09 - Viewing Projection

Rendering's three steps

(1) Viewing and projection, (2) hidden surface removal, (3) determining surface colour.

Photorealistic vs non-photorealistic rendering

Photorealistic looks like a camera image. Non-photorealistic = illustration, visualization, art, where realism can get in the way.

Full viewing chain

M = Mo Mp Mv (read right to left): Mv lookAt, Mp perspective, Mo orthographic + viewport.

Mprojection

Mo Mp together.

Canonical view volume

The cube [-1, +1]^3 centred at the origin.

Viewport transform (canonical -> screen)

A scale (size 2 -> nx) then a translate (origin to the lower-left). z passes through for the z-buffer.

Pixel centres and screen extent

Pixel centres are at integers 0..nx-1, so the screen covers -0.5 to nx - 0.5.

Display origin at top-left, y down

Flip y: multiply the y scale by -1.

Parallel projection properties

Not photorealistic, but preserves angles and lengths, so you can measure from it. Good for technical drawings.

Orthographic view volume

Axis-aligned box [l, r] x [b, t] x [n, f], bounded by 6 planes: left, right, bottom, top, near, far.

Camera viewing direction convention

Looks down -z (right-handed, +z out of the screen), so visible z values are negative and n > f even though |n| < |f|.

Morth

Maps the orthographic box to the canonical cube with a scale and a translate. Mo = Mvp Morth.

lookAt inputs

e = eye position, g = gaze direction, t = view-up vector.

Goal of the viewing transformation

Put the camera at the origin, looking along -z with y up.

w (camera frame)

w = -g / |g|. Points opposite the gaze. Maps to z.

u (camera frame)

u = (t x w) / |t x w|. Perpendicular to t and w. Maps to x.

v (camera frame)

v = w x u. The true up (t made perpendicular to the view direction). Maps to y.

Mv construction

Rotation with rows u, v, w times translate by -e. Translate the eye to the origin first, then rotate uvw onto xyz.

Mv is the transformation from…

Model (world) space to eye space.

Orthographic viewing matrix

M = Mo Mv

Perspective projection

Farther objects look smaller; size scales with distance. How eyes and cameras see.

Perspective by similar triangles

Screen at distance d: ys = (d / z) y. Perspective = divide by z.

Why homogeneous coordinates for perspective?

Matrices can't divide. Put z into the w slot; the final divide by w does the divide by z.

Simple perspective matrix effect

(x, y, z, 1) -> (x, y, z, z/d), then divide by w -> (dx/z, dy/z, d).

Mp (perspective matrix)

Turns the frustum into the orthographic box. Leaves the z = n plane alone and squashes the far end to the same size.

Why that particular Mp?

Any matrix that divides by z gives perspective; you pick the convenient one that maps to the ortho box.

z' after perspective divide

z' = n + f - fn/z. Non-linear in z; it causes z-buffer precision problems.

x and y after perspective divide

xs = nx/z, ys = ny/z

Why GLM gives lookAt and perspective

They build Mv and Mp (Mprojection) for you.

Parallel vs perspective (review Q)

Parallel: projection lines parallel, size independent of distance, angles/lengths preserved, not realistic. Perspective: lines converge at the eye, divide by z, distant objects smaller, needs homogeneous coords (w = z).