Introduction to Computer Graphics CS 445 / 645

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Introduction to Computer GraphicsCS 445 / 645

• Lecture 6
• Geometric primitives andthe rendering pipepline

M.C. Escher Smaller and Smaller (1956)
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Rendering geometric primitives

• Describe objects with points, lines, and surfaces
  
• Compact mathematical notation
• Operators to apply to those representations
• Render the objects
• The rendering pipeline
• Appendix A1-A5

HB Figure 109
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Rendering

• Generate an image from geometric primitives

Rendering
Geometric Primitives
Raster Image
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3D Rendering Example
What issues must be addressed by a 3D rendering
system?
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Overview

• 3D scene representation
• 3D viewer representation
• Visible surface determination
• Lighting simulation

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Overview

• 3D scene representation
• 3D viewer representation
• Visible surface determination
• Lighting simulation

How is the 3D scene described in a computer?
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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

(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)
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Vector Addition/Subtraction

• operation u v, with
• Identity 0 v 0 v
• Inverse - v (-v) 0
• Addition uses the parallelogram rule

v
u
-v
v
u-v
-v
u
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Vector Space

• Vectors define a vector space
• They support vector addition
• Commutative and associative
• Possess identity and inverse
• They support scalar multiplication
• Associative, distributive
• Possess identity

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Affine Spaces

• Vector spaces lack position and distance
• They have magnitude and direction but no location
• Combine the point and vector primitives
• Permits describing vectors relative to a common
  location
• A point and three vectors define a 3-D coordinate
  system
• Point-point subtraction yields a vector

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Coordinate Systems

• Grasp z-axis with hand
• Thumb points in direction of z-axis
• Roll fingers from positive x-axis towards
  positive y-axis

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Points Vectors

• 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
  defined

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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)

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 ?)

V
P1
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3D Line

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

V
P1
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3D Line Slope Intercept

• Slope m
• rise / run
• Slope (y - y1) / (x - x1) (y2 - y1) / (x2
  - x1)
• Solve for y
• y (y2 - y1)/(x2 - x1)x -(y2-y1)/(x2 -
  x1)x1 y1
• or y mx b

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Euclidean Spaces

• Q What is the distance function between points
  and vectors in affine space?
• A Dot product
• Euclidean affine space affine space plus dot
  product
• Permits the computation of distance and angles

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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)

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Dot Product

• Useful for many purposes
• Computing the length (Euclidean Norm) of a
  vector
• length(v) v sqrt(v v)
• Normalizing a vector, making it unit-length v
  v / v
• Computing the angle between two vectors
• u v u v cos(?)
• Checking two vectors for orthogonality
• u v 0.0

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Dot Product

• Projecting one vector onto another
• If v is a unit vector and we have another vector,
  w
• We can project w perpendicularly onto v
• And the result, u, has length w v

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Dot Product

• Is commutative
• u v v u
• Is distributive with respect to addition
• u (v w) u v u w

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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

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Cross Product Right Hand Rule

• See http//www.phy.syr.edu/courses/video/RightHan
  dRule/index2.html
• Orient your right hand such that your palm is at
  the beginning of A and your fingers point in the
  direction of A
• Twist your hand about the A-axis such that B
  extends perpendicularly from your palm
• As you curl your fingers to make a fist, your
  thumb will point in the direction of the cross
  product

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Cross Product Right Hand Rule

• See http//www.phy.syr.edu/courses/video/RightHan
  dRule/index2.html
• Orient your right hand such that your palm is at
  the beginning of A and your fingers point in the
  direction of A
• Twist your hand about the A-axis such that B
  extends perpendicularly from your palm
• As you curl your fingers to make a fist, your
  thumb will point in the direction of the cross
  product

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Cross Product Right Hand Rule

• See http//www.phy.syr.edu/courses/video/RightHan
  dRule/index2.html
• Orient your right hand such that your palm is at
  the beginning of A and your fingers point in the
  direction of A
• Twist your hand about the A-axis such that B
  extends perpendicularly from your palm
• As you curl your fingers to make a fist, your
  thumb will point in the direction of the cross
  product

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Cross Product Right Hand Rule

• See http//www.phy.syr.edu/courses/video/RightHan
  dRule/index2.html
• Orient your right hand such that your palm is at
  the beginning of A and your fingers point in the
  direction of A
• Twist your hand about the A-axis such that B
  extends perpendicularly from your palm
• As you curl your fingers to make a fist, your
  thumb will point in the direction of the cross
  product

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Cross Product Right Hand Rule

