Computer Graphics

CSE167: Computer Graphics

Instructor: Ronen Barzel

UCSD, Winter 2006

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CSE167 Computer Graphics Instructor: Ronen Barzel (ronen@graphics.ucsd.edu)

TAs: Alex Kozlowski (kman@graphics.ucsd.edu) Cameron Chrisman (cchrisman@cs.ucsd.edu)

Lecture: H&SS 1330 (TTh 8:00am-9:20am) Section: TBD Office: EBU3B 2104 (T 9:45-10:45am; Th 10:30-11:30am)

Web page: http://graphics.ucsd.edu/courses/cse167_w06

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What is computer graphics?

Large field… 3D Modeling Image Synthesis (AKA Rendering) Animation Simulation Interaction Image Processing Vision Perception Hardware design Etc…

Way too much to cover in one quarter

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What you’ll learn in this class

The Basics… How 3D models are defined

• Polygonal models• Smooth surfaces

How to render 3D models• Camera simulation• Lighting, shading

(Won’t make pretty pictures – not an art class)

Experience with Linear Algebra, C++, OpenGL

Basis for going on to advanced topics, game development, etc.

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Prerequisites

Basic Familiarity with: Linear Algebra

• Vectors: dot products, cross products…• Matrices : matrix multiplication

C++ • (if you know Java you’ll be able to adapt)

Object oriented programming

You’ll get more comfortable with them. (And I’ll review Vectors & Matrices today)

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Reading

3D Computer Graphics: A Mathematical Introduction with OpenGL (Buss)

Required pages are listed on the class web page

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Programming ProjectsNOTE: Details of this may change. Check the web page for

updates Projects due every Tuesday Project 1: Matrices and basic graphics program Project 2: Matrix stack and hierarchical modeling Project 3: Scene graph & traversal Project 4: Bézier Curves & Surfaces Project 5: Camera Perspective & Clipping Project 6: Scan Conversion & Z-Buffer Project 7: Ray Casting & Lighting Final Project (due last day of class): Choose one of

the following Water fountain (particle system) Ray tracer Procedural tree Choose your own project (but talk to me first)

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Tests

Midterm (in class) Thurs, Feb 9, 8:00am-9:20am H&SS 1330

Final Date & Time: TBD Location: TBD

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Grading

5% Project 1 5% Project 2 8% Project 3 8% Project 4 8% Project 5 8% Project 6 8% Project 7 15% Final Project 15% Midterm 20% Final

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About me

Pixar animation studios (12 years) Toy Story

•modeling, lighting, software Software tools development

Research Physically-based modeling Non-photorealistic rendering

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Course Outline Introduction / Linear Algebra Review Geometry and Homogeneous Coordinates Hierarchical Transforms / Geometrical Calculations TA Lecture: C++, OpenGL, project implementation Hierarchical Modeling Cubic Curves Curved Surfaces Perspective & Viewing Clipping & Scan Conversion Lighting Ray Tracing Texture Mapping Antialiasing Advanced Topics, Guest Lecturers TBD Review

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Outline for Today

1. Overview of the class2. Fundamental concepts of

graphics3. Linear Algebra Review

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Fundamental concepts of CG

Modeling

Rendering

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Modeling Creating 3D geometric data, AKA the “model” or the “scene”

By hand, often with an interactive editing program• Maya, Blender, AutoCAD, LightWave 3D, …

Procedurally, i.e. by writing programs Scanning real-world objects

Manipulating the data Deforming or editing the data

• Change this over time: you have animation! Converting one form into another.

• Convert higher order descriptions into things we know how to render

• Convert complex objects into simpler ones that we can render more quickly

Makes heavy use of differential geometry and computational geometry

Ultimately, typically defined as or reduced to polygonal models: Triangles Quadrilaterals Meshes

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Modeling

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

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Modeling Primitives

Complex scenes: Usually built up from simpler objects Objects are built from individual primitives

Most fundamental and useful: 3D triangle Points and lines are also useful primitives

May have more complex “primitives”:•Polygons, spheres, curves, curved surfaces, …•Often automatically tessellated into triangles before rendering.

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Rendering

Fancier term: Image Synthesis Synthesis of a 2D image from a 3D scene description

Result is a 2D array of pixels Red, Green, Blue values (range 0-255 or 0.0-1.0)

Can also have: opacity (“alpha”), depth (“Z”), …

Rendering style often classified as “Photorealistic”

• simulate light & camera “Nonphotorealistic” AKA stylized

• aesthetics, communication

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Photoreal Rendering

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QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Non-Photoreal Rendering

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CSE167 Rendering :)

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Rendering a scene

Define a “virtual camera” Tells renderer where to look in the scene

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

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Rendering a Scene, cont’d.

