of rotation one of the Cartesian axes! –  Doing the rotation (about the chosen axis)! –  Restoring the line to its original place! Graphics Lecture 2: Slide 15 ! Rotation about a general line ! •  The first part is achieved using the same matrices that we derived for the flying sequences ! CBA •  This rotates the general line so it is aligned with the z-axis.! •  We then carry out the rotation about the z-axis then follow this by the inversion of the initial matrices.! •  So the full matrix T of the combined transformation is! T = A−1B−1C−1RzCBA Graphics Lecture 2: Slide 16 ! Other effects ! •  Similar effects can be created using this approach ! •  e.g. to make an object shrink (and stay in place) ! 1.  Move the object to the origin! 2.  Apply a scaling matrix! 3.  Move the object back to where it was ! Graphics Lecture 2: Slide 17 ! Projection by matrix multiplication ! •  Usually projection and drawing of a scene comes after the transformation(s). ! •  It is therefore convenient to combine the projection with the other parts of the transformation ! •  So it is useful to have matrices for the projection operation ! Graphics Lecture 2: Slide 18 ! Orthographic Projection Matrix Orthographic projection matrix ! For For (canonical) orthographic projection, wedrop the drop •  (canonical) orthographic projection, we simply simply z the z-coordinate: ! coordinate: ! Graphics Lecture 2: Slide 19 ! 0 1 1000 B0 1 0 0C C Mo = B @0 0 0 0A 0001 01 01 x x By C By C Mo B C = B C @z A @0A 1 1 Perspective Projection Matrix Perspective projection matrix ! •  Perspective projection of homogenous coor...
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