Lecture 2 - Transformations for animation (slides)

0 0 1 position vector 1 in the last ordinate1 all

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Unformatted text preview: z C z A @⇤ A @⇤ A qx rx sx Cx ⇤ ⇤ Bqy ry sy Cy C B⇤C B⇤C B y y y 1 C B0 C B0 C Bq0 r0 s0 Cy C B⇤C = B⇤C @ A@ A @ A B qz rz sz CzC B ⇤C = B ⇤C z z z z Position vector: @q0 the last ordinate ) all vectors will have the 1 in r0 s0 C1 A @⇤A @⇤A 0 0 0 0 0 same displacement. 0 0 1 Position vector: 1 in the last ordinate1 all vectors1 have the will 0 1 in the last ordinate ⇒)ll all vectors will have the 1 0 ) vectors will have the 0 Position vector: q 1 in last ordinate a •same displacement. rthe sx Cx   Position vector: x ⇤ ⇤ + Cx x same displacement. r! s C C B⇤C B⇤ + C C same displacement. B qy y y 0 10 1 0 1 B q r s Cy1 0 ⇤1 = 0 ⇤ + Cy1 CB C B C 0x x x x x @q z r z s z C z A @⇤ A @⇤ + C z A qx rx sx Cx C B⇤ C B⇤ + Cx C Bqy ry sy Cy ⇤ ⇤ + Cy C B Bq0 r0 s0 Cy C B⇤C = B⇤ +1Cy C 1 C B1 C B @y y y A@ A @ A B qz rz sz CzC B ⇤C = B ⇤ + CzC @qz rz sz Cz A @⇤A @⇤ + Cz A If we do not shear the object the three vectors q, r and s will 0001 1 1 00 1 1 remaindo not shear the object the three vectors q, r and s will remain orthogonal, ie: 0 1 •  If we If we do not shear the object the three vectors q, r and s will If orthogonal, shear the object the three vectors q, r and s will we doorthogonal, ie: not ie: ! remain q·r=r·s=q·s=0 remain orthogonal, ie: q·r=r·s=q·s=0 28 / 45 What do the individual columns mean? What do the individual columns mean? ! To see this, consider the e↵ect of the transformation in simple •  To cases. see this, consider the effect of the transformation in simple cases. ! •  example take take the unit direction along the Cartesian For For example the unit direction vectors vectors along the C axes artesian axes ! –  e.g. along the x-axis, i = (1, 0, 0, 0)T E.g. along the x-axis, i = (1, 0, 0, 0)T 0 10 1 0 1 qx rx sx Cx 1 qx Bqy ry sy Cy C B0C Bqy C B CB C B C @qz rz sz Cz A @0A = @qz A 0001 0 0 Graphics Lecture 2: Slide 30 ! 29 / 45 What do the individual columns mean? What do the individual columns mean?...
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This document was uploaded on 03/26/2014.

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