• See http//www.phy.syr.edu/courses/video/RightHan
  dRule/index2.html
• Orient your right hand such that your palm is at
  the beginning of A and your fingers point in the
  direction of A
• Twist your hand about the A-axis such that B
  extends perpendicularly from your palm
• As you curl your fingers to make a fist, your
  thumb will point in the direction of the cross
  product

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Other helpful formulas

• Length sqrt (x2 - x1)2 (y2 - y1)2
• Midpoint, p2, between p1 and p3
• p2 ((x1 x3) / 2, (y1 y3) / 2))
• Two lines are perpendicular if
• M1 -1/M2
• cosine of the angle between them is 0
• Dot product 0

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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
• ax by cz d 0, or
• PN d 0
• N is the plane normal
• Unit-length vector
• Perpendicular to plane

N (a,b,c)
P3
P2
P1
d
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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

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3D Geometric Primitives

• More detail on 3D modeling later in course
• Point
• Line segment
• Polygon
• Polyhedron
• Curved surface
• Solid object
• etc.

HB Figure 10.46
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Take a breath

• Were done with the primitives
• Now were moving on to study how graphics uses
  these primitives to make pretty pictures

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Rendering 3D Scenes
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Camera Models

• The most common model is pin-hole camera
• All captured light rays arrive along paths toward
  focal point without lens distortion (everything
  is in focus)
• Sensor response proportional to radiance

View plane
Other models consider ... Depth of field Motion
blur Lens distortion
Eye position (focal point)
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Camera Parameters

• What are the parameters of a camera?

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Camera Parameters

• Position
• Eye position (px, py, pz)
• Orientation
• View direction (dx, dy, dz)
• Up direction (ux, uy, uz)
• Aperture
• Field of view (xfov, yfov)
• Film plane
• Look at point
• View plane normal

View Plane
Up direction
Look at Point
back
View direction
Eye Position
right
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Moving the camera
Up
Back
Towards
Right
View Frustum
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The Rendering Pipeline
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Rendering Transformations

• Weve learned about transformations
• But they are used in three ways
• Modeling transforms
• Viewing transforms (Move the camera)
• Projection transforms (Change the type of camera)

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The Rendering Pipeline 3-D
Scene graphObject geometry

• Result
• All vertices of scene in shared 3-D world
  coordinate system
• Vertices shaded according to lighting model
• Scene vertices in 3-D view or camera
  coordinate system
• Exactly those vertices portions of polygons in
  view frustum
• 2-D screen coordinates of clipped vertices

ModelingTransforms
LightingCalculations
ViewingTransform
Clipping
ProjectionTransform
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The Rendering Pipeline 3-D
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Rendering Transformations

• Modeling transforms
• Size, place, scale, and rotate objects and parts
  of the model w.r.t. each other
• Object coordinates -gt world coordinates

Y
Z
X
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The Rendering Pipeline 3-D
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Lighting Simulation

• Lighting parameters
• Light source emission
• Surface reflectance
• Atmospheric attenuation
• Camera response

Light Source
Surface
Camera
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Lighting Simulation

• Direct illumination
• Ray casting
• Polygon shading
• Global illumination
• Ray tracing
• Monte Carlo methods
• Radiosity methods

Light Source
Surface
N
More on these methods later!
Camera
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The Rendering Pipeline 3-D
Scene graphObject geometry

• Result
• Scene vertices in 3-D view or camera
  coordinate system

ModelingTransforms
LightingCalculations
ViewingTransform
Clipping
ProjectionTransform
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Rendering Transformations

• 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 a view coordinates

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The Rendering Pipeline 3-D
Scene graphObject geometry

• Result
• Remove geometry that is out of view

ModelingTransforms
LightingCalculations
ViewingTransform
Clipping
ProjectionTransform
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Assignment 2

• Due two and a half weeks from today
• Project description available online
• Well discuss details in class on Monday

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The Rendering Pipeline 3-D
Scene graphObject geometry

• Result
• 2-D screen coordinates of clipped vertices

ModelingTransforms
LightingCalculations
ViewingTransform
Clipping
ProjectionTransform
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Rendering Transformations

• Projection transform
• Apply perspective foreshortening
• Distant small the pinhole camera model
• View coordinates a screen coordinates

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Rendering Transformations

• Perspective Camera
• Orthographic Camera

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Rendering 3D Scenes
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Rasterize

• Convert screen coordinates to pixel colors

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Summary

• Geometric primitives
• Points, vectors
• Operators on these primitives
• Dot product, cross product, norm
• The rendering pipeline
• Move models, illuminate, move camera, clip,
  project to display, rasterize