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Lecture 2 - Transformations for animation (slides)

# Let q be some vector in vertical direction we we then

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Unformatted text preview: ome vector p in the in the We  can write write termsterms of vector p in thephorizontal direction horizontal direction ! p direction u= p u = |p| |p| • To ensure that p is horizontal To   ensure that p is horizontal we set we set ! To ensure that p is horizontal we set py = 0 py = 0   so p has has no vertical component ! so•that that p no vertical component so that p has no vertical component Graphics Lecture 2: Slide 41 ! 40 / 45 40 / 45 And the vertical direction And the vertical direction And the vertical direction ! •  Let q be some vector in vertical direction, we we then Let b LetLet qqbe e somevector in the the vertical direction,canthen vector in thevertical direction, we can can Let some then write v as ! write as write vvas qq )= ) vv= |q| |q| •  q must have a positive y component, so we can say that ! must have positive component, so we can say that qqmust have aapositive yycomponent, so we can say that = qqy= 11 y Graphics Lecture 2: Slide 42 ! 41 / 45 41 / 45 So we have four unknowns So we have four unknowns So we have four unknowns ! p , ,p new horizontal pp == [p[x ,x0,0pz ]z ] new horizontal q, , new vertical qq == [q[x ,x1,1qzq]z ] new vertical To solve for these we use the cross product and dot To solve for these we use the cross product and dot product. To solve for !these we use the cross product and dot product. product. We can write the view direction d, which is along the new z We can write the view direction , , which along the new z axis, We xis, as ! the view direction ddwhich isis along the new z axis, a can write as as dd = p ⇥ q =p⇥q ! (We can do this because the magnitude p is not yet set) (We can do this because the magnitude ofof p is not yet set) (We can do this because the magnitude of p is not yet set)! Graphics Lecture 2: Slide 43 ! 42 42 / 45 / 45 Evaluating the cross-product! Evaluatingthe cross product Evaluating the 0 1 product cross dx ijk @1 d = 0 d y A = p ⇥ q = px 0 p z dx i qj 1 q k dz x z d = @ d y A = p ⇥ q = px...
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