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3D Rendering Pipeline (II)

Prof. Hsien-Hsin Sean Lee School of Electrical and Computer

Engineering Georgia Institute of Technology

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Projection from 3D space •  Projection transform 3D geometry into a form that can be

rendered as a 2D image

Adapt from Joe Farrell’s article (http://www.codeguru.com/cpp/misc/misc/math/article.php/c10123__1/ )

x

y

z

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Projection Basics •  Projection transforms your geometry into a canonical view

volume (e.g., in D3D or XNA) • Only X- and Y-coordinates will be mapped onto the screen •  Z will be almost useless but used for depth test

(-1, -1, 0)

(1, 1, 1)

x

y

z

(0, 0, 0)

Canonical view volume (LHS) (-1, -1, 0) to (1,1,1) used by D3D/XNA

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Orthographic (or Parallel) Projection

Viewer’s position

View plane

• Project from 3D space to the viewer’s 2D space

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Orthographic Projection

View Volume (an axis-aligned box)

(l, b, n)

(r, t, f)

(-1, -1, 0)

(1, 1, 1)

x

y

z

Canonical view volume 6

Orthographic Projection • Derive x’ and y’

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Orthographic Projection • Derive z’ (slightly different for the range)

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Orthographic Projection • Put all together

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Orthographic Projection • In orthographic projection

– Z-axis passes through the center of your view volume – Field of view (FOV) extends equally far

• To the left as to the right (i.e., r = -l) • To the top as to the below (i.e., t=-b)

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Orthographic Projection

•  Same size in 2D and 3D • No distance feel •  Parallel lines remain parallel • Good for tile-based games where camera is in fixed location

(e.g., Mahjong or 3D Tetris)

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Perspective Projection

•  The farther the object is, the smaller it appears •  Some photo editing software allows you to perform

“Perspective Correction”

Viewer’s position

View plane

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Viewing Frustum

Far plane

Viewer’s position

Near plane

Think about looking through a window in a dark room

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Viewing Frustum

Near plane

Far plane

Viewer’s position

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Perspective Projection

(-1, -1, 0)

(1, 1, 1)

x

y

z

Canonical view volume

(l, b, n)

(r, t, f)

View Frustum (a truncated pyramid)

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Perspective Projection • Given a point (x,y,z) within the view frustum, project it onto

the near plane z=n –  x ∈ [l, r] and y ∈ [b, t]

• We will map x from [l,r] to [-1,1] and y from [b,t] to [-1,1]

(-1, -1, 0)

(1, 1, 1)

x

y

z

Canonical view volume

(l, b, n)

(r, t, f)

View Frustum

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Perspective Projection Math (x,y,z)

(x’,y’,n)

n x’

y’

x

y

z

(0,0,0) (0,0,n)

To calculate new coordinates of x’ and y’ f

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Perspective Projection Math (x,y,z)

(x’,y’,n)

n x’

y’

x

y

z

(0,0,0) (0,0,n)

f

⇒ Apply the formula we derived in orthographic projection

Now let’s tackle at z’ component 18

Perspective Projection Math

• We know z (depth) transformation has nothing to do with x and y

• the desired z’z function is shown above

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Perspective Projection Math

• We know for a fact that (boxed equations above) – z’ = 0 when z=n (near plane) – z’ = 1 when z=f (far plane)

⇒

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Perspective Projection Math

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Perspective Projection Math

•  At the end, we will have to divide by z to obtain [x’, y’, z’, w’] •  Similar to orthographic projection ⎯ we keep equal FOV

extension, thus replace l, r, t, b with w and h

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Define Viewing Frustum

Viewer’s position

+z

Parameters: FOV: Field of View Aspect ratio = Width/Height Near z Far z

fov/2

+y

Far (f)

Near (n)

Height

XNA: Matrix.CreatePerspectiveFieldOfView

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Perspective Projection

+z

+y

a h/2

h/2

n

Need to replace w and h with FOV and aspect ratio

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Final Matrix of Perspective Projection

Don’t worry. XNA API did this for you: Matrix.CreatePerspectiveFieldOfView

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Viewport Transformation

• The actual 2D projection to the viewer • Copy to your back buffer (frame buffer) • Can be programmed, scaled, ..

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Other Geometry Optimizations • Backface Culling

• Clipping

• Hidden surface removal (Occlusion)

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Backface Culling

(c1, c2, c3) (b1, b2, b3)

(a1, a2, a3)

Cross product

2 Vectors

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Compute the Surface Normal for a Triangle

(0, 0, 0) (4, 0, 0)

(3, 3, 0)

Compute Normals (Clockwise vectors)

v2 x

z

y

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Compute the Surface Normal for a Triangle

(0, 0, 0) (4, 0, 0)

(3, 3, 0)

Compute Normals

v1 x

z

y

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Backface Culling Method

Surface vectors

Eye vector

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Backface Culling Method (dot product)

θ

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Backface Culling Method (dot product)

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When to Perform Backface Culling?

