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Unformatted text preview: //multiply with translation matrix
//multiply with rotation matrix
//multiply with scaling matrix Chapter 4: Object Transformation 9 A closer look at geometry
• How do transformations work mathematically?
• To describe and represent transformations, we need to establish a formal mathematical framework
• Transformations are closely related to what you know from typical Linear Algebra courses
• Transformations are mappings between coordinate systems (note also: coordinate system handedness) ECS 175 Chapter 4: Object Transformation 10 Geometry and Linear Algebra
• Vector: encodes magnitude and direction
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v1 =
3
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v2 =
5 ECS 175 v 1 = √ 2·2+3·3 v2 = 2 · v1 Chapter 4: Object Transformation multiplication with scalars 11 Geometry and Linear Algebra
• Vectors connect/displace points (locations in space)
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P=
1 ECS 175
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Q=
4 Chapter 4: Object Transformation
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v =Q−P =
3 Q=P +v 12 Geometry and Linear Algebra
• Vectors can be added and subtracted
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v1 =
3
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v2 =
1
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v3 = v1 + v2 =
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v4 = v1 − v2 inverse and zero exist ECS 175 Chapter 4: Object Transformation 13 Geometry and Linear Algebra
• Vector orientations can be compared
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v1 =
2 dot product angle ECS 175 v2 = 2
−1 v1 · v2 = 1 · 2 + 2 · (−1) = 0
v1 · v2
cos(α) =
v 1 · v 2 Chapter 4: Object Transformation 14 Geometry and Linear Algebra
• Vectors can be projected
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 Spring '08
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