Lecture_Chapter4_a

Transformations work mathematically to describe and

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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 ￿￿ 2 v1 = 3 ￿￿ 4 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) ￿￿ 1 P= 1 ECS 175 ￿￿ 2 Q= 4 Chapter 4: Object Transformation ￿￿ 1 v =Q−P = 3 Q=P +v 12 Geometry and Linear Algebra •  Vectors can be added and subtracted ￿￿ 2 v1 = 3 ￿￿ 3 v2 = 1 ￿￿ 5 v3 = v1 + v2 = 4 v4 = v1 − v2 inverse and zero exist ECS 175 Chapter 4: Object Transformation 13 Geometry and Linear Algebra •  Vector orientations can be compared ￿￿ 1 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 ￿￿ 1 v1...
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