Lecture_Chapter4_b

P 1 v 1 2 v 2 3 v 3 p0 ecs 175 chapter 4 object

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Unformatted text preview: Chapter 4: Object Transformation 37 Homogeneous Coordinates •  Distinguishing points and vectors In world frame vectors are defined by a local coordinate frame. v = α1 v1 + α2 v2 + α3 v3 In world frame points are defined by a local coordinate frame and origin. P = α 1 v 1 + α 2 v 2 + α 3 v 3 + P0 ECS 175 Chapter 4: Object Transformation 38 Homogeneous Coordinates •  Distinguishing points and vectors v1 v2 Local coordinate system as rows in matrix v3 P0 v = α1 v1 + α2 v2 + α3 v3 P = α 1 v 1 + α 2 v 2 + α 3 v 3 + P0 ECS 175 Chapter 4: Object Transformation v1 v2 v = (α1 , α2 , α3 , 0) · v3 P0 v1 v2 P = (α1 , α2 , α3 , 1) · v3 P0 39 Transformation in Homogeneous Coordinates •  Affine transformation now represented by 4x4 matrix α11 α21 A= α31 0 α12 α22 α32 0 α13 α23 α33 0 α14 α24 α34 1 x x y and vectors v = y A transforms points p = z z 1 0 ECS 175 Chapter 4: Object Transformation 40 Translation P￿ = P + v x y P = z 1 vx vy v= vz 0 100 0 1 0 ￿ P = 0 0 1 000 ￿ x + vx x y ￿ y + vy ￿ P = ￿ = z z + vz 1 1 vx vy ·P vz 1 translation matrix T ECS 175 Chapter 4: Object Transformation 41 Transformation in Homogeneous Coordinates •  Example coordinate transformation (scaling and translation) vworld = T (p)S (0.5, 0.5, 0.5) · vlocal ECS 175 Chapter 4: Object Transformation 0.5 0 = 0 0 0 0.5 0 0 0 0 0.5 0 p1 p2 · vlocal p3 1 42 Rotation in 3D cos(α) sin(α) Rz ( α ) = 0 0 − sin(α) cos(α) 0 0 1 0 0 cos(α) Rx ( α ) = 0 sin(α) 0 0 cos(α) 0 Ry ( α ) = − sin(α) 0 ECS 175 0 0 1 0 0 − sin(α) cos(α) 0 0 sin(α) 1 0 0 cos(α) 0 0 Chapter 4: Object Transformation 0 0 upper left sub-matrix 0 identical to 2D matrix 1 0 0 0 1 0 0 0 1 inversion: R ( α ) −1 = R ( − α ) cos(−α) = cos(α) sin(−α) = − sin(α) R ( α ) −1 = R ( α ) T 43 Rotation: Fixed Point •  Rotation always performed around origin •  Changing the fixed point requires translations: vworld = T (p)Rz (α)T (−p) · vlocal vworld ECS 175 cos(α) sin(α) = 0 0 − sin(α) cos(α...
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