CS 655 Advanced Computer Graphics

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CS 655 Advanced Computer Graphics

• Introduction

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Intent of the Course

• Look at the entire graphics pipeline
• Basic modeling
• Solid modeling
• Modeling natural phenomena
• Fractals
• L-systems
• Particle systems
• Volumetric modeling and rendering
• Texture and other mapping
• Ray tracing
• Radiosity
• Global illumination
• Animation

3
Assumptions

• You have had CS 455 and understand
• Basic scan conversion
• Geometric transformations
• Viewing transformations
• Illumination models
• Lighting
• Shading algorithms
• Flat shading
• Gouraud shading
• Phong shading
• Ray-tracing (basic)
• Curves and surfaces (basic)
• Linear algebra
• Various spaces and coordinate systems
  (modeling,world, view reference)

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Object Representations

• Non-parametric or implicit
• 2D curve f(x, y) 0 or y y(x)
• 3D curve f1(x, y, z) 0, f2(x, y, z) 0, or
• y y(x), z z(x)
• Surface f(x, y, z) 0 or z z(x, y)
• Solid specify with inequalities (also need
  boundary conditions)
• e.g.
• y mx b straight line
• x2 y2 r2 circle
• Ax By Cz D 0 plane
• x2 y2 z2 - r2 0 spherical surface
• x2 y2 z2 r2 sphere
• Useful for scan conversion algorithms,
  determining object boundaries

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Object Representations

• Parametric
• Curve x x(u), y y(u), z z(u) or
• P(u) (x(u), y(u), z(u))
• typically with 0 u 1
• Surface P(u, v) (x(u,v), y(u,v), z(u,v))
• typically with 0 u, v 1
• Solid P(u, v, w) (x(u,v,w), y(u,v,w),
  z(u,v,w))
• typically with 0 u, v, w 1

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Object Representations

• Advantages of Parametric Representation
• provide more degrees of freedom than implicit
• coordinate independent representation
• easier to deal with for some applications
• object bounds specified by limits on u, v, w
  instead of on coordinate
• Easily expressed in vector or matrix form
• simplifies object descriptions
• Transformations can be applied directly to the
  parametric equation
• Easy to extend to higher dimensions

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Geometric Transformations

• Homogeneous coordinate form
• Translation Matrix
• Scale Matrix
• Scaled by (Sx, Sy, Sz)
• about the fixed point
• (xF, yF, zF)

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Geometric Transformations

• Rotations
• Right handed coordinate system
• Positive rotations are counter-
• clockwise looking down the
• positive axis towards the origin

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Rotation about Arbitrary Axis

• Basic Idea
• Move a point on the line to the origin
• Rotate about some axis to put l into a coordinate
  plane
• Rotate about another axis to align l with a
  coordinate axis
• Rotate about that axis by q
• Reverse rotation about the coordinate axis in
  step 3
• Reverse rotation about the coordinate axis in
  step 2
• