08viewing

# l yh l zh x x x l x h x h x l h l 0 0 x h x l x h

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Unformatted text preview: ves no doubt as to t cor can implement . •heWerectness of the resultthis by mapping the view volume A exa canonical us co struction tonthe ctly analogoviewnvolume. can be used to deﬁne a 3D windowing transformation, which maps the box [xl , xh ]×[yl , yh ]×[zl , zh ] to the box [xl , xh ]× •y ,This iszjust ]a 3D windowing transformation! [ l yh ] × [ l , zh x −x x l x h −x h x l h l 0 0 x h −x l x h −x l y l y h −y h y l y h −y l 0 0 y h −y l y h −y l . ( 6 .7 ) z l z h −z h z l z h −z l 0 0 z h −z l z h −z l 0 0 0 1 r+ ⇥ It is interesting to note 2hat if we multiply an arbitrlary matrix composed of t 0 0 rl rl scales, shears and rotations with a s2mple translation t+an⌃ation comes second), i (tr b sl ⇧0 0 tb t b⌃ we g e t Morth = ⇧ n 2 ⇤ n+f ⌅ 0 0 n0 f fa a11 a12 a13 a11 a13 xt 1 0 0 xt 12 0 1 0 yt a20 a22 0 a23 0 a21 a22 a23 yt 0 1 1 0 0 1 zt a31 a32 a33 0 = a31 a32 a33 zt . Cornell CS4620 Fall 2013 • Lecture 8 © 2013 Steve Marschner • 15 Camera and modeling matrices • We worked out all the preceding transforms starting from eye coordinates – before we do any of this stuff we need to transform into that space • Transform from world (canonical) to eye space is traditionally called the viewing matrix – it is the canonical-to-frame matrix for the camera frame – that is, Fc–1 • Remember that geometry would originally have been in the object’s local coordinates; transform into world coordinates is called the modeling matrix, Mm • Note many programs combine the two into a modelview matrix and just skip world coordinates Cornell CS4620 Fall 2013 • Lecture 8 © 2013 Steve Marschner • 16 Viewing transformation the camera matrix rewrites all coordinates in eye space Cornell CS4620 Fall 2013 • Lecture 8 © 2013 Steve Marschner • 17 Orthographic transformation chain • Start with coordinates in object’s local coordinates • Transform into world coords (modeling transform, Mm) • Transform into eye coords (camera xf., Mcam = Fc–1) • Orthographic projection, Morth • Viewport transform, Mvp ps = Mvp Morth Mcam Mm po ⇤ ⌅ ⇤ nx xs 2 ⌥ ys ⌥0 ⌥ ⌥ ⇧ zc ⌃ = ⇧ 0 1 0 0 ny 2 0 0 0 0 1 0 nx 1 ⌅ ⇤ 2 rl 2 ny 1 ⌥ ⌥0 2 0 ⌃⇧ 0 1 0 Cornell CS4620 Fall 2013 • Lecture 8 0 2 tb 0 0 0 0 2 nf 0 r +l ⌅ rl t+b tb n+f ⌃ nf 1 uv 00 w 0 ⇤⌅ xo ⇥1 ⌥ yo e Mm ⌥ ⌃ ⇧ zo 1 1 © 2013 Steve Marschner • 18 Perspective projection similar triangles: Cornell CS4620...
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## This document was uploaded on 03/20/2014.

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