Viewing The Camera and Projection Gail Carmichael gailcscs

Viewing The Camera and Projection Gail Carmichael (gail_c@scs. carleton. ca)

The Goal Understand the process of getting from 3 D line segments to images of these lines on the screen.

Canonical View Volume Windowing transform brings points to pixels: MW

Canonical View Volume xpixel ypixel xcanonical = ycanonical 1 1 Mw

Orthographic Projection Orthographic Perspective

Orthographic Viewing Volume

Orthographic View to Canonical View x Scale Move to Origin y z 1 World to Canonical Coordinates

Orthographic View to Canonical View 2 /(r-l) 0 0 0 1 0 0 -(l+r) /2 x 0 2 /(t-b) 0 0 0 1 0 -(b+t) /2 y 0 0 2 /(n-f) 0 0 0 1 -(n+f) /2 z 0 0 0 1 1 World to Canonical Coordinates

Drawing Lines in Orthographic View Mo=Mw Mscale Mmove_to_origin xpixel x ypixel y zcanonical 1 = Mo z 1

Arbitrary View Positions Top of camera goes this way Camera is centered here Camera is looking this way

Arbitrary View Positions w = - (g / ||g||) u = (t × w) / || t × w || v= w×u

Arbitrary View Positions

Coordinate Transformations

Coordinate Transformations

Coordinate Transformations

Coordinate Transformations p = (xp, yp) ≡ o + xpx + ypy p = (up, vp) ≡ e + upu + vpv

Coordinate Transformations p = (xp, yp) ≡ o + xpx + ypy p = (up, vp) ≡ e + upu + vpv

Coordinate Transformations p = (xp, yp) ≡ o + xpx + ypy p = (up, vp) ≡ e + upu + vpv xp yp 1 up = ? ? vp 1

Coordinate Transformations p = (xp, yp) ≡ o + xpx + ypy p = (up, vp) ≡ e + upu + vpv xp yp 1 = 1 0 xe xu xv 0 up 0 1 ye yu yv 0 vp 0 0 1 1

Camera Coordinate Transform

Camera Coordinate Transform Mv = xu yu zu 0 1 0 0 -xe xv yv zv 0 0 1 0 -ye xw yw zw 0 0 0 1 -ze 0 0 0 1

Drawing with Arbitrary View and Orthographic Projection xpixel x ypixel y zcanonical 1 = M o Mv z 1

Perspective Projection ys = y(d/z)

Perspective Via Orthographic

Perspective Via Orthographic

Perspective Via Orthographic

Perspective Transform Mp = 1 0 0 0 (n+f) /n 1 /n -f 0

Perspective Transform Mp x x nx/z y y ny/z z[(n+f)/n] - f n + f – (fn/z) z/n 1 z 1 =

Perspective Transform Mp = n 0 0 0 0 (n+f) -fn 0 0 1 0

Drawing with Arbitrary View and Perspective Projection xpixel x ypixel y zcanonical 1 = M o Mp Mv z 1

CAUTION!! Everything up until now used the more common right-hand coordinate system. Direct 3 D uses the left-hand coordinate system. See: http: //msdn. microsoft. com/enus/library/windows/desktop/bb 204853%28 v=vs. 85%29. aspx