Specify “virtual lights” (and shadows):

Specify “shaders” for model surfaces:

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

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Hardware vs. Software Rendering

Highest quality rendering is done by software Algorithms such as “ray tracing”, “photon maps”, etc… Fanciest lighting, shadowing, surface shading, smoke &

fog, etc. Can take minutes or hours to compute an image RenderMan (Pixar), Dali (Henrik Wann Jensen), RADIANCE,

POVRay, …

Modern computers often have special-purpose 3D rendering hardware. “GPU” == Graphics Processing Unit. (Nvidia, ATI) Hardware implements the traditional 3D graphics

rendering pipeline Very fast, but relatively simple algorithm:

• Limits ability to get subtle shadows, reflections, etc.• Limits on complexity of surface shading description, etc.

Continually improving, driven by games industry.

(Modern graphics hardware is programmable, blurring the distinction between hardware & software rendering.)

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OpenGL An API for rendering

Widely supported on many platforms Provides a standard interface to (platform-specific)

hardware. Other APIs: Java3D, Direct3D, …

Example–render a red triangle (this doesn’t include the ‘setup’ code, or camera or lights):glBegin(GL_TRIANGLES);glColor3f(1.0, 0.0, 0.0); // redglVertex3f(-4.0, -2.0, 0.0);glVertex3f(4.0, -2.0, 0.0);glVertex3f(0.0, 5.0, 0.0);glEnd();

Programming projects will use OpenGL. Lectures won’t focus on OpenGL Bulk of material is about the theory and algorithms (See Buss, ch. 1 for an intro on rendering triangles in

OpenGL.)

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3-D Graphics Rendering Pipeline

Modeling Transformation

Viewing Transformation

Lighting & Shading

Clipping

Projection

Scan conversion, Hiding

Primitives

Image

Object coordinates

World space

Eye space

Normalized view space

Device coordinates

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Outline for Today

1. Overview of the class2. Fundamental concepts of

graphics3. Linear Algebra Review

• Vectors• Matrices

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Coordinate Systems

Right handed coordinate systems

(more on coordinate systems next class)

X

Y

Z

X

Y

Z

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Vector Arithmetic

a =ax

ay

az

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

b =bx

by

bz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

a + b =ax + bx

ay + by

az + bz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

a −b =ax −bx

ay −by

az −bz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

sa =sax

say

saz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

−a =

−ax

−ay

−az

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

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Vector Magnitude

The magnitude (length) of a vector is:

A vector with length=1.0 is called a unit vector

We can also normalize a vector to make it a unit vector:

v2 =vx

2 + vy2 + vz

2

v = vx2 + vy

2 + vz2

v

v

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Dot Product

a ⋅b = aibi∑a⋅b =axbx + ayby + azbz

a⋅b = a b cosθ

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Dot Product

a ⋅b = aibi∑a⋅b =axbx + ayby + azbz

a⋅b = ax ay az⎡⎣ ⎤⎦

bx

by

bz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

a⋅b =aTb

a⋅b = a b cosθ

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Example: Angle Between Vectors

How do you find the angle θ between vectors a and b?

a

b θ

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Example: Angle Between Vectors

a

b θ

a ⋅b = a b cosθ

cosθ =a⋅ba b

⎛

⎝⎜⎞

⎠⎟

θ =cos−1 a⋅ba b

⎛

⎝⎜⎞

⎠⎟

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Dot Product Properties

The dot product is a scalar value that tells us something about the relationship between two vectors If a·b > 0 then θ < 90º

•Vectors point in the same general direction

If a·b < 0 then θ > 90º•Vectors point in opposite direction

If a·b = 0 then θ = 90º •Vectors are perpendicular•(or one or both of the vectors is degenerate (0,0,0))

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Dot Products with One Unit Vector

a

u

a·u

If |u|=1.0 then a·u is the length of the projection of a onto u

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Dot Products with Unit Vectors

bθ a

a·b = 00 < a·b < 1

a·b = -1

a·b = 1

-1 < a·b < 0 a·b a = b =1.0

a⋅b =cos θ( )

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Cross Product

a ×b =i j kax ay az

bx by bz

a×b = aybz −azby azbx −axbz axby −aybx⎡⎣ ⎤⎦

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Properties of the Cross Product

area of parallelogram ab

is a vector perpendicular to both a and b, in the direction defined by the right hand rule

if a and b are parallel(or one or both degenerate)

a ×b

a×b = a b sinθa×b =

a×b =0

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Example: Align two vectors

We are heading in direction h. We want to rotate so that we will align with a different direction d. Find a unit axis a and an angle θ to rotate around.

h

d

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Example: Align two vectors

h

d

θ

a

a =h×dh×d

θ =sin−1 h×dh d

⎛

⎝⎜⎞

⎠⎟

θ =cos−1 h⋅dh d

⎛

⎝⎜⎞

⎠⎟

θ =tan−1 h×dh⋅d

⎛⎝⎜

⎞⎠⎟

t het a =at an2 h×d , h⋅d( )

