Lecture 2 - Transformations for animation (slides)

Product as a projection graphics lecture 2 slide 35

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Unformatted text preview: mation matrix that moves the scene scene to that coordinate system, we need to find it to that coordinate system, we need to find it ! I We are given a view direction d and location C –  We are given a view direction d and location C ! To see how to get the matrix, we introduce the idea of the dot product as a To see how to get the matrix, we introduce the idea of the dot projection ! product as a projection Graphics Lecture 2: Slide 35 ! 34 / 45 The dot product asas projection ! The dot product a projection The dot product is defined as •  The dot product is defined as !P · u = |P||u| cos ✓ P · u = |P||u| cos θ If u is •  If u is ! I a unit vector –  a unit vector then P · then|P| c·os θ |P| cos ✓ u= P u= I along a co-ordinate axis then P · u is the ordinate of P in the –  along a co-ordinate axis then P · u is the ordinate of P in the directiondirection of u of u Graphics Lecture 2: Slide 36 ! 35 / 45 Changing axes projection Changingaxes byby projection ! Extending the idea to three dimensions we see see that •  Extending the idea to three dimensions we can canthat a change aof hange ofbe expressed asexpressed as projections using c axes can axes can be projections using the dot product the dot product ! For example, call the first coordinate of P in the new For example, call the first t system Px coordinate of P in the new t system Pxt ! (P C) · u Px = ! = P·u C·u Pxt = (P−C)·u = P·u−C·u Graphics Lecture 2: Slide 37 ! Transforming point P Transforming P Transforming point point P Given point P in the (x, y, z ) axis system, we can calculate the corresponding point in(x, y,(z ) v, w)system, we can calculate the the Given pointpoint then the (u, y, z) asystem as: we can calculate •  Given P in P i x, axis xis system, corresponding point in the (u, v, w) system w) system as:
 the corresponding point)inu = (u,· v, as: C · u t Px = (P C · the P u ! Cu Ptt = P C u = P u C ·· v Pyx = ((P C))··v = P ··v Ptt = P Pzy = ((P t Pz = ( P C v =P v C))··w = P ·· w C)...
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