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# The centre of the object is located at position p 10 0

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Unformatted text preview: ◆ ✓◆ ✓ ◆ 1 1 1 11/2 1 5/2 4 1 25/8 P = + + + = 1 5/2 3 26/8 2 8 4 2 3/2 The gradient is given by P 0 ( µ ) = 3a3 µ2 + 2a2 µ + a1 which, at µ = 1/2, is ✓◆ 1 3 P0 = a3 + a2 + a1 2 4 and substituting the values found for a3 a2 a1 a0 gives ✓◆ ✓◆✓ ◆✓ ◆✓ ◆ 1 3 4 11/2 5/2 5 P0 = + + = 1 5/2 3/2 1/4 2 4 Marks: 4 Transformations" 1 Transformations of Graphics Scenes a In a computer graphics animation scene an object is deﬁned as a planar polyhedron. The centre of the object is located at position P = (10, 0, 10), and the scene is drawn, as usual, in perspective projection with the viewpoint at the origin and the view direction along the z-axis. Calculate the transformation matrix M that will shrink the object in size by a factor of 0.9 towards its centre point. b In a different animation, the object deﬁned in part a is required to rotate around an axis parallel to the z-axis and shrink. The rotation is anti-clockwise when viewed along the axis. In each successive frame it should rotate by 10 and shrink to 0.9 of its original size. As before, the shrinkage is towards the object’s centre. Assume that cos(10 ) = 0.98 and sin(10 ) = 0.17. What is the transformation matrix that will achieve this animation? c For another sequence the object is to shrink, as deﬁned in part a, and to drop vertically downwards by 1 unit each frame. If the animation sequence is made up of a number of frames, numbered consecutively from zero, what is the transformation matrix that should be the scene is drawn, as usual, in perspective projection with the viewpoint at the origin and the view direction along the z-axis. Calculate the transformation matrix M that will shrink the object in size by a factor of 0.9 towards its centre point. Transformations" In order, the transformation can be achieved in three steps: i) Translate the object to the origin ii) Perform a shrink relative to the origin iii) Translate the object back to its original centre In matrix terms: x ! Mx where 0 1 B0 M=B @0 0 0 1 0 0 10 0 10 0.9 0 00 100 0 0C B 0 0.9 0 0C B0 1 0 CB CB A@ 0 1 10 0 0.9 0A @0 0 1 01 0 0 01 000 Multiplying out, we get Marks: 10 0 1 10 0C C 10A 1 1 0.9 0 01 B 0 0.9 0 0C B C @0 0 0.9 1A 0 0 01 4 Multiplying out, we get Marks: b 0 1 0.9 0 01 B 0 0.9 0 0C B C @0 0 0.9 1A 0 0 01 In a different animation, the object deﬁned in part a is required to rotate around an axis parallel to the z-axis and shrink. The rotation is anti-clockwise when viewed along the axis. In each successive frame it should rotate by 10 and shrink to 0.9 of its original size. As before, the shrinkage is towards the object’s centre. Assume that cos(10 ) = 0.98 and sin(10 ) = 0.17. What is the transformation matrix that will achieve this animation? In order, the transformation can be achieved in four steps: i) Translate the object to the origin 4 b In a different animation, the object deﬁned in part a is required to rotate around an axis parallel to the z-axis and shrink. The rotation is anti-clockwise when viewed along the axis. In each successive frame it should rotate by 10 and shrink to 0.9 of its original size. As before, the shrinkage is towards the object’s centre. Department of Computing= 0.98 and sin(10 ) = 0.17. Session the transformation Conﬁdential Assume that cos(10 ) Examinations – 2011 - 2012 What is Transformations" matrix that will MODELthis animation? MARKING SCH...
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## This document was uploaded on 03/26/2014.

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