Welcome to CSc 830 Advanced Computer Graphics By Ilmi Yoon Based on Lecture note from David Luebke & Pradondet Nilagupta

Welcome toCSc 830 Advanced Computer Graphics

By Ilmi Yoon

Based on Lecture note from David Luebke & Pradondet Nilagupta

Where We’re GoingToday’s lectures:

Mathematical Foundations The graphics pipeline: the big picture Rigid-body transforms Homogeneous coordinates The viewing transform The projection transform

Mathematical Foundations

FvD appendix gives good reviewI’ll give a brief, informal review of some of the

mathematical tools we’ll employ Geometry (2D, 3D) Trigonometry Vector and affine spaces

• Points, vectors, and coordinates Dot and cross products Linear transforms and matrices

2D GeometryKnow your high-school geometry:

Total angle around a circle is 360° or 2π radians When two lines cross:

• Opposite angles are equivalent

• Angles along line sum to 180° Similar triangles:

• All corresponding angles are equivalent

• Corresponding pairs of sides have the same length ratio and are separated by equivalent angles

• Any corresponding pairs of sides have same length ratio

Trigonometry

Sine: “opposite over hypotenuse”Cosine: “adjacent over hypotenuse”Tangent: “opposite over adjacent”Unit circle definitions:

sin () = x cos () = y tan () = x/y Etc…

(x, y)

3D GeometryTo model, animate, and render 3D scenes, we

must specify: Location Displacement from arbitrary locations Orientation

We’ll look at two types of spaces: Vector spaces Affine spaces

We will often be sloppy about the distinction

Vector SpacesTwo types of elements:

Scalars (real numbers): … Vectors (n-tuples): u, v, w, …

Supports two operations: Addition operation u + v, with:

• Identity 0 v + 0 = v• Inverse - v + (-v) = 0

Scalar multiplication:• Distributive rule: (u + v) = (u) + (v)

( + )u = u + u

Vector SpacesA linear combination of vectors results in a new vector:

v = 1v1 + 2v2 + … + nvn

If the only set of scalars such that

1v1 + 2v2 + … + nvn = 0

is 1 = 2 = … = 3 = 0

then we say the vectors are linearly independent The dimension of a space is the greatest number of linearly

independent vectors possible in a vector set For a vector space of dimension n, any set of n linearly

independent vectors form a basis

Vector Spaces: A Familiar ExampleOur common notion of vectors in a 2D plane is

(you guessed it) a vector space: Vectors are “arrows” rooted at the origin Scalar multiplication “streches” the arrow, changing

its length (magnitude) but not its direction Addition uses the “trapezoid rule”:

u+vy

xu

v

Vector Spaces: Basis Vectors

Given a basis for a vector space: Each vector in the space is a unique linear

combination of the basis vectors The coordinates of a vector are the scalars from

this linear combination Best-known example: Cartesian coordinates

• Draw example on the board Note that a given vector v will have different

coordinates for different bases

Vectors And PointWe commonly use vectors to represent:

Points in space (i.e., location) Displacements from point to point Direction (i.e., orientation)

But we want points and directions to behave differently Ex: To translate something means to move it without

changing its orientation Translation of a point = different point Translation of a direction = same direction

Affine SpacesTo be more rigorous, we need an explicit notion of

position Affine spaces add a third element to vector spaces:

points (P, Q, R, …)Points support these operations

Point-point subtraction: Q - P = v• Result is a vector pointing from P to Q

Vector-point addition: P + v = Q• Result is a new point

Note that the addition of two points is not definedP

Q

v

Affine SpacesPoints, like vectors, can be expressed in

coordinates The definition uses an affine combination Net effect is same: expressing a point in terms of a

basis

Thus the common practice of representing points as vectors with coordinates (see FvD)

Analogous to equating points and integers in C Be careful to avoid nonsensical operations

Affine Lines: An Aside

Parametric representation of a line with a direction vector d and a point P1 on the line:

P() = Porigin + d

Restricting 0 produces a raySetting d to P - Q and restricting 0 1

produces a line segment between P and Q

Dot ProductThe dot product or, more generally, inner product of

two vectors is a scalar:v1 • v2 = x1x2 + y1y2 + z1z2 (in 3D)

Useful for many purposes Computing the length of a vector: length(v) = sqrt(v • v)

