Viewing - University of Calgary

CPSC 587/687

Graphics Pipeline : Geometric Operations

University of Calgary GraphicsJungle Project

CPSC 587 2005

page 1

CPSC 587/687

Viewing transformation

•

Tools for creating and manipulating a “camera” that produces pictures of a 3D scene

•

Viewing transformations and projections

•

Perform culling or back-face elimination


CPSC 587 2005

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Local coordinate space

Object definition Modeling transformation World coordinate space

Compose scene Define view reference Define lighting View transformation

View space

Projection

Cull Clip to 3D view volume 3D screen space Projection divide

Hidden surface removal Rasterization Shading Display space

3D Rendering Pipeline

CPSC 587/687

Modeling transformation Y

World coordinate space

( xw

w

yw

Compose scene Define view reference

X Z w

zw )

w

View space

( xv

yv

zv )

View transformation

⎛ xv ⎞ ⎛ xw ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ yv ⎟ ⎜ yw ⎟ ⎜ z ⎟ = Tview ⎜ z ⎟ ⎜⎜ v ⎟⎟ ⎜⎜ w ⎟⎟ ⎝1⎠ ⎝1⎠ University of Calgary GraphicsJungle Project

CPSC 587 2005

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Viewing transformation V View plane

U eye

CPSC 587/687

⎛ xv ⎞ ⎛ xw ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ yv ⎟ ⎜ yw ⎟ ⎜ z ⎟ = Tview ⎜ z ⎟ ⎜⎜ v ⎟⎟ ⎜⎜ w ⎟⎟ ⎝1⎠ ⎝1⎠

-N

Tview = RT

⎛U x ⎜ ⎜V R=⎜ x N ⎜⎜ x ⎝ 0

Uy Vy

Uz Vz

Ny

Nz

0

0

⎛1 0 0⎞ ⎜ ⎟ 0⎟ ⎜0 1 T =⎜ 0 0 0⎟ ⎜ ⎟⎟ ⎜0 0 ⎝ 1⎠

0 − eVec ⋅ U ⎞ ⎟ 0 − eVec ⋅ V ⎟ 1 − eVec ⋅ N ⎟ ⎟⎟ 0 1 ⎠

eVec = (eye − (0 0 0))


CPSC 587 2005

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Viewing Transforms ⎛U x ⎜ ⎜V R=⎜ x N ⎜⎜ x ⎝ 0

Uy Vy

Uz Vz

Ny

Nz

0

0

0⎞ ⎟ 0⎟ 0⎟ ⎟⎟ 1⎠

CPSC 587/687

N Lemma: Where does this matrix come from:

e = Eye Position g = gaze direction t = view up vector

g N= ||g|| U= t x N ||t x N||

V t

N

V= N x U e

g

U University of Calgary GraphicsJungle Project

CPSC 587 2005

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Special Orthogonal Matrics

CPSC 587/687

See 453 slides


CPSC 587 2005

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CPSC 587/687

Taxonomy of Projections

Parallel


Perspective CPSC 587 2005

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CPSC 587/687

Perspective Projections

X = xv Y = yv Z = zv

Tproj = Tpers

w = zv d University GraphicsJungle obs: d isofNCalgary in the pictureProject

⎛1 ⎜ ⎜0 =⎜ 0 ⎜⎜ ⎝0

CPSC 587 2005

0 1 0

0⎞ ⎟ 0⎟ 0⎟ ⎟⎟ 1⎠

0 0 1

0 1/ d page 9

Parallel Projections (orthographic)

Tproj

⎛1 ⎜ ⎜0 = Tort = ⎜ 0 ⎜⎜ ⎝0


CPSC 587/687

0 0 0⎞ ⎟ 1 0 0⎟ 0 0 0⎟ ⎟⎟ 0 0 1⎠ CPSC 587 2005

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CPSC 587/687

Perspective Divide Y

Modeling transformation

w


( xw

X Z

w

w

yw

zw )

Compose scene Define view View spacereference

( xv

yv

zv )

View transformation

Projection 3D screen space

(X

Z W)

Y

Projection Divide

( xs


ys

zs )

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CPSC 587/687

Perspective Divide

X = xv

xs = X Perspective

Parallel (orthographic)

w

ys = Y

w

zs = Z

w

where

Y = yv Z = zv w = zv d

x s = xv y s = yv zs = 0


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CPSC 587/687

Perspective Divide

See 453 slides for ramifications of divide by W


CPSC 587 2005

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CPSC 587/687

View Volume


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CPSC 587/687 We specify it!

V U View volume

eye -N

Yw

View plane window

What appears on the screen: projection of part of the scene contained within the view volume

Xw Zw University of Calgary GraphicsJungle Project

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CPSC 587/687

eye

View volume University of Calgary GraphicsJungle Project

View plane window CPSC 587 2005

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yvol =

hzv d

zv = d

h d

hzv xv = − d

hzv xv = d

f

hzv yv = − d

zv = f

CPSC 587/687

Perspective Matrix

Tpers

1 ⎛ ⎞ ⎛d ⎜ h 0 0 0 ⎟⎜ 0 ⎜ ⎟ ⎜ 0 d 0 0 ⎜ =⎜ h ⎟ 0 0 1 0 ⎟⎜ ⎜ 0 ⎜0 ⎟ ⎜ 0 0 0 1 ⎠⎜ ⎝ ⎝

Scaling (d/h in x and y) Truncated pyramid into a Regular pyramid University of Calgary GraphicsJungle Project

0

0

1 0

0 f

(f

0 − fd

−d)

(f

−d)