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Vector Class

class Vector3 {public:

Vector3() {x=0.0; y=0.0; z=0.0;}Vector3(float x0,float y0,float z0) {x=x0; y=y0; z=z0;}void Set(float x0,float y0,float z0) {x=x0; y=y0; z=z0;}void Add(Vector3 &a) {x+=a.x; y+=a.y; z+=a.z;}void Add(Vector3 &a,Vector3 &b) {x=a.x+b.x; y=a.y+b.y; z=a.z+b.z;}void Subtract(Vector3 &a) {x-=a.x; y-=a.y; z-=a.z;}void Subtract(Vector3 &a,Vector3 &b) {x=a.x-b.x; y=a.y-b.y; z=a.z-b.z;}void Negate() {x=-x; y=-y; z=-z;}void Negate(Vector3 &a) {x=-a.x; y=-a.y; z=-a.z;}void Scale(float s) {x*=s; y*=s; z*=s;}void Scale(float s,Vector3 &a) {x=s*a.x; y=s*a.y; z=s*a.z;}float Dot(Vector3 &a) {return x*a.x+y*a.y+z*a.z;}void Cross(Vector3 &a,Vector3 &b)

{x=a.y*b.z-a.z*b.y; y=a.z*b.x-a.x*b.z; z=a.x*b.y-a.y*b.x;}float Magnitude() {return sqrtf(x*x+y*y+z*z);}void Normalize() {Scale(1.0f/Magnitude());}

float x,y,z;};

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Matrix Arithmetic

Multiply a matrix by a vector:

Each entry is dot product of row of M with v

M =mxx mxy mxz

myx myy myz

mzx mzy mzz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥             v =

vx

vy

vz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Mv( )i = mij∑ vj

Mv =

mxj∑ vj

myj∑ vj

mzj∑ vj

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Mv =mxxvx + mxyvy + mxzvz

myxvx + myyvy + myzvz

mzxvx + mzyvy + mzzvz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

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Identity

Multiplication by the identity matrix does not affect the vector

I =1 0 00 1 00 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

I v =1 0 00 1 00 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

vx

vy

vz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

1* vx + 0 * vy + 0 * vz

0 * vx +1* vy + 0 * vz

0 * vx + 0 * vy +1* vz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

vx

vy

vz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=v

    

             v=I v

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A diagonal matrix scales a vector

If s>1, the vector will grow by a factor of s If s=1, the vector won’t change If 0<s<1, the vector will shrink If s<0, the vector will point in the opposite

direction

Uniform Scaling

S =s 0 00 s 00 0 s

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

S v =s 0 00 s 00 0 s

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

vx

vy

vz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

s* vx + 0 * vy + 0 * vz

0 * vx + s* vy + 0 * vz

0 * vx + 0 * vy + s* vz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

svx

svy

svz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=s

vx

vy

vz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

sv =S v

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Each dimension has its own scale factor

which leads to the equations:

Scaling by 0 in a dimension projects onto that plane

Scaling by -1 in a dimension reflects across that plane

Non-Uniform Scaling

′vx′vy′vz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

sx 0 0

0 sy 0

0 0 sz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥ 

vxvyvz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

′vx = sxvx′vy = syvy′vz = szvz

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Rotation

Rotate in the xy plane by an angle θ, spinning around the z axis:

a positive angle will rotate counterclockwise when the rotation axis is pointing towards the observer

Right-hand rule Food for thought:

What is the matrix when θ is 0? 90 degrees? 180 degrees? How does a rotation by -θ compare to rotation by θ?

Rz (θ) =cos θ( ) −sin θ( ) 0sin θ( ) cos θ( ) 0

0 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

′v =R z(θ)  v

′v =cos θ( )vx −sin θ( )vy

sin θ( )vx + cos θ( )vy

vz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

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Rotation about coordinate axes

We can define rotation matrices for each axis:

Rx θ( ) =1 0 00 cosθ −sinθ0 sinθ cosθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Rx θ( ) =1 0 00 cosθ −sinθ0 sinθ cosθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

R z θ( ) =cosθ −sinθ 0sinθ cosθ 00 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

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Rotation about arbitrary axis

This can be derived…

Note: a must be a unit vector!

Same right-hand rule applies for direction of rotation

R(a,θ) =1+ (1−cos(θ))(ax

2 −1) −azsin(θ) + (1−cos(θ))axay aysin(θ) + (1−cos(θ))axaz

azsin(θ) + (1−cos(θ))ayax 1+ (1−cos(θ))(ay2 −1) −axsin(θ) + (1−cos(θ))ayaz

−aysin(θ) + (1−cos(θ))azax axsin(θ) + (1−cos(θ))azay 1+ (1−cos(θ))(az2 −1)

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

a =1

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Next class

Finish up linear algebra review: Matrix Composition

Points Moving points around “Homogeneous Coordinates”