Normalizing a vector, making it unit-length Computing the angle between two vectors:

u • v = |u| |v| cos(θ) Checking two vectors for orthogonality Projecting one vector onto another

θu

v

Cross Product

The cross product or vector product of two vectors is a vector:

The cross product of two vectors is orthogonal to both

Right-hand rule dictates direction of cross product

1221

1221

1221

21 )(

y x y x

z x z x

z y z y

vv

Linear Transformations

A linear transformation: Maps one vector to another Preserves linear combinations

Thus behavior of linear transformation is completely determined by what it does to a basis

Turns out any linear transform can be represented by a matrix

Matrices

By convention, matrix element Mrc is located at row r and column c:

By (OpenGL) convention, vectors are columns:

mnm2m1

2n2221

1n1211

MMM

MMM

MMM

M

3

2

v

v

v

v

1

Matrices

Matrix-vector multiplication applies a linear transformation to a vector:

Recall how to do matrix multiplication

z

y

x

v

v

v

MMM

MMM

MMM

vM

333231

232221

131211

Matrix Transformations

A sequence or composition of linear transformations corresponds to the product of the corresponding matrices Note: the matrices to the right affect vector first Note: order of matrices matters!

The identity matrix I has no effect in multiplication

Some (not all) matrices have an inverse:

vvMM 1

3-D Graphics: A Whirlwind Tour

Transform

Illuminate

Transform

Clip

Project

Rasterize

Model & CameraModel & CameraParametersParameters Rendering PipelineRendering Pipeline FramebufferFramebuffer DisplayDisplay

The Display You Know

Transform

Illuminate

Transform

Clip

Project

Rasterize


The Framebuffer You Know

Transform

Illuminate

Transform

Clip

Project

Rasterize


The Rendering Pipeline

Transform

Illuminate

Transform

Clip

Project

Rasterize


Why do we call it a pipeline?

2-D Rendering You Know

Transform

Illuminate

Transform

Clip

Project

Rasterize


The Rendering Pipeline: 3-D

Transform

Illuminate

Transform

Clip

Project

Rasterize



ModelingTransforms

Scene graphObject geometry

LightingCalculations

ViewingTransform

Clipping

ProjectionTransform

Result:Result:

• All vertices of scene in shared 3-D “world” coordinate All vertices of scene in shared 3-D “world” coordinate systemsystem

• Vertices shaded according to lighting modelVertices shaded according to lighting model

• Scene vertices in 3-D “view” or “camera” coordinate Scene vertices in 3-D “view” or “camera” coordinate systemsystem

• Exactly those vertices & portions of polygons in view Exactly those vertices & portions of polygons in view frustumfrustum

• 2-D screen coordinates of clipped vertices2-D screen coordinates of clipped vertices

The Rendering Pipeline: 3-DScene graph

Object geometry


Clipping

Result:Result:






ModelingTransforms

ViewingTransform

ProjectionTransform

Rendering: Transformations

So far, discussion has been in screen spaceBut model is stored in model space

(a.k.a. object space or world space)Three sets of geometric transformations:

Modeling transforms Viewing transforms Projection transforms


Modeling transforms Size, place, scale, and rotate objects parts of the

model w.r.t. each other Object coordinates world coordinates

Z

X

Y

X

Z

Y


Viewing transform Rotate & translate the world to lie directly in

front of the camera• Typically place camera at origin

• Typically looking down -Z axis World coordinates view coordinates


Projection transform Apply perspective foreshortening

• Distant = small: the pinhole camera model View coordinates screen coordinates

Rendering: TransformationsAll these transformations involve shifting

coordinate systems (i.e., basis sets)That’s what matrices do…Represent coordinates as vectors, transforms as

matrices

Multiply matrices = concatenate transforms!

y

x

y

x

cossin

sincos

'

'


Homogeneous coordinates: represent coordinates in 3 dimensions with a 4-vector Denoted [x, y, z, w]

• Note that typically w = 1 in model coordinates To get 3-D coordinates, divide by w:

[x’, y’, z’] = [x/w, y/w, z/w]

Transformations are 4x4 matricesWhy? To handle translation and projection


OpenGL caveats: All modeling transforms and the viewing

transform are concatenated into the modelview matrix

A stack of modelview matrices is kept The projection transform is stored separately in

the projection matrix

See Chapter 3 of the OpenGL book


ModelingTransforms



ViewingTransform

Clipping

ProjectionTransform

Result:Result:






Rendering: Lighting

Illuminating a scene: coloring pixels according to some approximation of lighting Global illumination: solves for lighting of the

whole scene at once Local illumination: local approximation,

typically lighting each polygon separately

Interactive graphics (e.g., hardware) does only local illumination at run time


ModelingTransforms



ViewingTransform

Clipping

ProjectionTransform

Result:Result:






Rendering: Clipping

Clipping a 3-D primitive returns its intersection with the view frustum:

See Foley & van Dam section 19.1

Rendering: Clipping

Clipping is tricky!

In: 3 verticesIn: 3 verticesOut: 6 verticesOut: 6 vertices

Clip

Clip In: 1 polygonIn: 1 polygonOut: 2 polygonsOut: 2 polygons


Transform

Illuminate

Transform

Clip

Project

Rasterize


Modeling: The Basics

Common interactive 3-D primitives: points, lines, polygons (i.e., triangles)

Organized into objects Collection of primitives, other objects Associated matrix for transformations

Instancing: using same geometry for multiple objects 4 wheels on a car, 2 arms on a robot

Modeling: The Scene Graph

The scene graph captures transformations and object-object relationships in a DAG

Objects in black; blue arrows indicate instancing and each have a matrix

Robot

BodyHead

ArmTrunkLegEyeMouth

Modeling: The Scene Graph

Traverse the scene graph in depth-first order, concatenating transformations

Maintain a matrix stack of transformations

ArmTrunkLegEyeMouth

Head Body

Robot

Foot

MatrixMatrixStackStack

VisitedVisited

UnvisitedUnvisited

ActiveActive

Modeling: The Camera

Finally: need a model of the virtual camera Can be very sophisticated

• Field of view, depth of field, distortion, chromatic aberration…

Interactive graphics (OpenGL):• Pinhole camera model

– Field of view

– Aspect ratio

– Near & far clipping planes

• Camera pose: position & orientation

Modeling: The Camera

These parameters are encapsulated in a projection matrix Homogeneous coordinates 4x4 matrix! See OpenGL Appendix F for the matrix

The projection matrix premultiplies the viewing matrix, which premultiplies the modeling matrices Actually, OpenGL lumps viewing and modeling

transforms into modelview matrix

Rigid-Body Transforms

Goal: object coordinatesworld coordinates

Idea: use only transformations that preserve the shape of the object Rigid-body or Euclidean transforms Includes rotation, translation, and scale

To reiterate: we will represent points as column vectors:

z

y

x

zyx ),,(

Vectors and Matrices

Vector algebra operations can be expressed in this matrix form Dot product:

Cross product:• Note: use

right-handrule!

z

y

x

zyx

b

b

b

aaaba

0cb

0ca

cba

z

y

x

z

y

x

xy

xz

yz

c

c

c

b

b

b

aa

aa

aa

0

0

0

TranslationsFor convenience we usually describe objects in relation to

their own coordinate system Solar system example

We can translate or move points to a new position by adding offsets to their coordinates:

Note that this translates all points uniformly

z

y

x

t

t

t

z

y

x

z

y

x

'

'

'

2-D Rotation

(x, y)

(x’, y’)

x’ = x cos() - y sin()y’ = x sin() + y cos()

(Draw it)

2-D Rotations

Rotations in 2-D are easy:

3-D is more complicated Need to specify an axis of rotation Common pedagogy: express rotation about this axis

as the composition of canonical rotations• Canonical rotations: rotation about X-axis, Y-axis, Z-axis

y

x

y

x

cossin

sincos

'

'

3-D RotationsBasic idea:

Using rotations about X, Y, Z axes, rotate model until desired axis of rotation coincides with Z-axis

Perform rotation in the X-Y plane (i.e., about Z-axis) Reverse the initial rotations to get back into the initial

frame of reference

Objections: Difficult & error prone Ambiguous: several combinations about the canonical

axis give the same result

ScalingScaling a coordinate means multiplying each of

its components by a scalarUniform scaling means this scalar is the same for

all components:

2

Scaling

Non-uniform scaling: different scalars per component:

How can we represent this in matrix form?