0⎞ ⎟ 0⎟ 1⎟ ⎟ ⎟ 0⎟ ⎠

Regular pyramid into a box

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CPSC 587/687

Tpers

⎛d 0 ⎜ h 0 ⎜ 0 d 0 h ⎜ =⎜ f 0 0 (f −d) ⎜ ⎜⎜ 0 0 − fd (f −d) ⎝

0 ⎞⎟ 1⎟ ⎟ 1⎟ ⎟ 0 ⎟⎟ ⎠


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CPSC 587/687

Tpers

⎛d 0 ⎜ h 0 ⎜ 0 d 0 h ⎜ =⎜ f 0 0 (f −d) ⎜ ⎜⎜ 0 0 − fd (f −d) ⎝

xs = X

w

ys = Y

w

zs = Z

w

0 ⎞⎟ 1⎟ ⎟ 1⎟ ⎟ 0 ⎟⎟ ⎠

where


d X = xv h d Y = yv h fzv df Z= − (f −d) (f −d) w = zv

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OpenGL: Viewing Process, Graphics Pipeline CPSC 587/687 Scales and shifts each vertex so that all of them that lie inside the view volume will lie inside a standard cube

Provides the CT Combines two effects: Modeling transformations applied to objects + Transformation that orients and positions the camera in space University of Calgary GraphicsJungle Project

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CPSC 587/687


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OpenGL: Positioning and Aiming the Camera V U

View plane eye

-N

glMatrixMode (GL_MODELVIEW) glLoadIdentity();

glLookAt (eye.x, eye.y, eye.z, look.x, look.y, look.z, up.x, up.y, up.z);

CPSC 587/687

OpenGL: Setting the Camera (Parallel Projection)

glMatrixMode (GL_PROJECTION) glLoadIdentity();

glOrtho (left, right, bottom, top, near, far);


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CPSC 587/687

OpenGL: Setting the Camera (Perspective Projection)

glMatrixMode (GL_PROJECTION) glLoadIdentity();

glFrustum (left, right, bottom, top, near, far); more intuitive

or

gluPerspective (viewAngle, aspect, near, far);


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3D Rendering Pipeline

Local coordinate space

Object definition Modeling transformation

1.

INTRODUCTION

2.

IMAGING

3.

3D RENDERING OVERVIEW

4.

MODELING (1) shapes and transformations

5.

RENDERING (1, 2, 3, 4)


Compose scene Define view reference Define lighting View transformation

View space

Projection

Cull Clip to 3D view volume 3D screen space Projection divide

Hidden surface removal Rasterization Shading Display space

1.

Graphics Pipeline : Geometric Operations

6.

MODELING (2) Representation and Modeling

7.

ANIMATION

8.

TOPICS

Modeling transformation

Yw


( xw

yw

zw )

Compose scene Define view reference View space

Xw Zw

( xv

yv

zv )

View transformation

⎛ xv ⎞ ⎛ xw ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ yv ⎟ ⎜ yw ⎟ T = view ⎜ ⎜z ⎟ ⎟ z ⎜⎜ v ⎟⎟ ⎜⎜ w ⎟⎟ ⎝1⎠ ⎝1⎠

V U

View plane

Culling (back-face removal)

eye

-N

F is back face if

(P − eye ) dot mF > 0

Dot Product

n

d = v ⋅ w = ∑ vi wi i =1

∧ ∧

Angle between Two Vectors

cos(θ ) = b⋅ c

Sign and Perpendicularity

“perpendicular” orthogonal normal

Normal to a Plane

Finding the Normal Vectors V1

Flat Face:

m

V3

V2

m = (V1 − V2 ) × (V3 − V2 )

Two Problems:

V1

V3

V2 Vectors nearly parallel, the cross product will be very small, Numerical inaccuracies may result

Polygon is not perfect planar

Finding the Normal Vectors V1 Robust Method (Newell):

m V2

V3

N −1

mx = ∑ ( yi − ynext (i ) )(zi + znext (i ) ) i =0

N −1

m y = ∑ (zi − znext (i ) )(xi + xnext (i ) ) i =0

N −1

mz = ∑ (xi − xnext (i ) )( yi + ynext (i ) ) i =0

Vertex and Face Table

Each face lists vertex references Shared vertices Still no topology information

( x3 , y 3 , z 3 )

(x1 , y1 , z1 )

(x5 , y5 , z5 )

VERTEX TABLE V1

F3

F1

V3

F2

( x2 , y 2 , z 2 )

V2

( x4 , y 4 , z 4 )

V4 V5

(x1 , y1 , z1 ) ( x2 , y 2 , z 2 ) ( x3 , y 3 , z 3 ) ( x4 , y 4 , z 4 ) (x5 , y5 , z5 )

FACE TABLE F1

V1 V2 V3

F2

V3 V2 V4

F2

V3 V4 V5

Mesh pt

numVerts

Point3 x, y, z

norm

numNormals

face

Face numFaces

vert

Vector3 dx, dy, dz

nVerts

VertexID vertIndex normIndex

V U

View plane

Culling (back-face removal)

eye

-N

F is back face if

(P − eye ) dot mF > 0

void Mesh :: draw () { for (int f = 0; f < numFaces; f++) { glBegin (GL_POLYGON) for (int v = 0; v < face[f].nVerts; v++) { int iv = face[f].vert[v].vertIndex; glVertex3f(pt[iv].x, pt[iv].y, pt[iv].z); } glEnd(); } }

void Mesh :: draw () { for (int f = 0; f < numFaces; f++) {

If ( isBackFace ( f, eye ) ) continue; glBegin (GL_POLYGON) for (int v = 0; v < face[f].nVerts; v++) { int iv = face[f].vert[v].vertIndex; glVertex3f(pt[iv].x, pt[iv].y, pt[iv].z); } glEnd(); } }