Lecture 2 - Transformations for animation (slides)

Lastly we can use the factthat the vectors vectors p

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Unformatted text preview: 0 p z dz qx 1 qz = = p z i + (p z q x p z i + (p z q x i.e. i.e. 0 1 pz px px qz )j + px k0 @pz qx 1 qz A = pz px A @ px qz )j + px k = dx = pz dxdy= = ppz qx z pz q x px q z px px q z dy dz= = z qpx px qz px d z = px ! So we can write vector p completely in terms of d So we can write vector p completely in terms of d ! 0 1 So we can write vector p completely in terms of d dz 0 1 ! @ dz 0 A p= A p = @ 0 dx dx Graphics Lecture 2: Slide 44 ! 43 / 45 43 / 45 Using the dot product Using the dot product Using the dot product ! Lastly we can use the factthat the vectors vectors p a orthogonal •  Lastly we can use that the vectors p and q are orthogonal Lastly we can use the factthe fact that the p and q are nd q are orthogonal! pq p · ·q ) x qx + z qz ) ppx qx+ ppz qz = = = = 00 00 • from the the product (previous slide) AndAnd the cross cross product (previous And  from fromcrossproduct (previous slide) slide) ! ddy= ppz qx ppx qz y = z qx x qz Sowe have have two linear equations to solve for q and for q it So we two simple linear linear equations to solve write So • wehave two simple simpleequationsto solve for q and write it in terms write it in terms ofthe and of the components d in terms of the componentsofofd components of d ! Graphics Lecture 2: Slide 45 ! 44 / 45 44 / 45 The ﬁnal matrix The ﬁnal matrix ! The ﬁnal matrix •  Once we have expressions for p and q in terms of the Once we have expressions for p and q in terms of the given vector Once we vector d, we have ! and q in terms of the given vector given have expressions for p d,, we have d we have p q d p q u= v= w= d u = |p| v = |q| w = |d| |p| |q| |d| ! WeWe already know C as is also given. given. can we can write •  already know C as that that is also So we can write down We already know C as that is also given. So we So write down thedmatrix matrix ! own the the matrix 0 1 ! 0 1 ux uyy uzz C ··u ux u u Cu B vx vy vz C ··v C B C C vC B vx vy vz B C @wx wy wz A C ··w A @wx wy wz Cw 0 1 00 00 0 1 Graphics Lecture 2: Slide 46 !...
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