CS 445: Introduction to Computer Graphics

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Display Technologies, Mathematical Fundamentals

• CS 445 Introduction to Computer Graphics
• David Luebke
• University of Virginia

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Admin

• Call roll
• Introductions Sam Guarnieri
• Office hours
• M 11-12 am
• W 10-11 am
• Assignment 1 out

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Demo

• Soap-bubble bunny

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Mathematical Foundations

• A very brief review of some mathematical tools
  well employ
• Geometry (2D, 3D)
• Trigonometry
• Vector and affine spaces
• Points, vectors, and coordinates
• Dot and cross products
• Linear transforms and matrices

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3D Geometry

• To model, animate, and render 3D scenes, we must
  specify
• Location
• Displacement from arbitrary locations
• Orientation
• Well look at two types of spaces
• Vector spaces
• Affine spaces
• We will often be sloppy about the distinction

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Vector Spaces

• 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
• Note that a given vector will have different
  coordinates for different bases

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Vectors And Points

• We commonly use vectors to represent
• Direction (i.e., orientation)
• Points in space (i.e., location)
• Displacements from point to point
• 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

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Affine Spaces

• To 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
• P 0 P
• Note that the addition of two points is not
  defined

Q
v
P
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Affine Spaces

• Points, 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
• Be careful to avoid nonsensical operations
• Point point
• Scalar point

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Affine Lines An Aside

• Parametric representation of a line with a
  direction vector d and a point P1 on the line
• P(a) Porigin ad
• Restricting 0 ? a produces a ray
• Setting d to P - Q and restricting 0 ? a ? 1
  produces a line segment between P and Q

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Dot Product

• The 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(?)
• Checking two vectors for orthogonality
• Projecting one vector onto another

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Cross Product

• The cross product or vector product of two
  vectors is a vector
• Cross product of two vectors is orthogonal to
  both
• Right-hand rule dictates direction of cross
  product
• Cross product is handy for finding surface
  orientation
• Lighting
• Visibility

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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

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Matrices

• By convention, matrix element Mrc is located at
  row r and column c
• By (OpenGL) convention, vectors are columns

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Matrices

• Matrix-vector multiplication applies a linear
  transformation to a vector
• Recall how to do matrix multiplication

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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

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3D Scene Representation

• Scene is usually approximated by 3D primitives
• Point
• Line segment
• Polygon
• Polyhedron
• Curved surface
• Solid object
• etc.

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3D Point

• Specifies a location

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3D Point

• Specifies a location
• Represented by three coordinates
• Infinitely small

typedef struct Coordinate x Coordinate
y Coordinate z Point
(x,y,z)
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3D Vector

• Specifies a direction and a magnitude

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3D Vector

• Specifies a direction and a magnitude
• Represented by three coordinates
• Magnitude V sqrt(dx dx dy dy dz dz)
• Has no location

(dx,dy,dz)
typedef struct Coordinate dx Coordinate
dy Coordinate dz Vector
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3D Vector

• Specifies a direction and a magnitude
• Represented by three coordinates
• Magnitude V sqrt(dx dx dy dy dz dz)
• Has no location
• Dot product of two 3D vectors
• V1V2 dx1dx2 dy1dy2 dz1dz2
• V1V2 V1 V2 cos(Q)

(dx1,dy1,dz1)
typedef struct Coordinate dx Coordinate
dy Coordinate dz Vector
(dx2,dy2 ,dz2)
Q
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3D Line Segment

• Linear path between two points

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3D Line Segment

• Use a linear combination of two points
• Parametric representation
• P P1 t (P2 - P1), (0 ? t ? 1)

typedef struct Point P1 Point P2 Segment
P2
P1
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3D Ray

• Line segment with one endpoint at infinity
• Parametric representation
• P P1 t V, (0 lt t lt ?)

typedef struct Point P1 Vector V Ray
V
P1
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3D Line

• Line segment with both endpoints at infinity
• Parametric representation
• P P1 t V, (-? lt t lt ?)

typedef struct Point P1 Vector V Line
V
P1
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3D Plane

• A linear combination of three points

P3
P2
P1
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3D Plane

• A linear combination of three points
• Implicit representation
• PN d 0, or
• ax by cz d 0
• N is the plane normal
• Unit-length vector
• Perpendicular to plane

N (a,b,c)
typedef struct Vector N Distance d Plane
P3
P2
P1
d
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3D Polygon

• Area inside a sequence of coplanar points
• Triangle
• Quadrilateral
• Convex
• Star-shaped
• Concave
• Self-intersecting
• Holes (use gt 1 polygon struct)

typedef struct Point points int npoints
Polygon
Points are in counter-clockwise order
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3D Sphere

• All points at distance r from point (cx, cy,
  cz)
• Implicit representation
• (x - cx)2 (y - cy)2 (z - cz)2 r 2
• Parametric representation
• x r cos(?) cos(?) cx
• y r cos(?) sin(?) cy
• z r sin(?) cz

typedef struct Point center Distance
radius Sphere
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Display Technologies

• Cathode Ray Tubes (CRTs)
• Most common display device today
• Evacuated glass bottle (lastof the vacuum tubes)
• Heating element (filament)
• Electrons pulled towards anode focusing cylinder
• Vertical and horizontal deflection plates
• Beam strikes phosphor coating on front of tube

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Display Technologies CRTs

• Vector Displays
• My childhood Battlezone, Tempest

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Display Technologies CRTs

• Vector displays
• Early computer displays basically an
  oscilloscope
• Control X,Y with vertical/horizontal plate
  voltage
• Often used intensity as Z
• Show http//graphics.lcs.mit.edu/classes/6.837/F9
  8/Lecture1/Slide11.html
• Name two disadvantages
• Just does wireframe
• Complex scenes ? visible flicker

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Display Technologies CRTs

• Black and white television an oscilloscope with
  a fixed scan pattern left to right, top to
  bottom
• Paint entire screen 30 times/sec
• Actually, TVs paint top-to-bottom 60 times/sec,
  alternating between even and odd scanlines
• This is called interlacing. Its a hack. Why do
  it?
• To paint the screen, computer needs to
  synchronize with the scanning pattern of raster
• Solution special memory to buffer image with
  scan-out synchronous to the raster. We call this
  the framebuffer.