X 2,Y 0.5

Scaling

Scaling operation:

Or, in matrix form:

cz

by

ax

z

y

x

'

'

'

z

y

x

c

b

a

z

y

x

00

00

00

'

'

'

scaling matrix

2-D Rotation

This is easy to capture in matrix form:

3-D is more complicated Need to specify an axis of rotation Simple cases: rotation about X, Y, Z axes

y

x

y

x

cossin

sincos

'

'

3-D Rotation

What does the 3-D rotation matrix look like for a rotation about the Z-axis? Build it coordinate-by-coordinate

z

y

x

z

y

x

100

0)cos()sin(

0)sin()cos(

'

'

'

3-D Rotation

What does the 3-D rotation matrix look like for a rotation about the Y-axis? Build it coordinate-by-coordinate

z

y

x

z

y

x

)cos(0)sin(

010

)sin(0)cos(

'

'

'

3-D Rotation

What does the 3-D rotation matrix look like for a rotation about the X-axis? Build it coordinate-by-coordinate

z

y

x

z

y

x

)cos()sin(0

)sin()cos(0

001

'

'

'

3-D Rotation

General rotations in 3-D require rotating about an arbitrary axis of rotation

Deriving the rotation matrix for such a rotation directly is difficult But possible, see McMillan’s lectures

Standard approach: express general rotation as composition of canonical rotations Rotations about X, Y, Z

Composing Canonical Rotations

Goal: rotate about arbitrary vector A by Idea: we know how to rotate about X,Y,Z So, rotate about Y by until A lies in the YZ

plane, Then rotate about X by until A coincides with +Z, Then rotate about Z by

Then reverse the rotation about X (by -) Then reverse the rotation about Y (by -)

Composing Canonical RotationsFirst: rotating about Y by until A lies in YZ

Draw it…

How exactly do we calculate ? Project A onto XZ plane Find angle to X:

= -(90° - ) = - 90 °

Second: rotating about X by until A lies on ZHow do we calculate ?

Composing Canonical Rotations

Why are we slogging through all this tedium?

A: Because you’ll have to do it on the test

3-D Rotation Matrices

So an arbitrary rotation about A composites several canonical rotations together

We can express each rotation as a matrixCompositing transforms == multiplying matricesThus we can express the final rotation as the

product of canonical rotation matricesThus we can express the final rotation with a

single matrix!

Compositing Matrices

So we have the following matrices:

p: The point to be rotated about A by Ry : Rotate about Y by

Rx : Rotate about X by

Rz : Rotate about Z by

Rx -1: Undo rotation about X by

Ry-1

: Undo rotation about Y by In what order should we multiply them?

Compositing Matrices

Short answer: the transformations, in order, are written from right to left In other words, the first matrix to affect the

vector goes next to the vector, the second next to the first, etc.

So in our case:

p’ = Ry-1 Rx

-1 Rz Rx Ry p

Rotation Matrices

Notice these two matrices:



How can we calculate Rx -1?

Rotation Matrices





Obvious answer: calculate Rx (-)

Clever answer: exploit fact that rotation matrices are orthonormal

Rotation Matrices





Obvious answer: calculate Rx (-)

Clever answer: exploit fact that rotation matrices are orthonormal

What is an orthonormal matrix?What property are we talking about?

Rotation Matrices

Orthonormal matrix: orthogonal (columns/rows linearly

independent) Columns/rows sum to 1

The inverse of an orthogonal matrix is just its transpose:

jfc

ieb

hda

jih

fed

cba

jih

fed

cbaT1

Translation Matrices?

We can composite scale matrices just as we did rotation matrices

But how to represent translation as a matrix?

Answer: with homogeneous coordinates

Homogeneous Coordinates

Homogeneous coordinates: represent coordinates in 3 dimensions with a 4-vector

(Note that typically w = 1 in object coordinates)

w

z

y

x

wz

wy

wx

zyx

1

/

/

/

),,(


Homogeneous coordinates seem unintuitive, but they make graphics operations much easier

Our transformation matrices are now 4x4:

1000

0)cos()sin(0

0)sin()cos(0

0001

xR




1000

0)cos(0)sin(

0010

0)sin(0)cos(

yR




1000

0100

00)cos()sin(

00)sin()cos(

zR




1000

000

000

000

z

y

x

S

S

S

S


How can we represent translation as a 4x4 matrix?

A: Using the rightmost column:

1000

000

000

000

z

y

x

T

T

T

T

Documents

Welcome to CSc 830 Advanced Computer Graphics By Ilmi Yoon Based on Lecture note from David Luebke & Pradondet Nilagupta