Lecture 2 - Transformations for animation (slides)

What do the individual columns mean transforming the

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Unformatted text preview: ! •  The other axis transformations: ! The other axis Transformations: •  Similarly, we find the following transformations of unit vectors j and k! Similarly, we find the following transformations of unit vectors j and k 01 01 01 01 0 rx 0 sx B1C BC B0C BC B C ! B ry C B C ! Bsy C [email protected] A [email protected] A @ rz A @sz A 0 1 0 0 0 0 Graphics Lecture 2: Slide 31 ! 30 / 45 What do the individual columns mean? What do the individual columns mean? ! Transforming the Origin:origin: ! •  Transforming the –  If we transform the origin, (0, 0, 0, 1)T, we end up with the last If we transform of the transformation1)T , we! end up with the last column the origin, (0, 0, 0, matrix column of the transformation 0 qx rx sx Bqy ry sy B @qz rz sz 000 Graphics Lecture 2: Slide 32 ! matrix 10 1 0 1 Cx 0 Cx Cy C B0C BCy C CB C = B C Cz A @0A @Cz A 1 1 1 31 / 45 The meaning of a transformation matrix ! The meaning of a transformation matrix Putting everything together …! ! Putting everything together . . . The columns are the original axis system after transforming to the The columns are the original axis system after transforming to system coordinate system ! new coordinate the new 0 1 qx rx sx Cx Bqy ry sy Cy C B C @qz rz sz Cz A 0001 #### qrsC Graphics Lecture 2: Slide 33 ! q r s C transformed transformed transformed transformed x-axis y -axis z -axis origin E↵ect of a transformation matrix E↵ect of a transformation matrix Effect of a transformation matrix ! ! ! ! ! ! ! ! Tells Tells us the axesand origin in the new coordinate system. Tells usus the oldaxes axes and origin inthenew coordinate system. ! the old old and origin in the new coordinate system. 0 0 qx qx Bqyx r Bqy Bqzy B @r @ q z rz 0 0 Graphics Lecture 2: Slide 34 ! 0 1 rx sx C1 x sx sy Cx y C ⇥ ⇤ ry CC =q r s C ⇤ s rz y sz Cy C Cz A ⇥ C= q r s C A 0 z 0 Cz1 s 0 1 33 / 45 33 / 45 What we want is the other way round What we want is the other way round ! Normally, •  I We are !not given the transformation matrix that moves the Normally, –  We are not given the transfor...
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