Clipp

ing

Persp

ectiv

e Divi

de

View

port

Tran

sform

Raste

rizati

on

Proje

ction

Tran

sform

Wor

ld Tr

ansfo

rm

View

Tran

sform

Lighti

ng

Back

face C

ulling

Clipp

ing

Persp

ectiv

e Divi

de

View

port

Tran

sform

Raste

rizati

on

Proje

ction

Tran

sform

Wor

ld Tr

ansfo

rm

View

Tran

sform

Lighti

ng

Back

face C

ulling

How?

Transform your camera to the world space first! 34

3D Clipping

•  Test 6 planes if a triangle is inside, outside, or partially inside the view frustum

•  Clipping creates new triangles (triangulation) –  Interpolate new vertices info

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How to Clip Against a Plane? • Test each vertex of a triangle

– Outside – Inside – Partially inside

• Incurred computation overhead • Save unnecessary computation (and

bandwidth) later • Need to know how to determine a plane • Need to know how to determine a vertex is

inside or outside a plane 36

What it Takes to Determine a Plane?

• You need two things to specify a plane – A point on the plane (p0, p1, p2) – A vector (normal) perpendicular to the plane (a, b, c) – Plane a*(x – p0) + b*(y – p1) + c*(z - p2) = 0

V=(5, 6, 7)

P=(1, 2, 3) K=(x, y, z)

Plane equation

5*(x-1)+6*(y-2)+7*(z-3)=0

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Distance Calculation from a Plane

• Given a point R, calculate the distance –  Distance > 0 inside the plane –  Distance = 0 on the plane –  Distance < 0 outside the plane

Normal R υ →

d

d θP

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Distance Calculation from a Plane

R

υ →

d θ

P 180-θ

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Triangulation using Interpolation

(a2, b2, c2)

(a1, b1, c1)

(x, y, z)

d1 d2

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Illumination Models

• It won’t look 3D without lighting • Part of geometry processing • Illumination Types

– Ambient – Diffuse – Specular – Emissive

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Local vs. Global Illumination • Local Illumination

– Direct illumination: Light shines on all objects without blocking or reflection

– Used in most games

• Global Illumination – Indirect illumination: Light bounces from one

object to other objects – Add more realism (non real-time rendering) – Computationally much more expensive – Ray tracing, radiosity

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Common Light Sources

Directional Light (Infinitely far away)

Point Light (Emit in all directions)

Spot Light (Emit within a cone)

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Illumination: Ambient Lighting • Not created by any light source • A constant lighting from all directions • Contributed by scattered light in a surrounding

RGB multiplies separately Material Color Light Color

⊗

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Illumination: Diffuse Lighting • Light sources are given • Assume light bounces in all directions

Refelcted Light will reach the eyes no matter where the camera is!

Diffuse surface

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Reflected Light Intensity Calculation •  Reflectivity ∝ the entry angle •  Use Lambert’s cosine Law intensity

F(θ)=cos(θ)

Surface Normal

n →

Surface Normal

n → θ

Surface Normal

n → θ Dot product

Reflection Full

None

Partial

L →

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Diffuse

Illumination: Diffuse Lighting

Material Color Light Color Ambient +

Diffuse

Ambient + Diffuse

⊗

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Illumination: Specular Lighting •  Create shining surface (surface perfectly reflects) •  Viewpoint dependent

Ambient + Diffuse + Specular Material Color Light Color

Specular

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Jim Blinn’s Specular Model

E →

L →

n →

R→ Half-way vector

H→

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Brightness Effect (S)

L1

L2

n1

n2

H1

H2

More shining

E1

E2

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Brightness Effect (S)

intensity

cosθ

cos2θ

cos8θ

cos64θ

cosθ > cosω

~0.65

~0.95

cos8θ >>> cos8ω

~0.62

n1

n2

H1 H2

θ

ω

S=1

n1

n2

H1 H2

θ

ω

S=8

~0.05 radian

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Specular Lighting Effect

•  A larger S shows more concentration of the reflection

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Illumination: Emissive Lighting

• Color is emitted by the material only

Ambient + Diffuse +

Specular + Emissive

Material Emissive

Color

+

Ambient + Diffuse + Specular

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Common Light Sources

Directional Light (Infinitely far away)

Point Light (Emit in all directions)

Spot Light (Emit within a cone)

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Light Source Properties • Position • Range

– Specifying the visibility • Attenuation

– The farther the light source, the dimmer the color

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Falloff effect

Spotlight Effect

•  Similar idea to specular lighting •  Falloff factor determines the fading effect of a spotlight •  “f” exponentially decreases the cosα value

L → d → α