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 zaxis.
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 zaxis and shrink. The rotation is anticlockwise 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 zaxis.
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 zaxis and shrink. The rotation is anticlockwise 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 zaxis and shrink. The rotation is anticlockwise 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.
 Spring '14

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