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Display Technologies CRTs

• Raster Displays
• Raster A rectangular array of points or dots
• Pixel One dot or picture element of the raster
• Scanline A row of pixels
• Rasterize find the set of pixels corresponding
  to a 2D shape (line, circle, polygon)

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Display Technologies CRTs

• Raster Displays
• Frame must be refreshed to draw new images
• As new pixels are struck by electron beam, others
  are decaying
• Electron beam must hit all pixels frequently to
  eliminate flicker
• Critical fusion frequency
• Typically 60 times/sec
• Varies with intensity, individuals, phosphor
  persistence, lighting...

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Display Technology Color CRTs

• Color CRTs are much more complicated
• Requires manufacturing very precise geometry
• Uses a pattern of color phosphors on the screen
• Why red, green, and blue phosphors?

Delta electron gun arrangement
In-line electron gun arrangement
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Display Technology Color CRTs

• Color CRTs have
• Three electron guns
• A metal shadow mask to differentiate the beams

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Display Technology Raster CRTs

• Raster CRT pros
• Allows solids, not just wireframes
• Leverages low-cost CRT technology (i.e., TVs)
• Bright! Display emits light
• Cons
• Requires screen-size memory array
• Discreet sampling (pixels)
• Practical limit on size (call it 40 inches)
• Bulky
• Finicky (convergence, warp, etc)

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CRTs Overview

• CRT technology hasnt changed much in 50 years
• Early television technology
• high resolution
• requires synchronization between video signal and
  electron beam vertical sync pulse
• Early computer displays
• avoided synchronization using vector algorithm
• flicker and refresh were problematic

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CRTs Overview

• Raster Displays (early 70s)
• like television, scan all pixels in regular
  pattern
• use frame buffer (video RAM) to eliminate sync
  problems
• RAM
• ¼ MB (256 KB) cost 2 million in 1971
• Do some math
• 1280 x 1024 screen resolution 1,310,720 pixels
• Monochrome color (binary) requires 160 KB
• High resolution color requires 5.2 MB

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Display Technology LCDs

• Liquid Crystal Displays (LCDs)
• LCDs organic molecules, naturally in crystalline
  state, that liquefy when excited by heat or E
  field
• Crystalline state twists polarized light 90º.

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Display Technology LCDs

• Liquid Crystal Displays (LCDs)
• LCDs organic molecules, naturally in crystalline
  state, that liquefy when excited by heat or E
  field
• Crystalline state twists polarized light 90º

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Display Technology LCDs

• Transmissive reflective LCDs
• LCDs act as light valves, not light emitters, and
  thus rely on an external light source.
• Laptop screen backlit, transmissive display
• Palm Pilot/Game Boy reflective display

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Display Technology Plasma

• Plasma display panels
• Similar in principle to fluorescent light tubes
• Small gas-filled capsules are excited by
  electric field,emits UV light
• UV excites phosphor
• Phosphor relaxes, emits some other color

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Display Technology

• Plasma Display Panel Pros
• Large viewing angle
• Good for large-format displays
• Fairly bright
• Cons
• Expensive
• Large pixels (1 mm versus 0.2 mm)
• Phosphors gradually deplete
• Less bright than CRTs, using more power

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Display Technology DMDs

• Digital Micromirror Devices (projectors)
• Microelectromechanical (MEM) devices, fabricated
  with VLSI techniques

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Display Technology DMDs

• DMDs are truly digital pixels
• Vary grey levels by modulating pulse length
• Color multiple chips, or color-wheel
• Great resolution
• Very bright
• Flicker problems

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Display Technologies Organic LED Arrays

• Organic Light-Emitting Diode (OLED) Arrays
• The display of the future? Many think so.
• OLEDs function like regular semiconductor LEDs
• But with thin-film polymer construction
• Thin-film deposition of organic, light-emitting
  molecules through vapor sublimation in a vacuum.
• Dope emissive layers with fluorescent molecules
  to create color.
• Not grown like a crystal, no high-temperature
  doping
• Thus, easier to create large-area OLEDs

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Display Technologies Organic LED Arrays

• OLED pros
• Transparent
• Flexible
• Light-emitting, and quite bright (daylight
  visible)
• Large viewing angle
• Fast (lt 1 microsecond off-on-off)
• Can be made large or small

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Display Technologies Organic LED Arrays

• OLED cons
• Not quite there yet (96x64 displays) except niche
  markets
• Cell phones (especially back display)
• Car stereos
• Not very robust, display lifetime a key issue
• Currently only passive matrix displays
• Passive matrix Pixels are illuminated in
  scanline order (like a raster display), but the
  lack of phosphorescence causes flicker
• Active matrix A polysilicate layer provides thin
  film transistors at each pixel, allowing direct
  pixel access and constant illumination
• See http//www.howstuffworks.com/lcd4.htm for
  more info
• Hard to compete with LCDs, a moving target

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Framebuffers

• So far weve talked about the physical display
  device
• How does the interface between the device and the
  computers notion of an image look?
• Framebuffer A memory array in which the computer
  stores an image
• On most computers, separate memory bank from main
  memory (why?)
• Many different variations, motivated by cost of
  memory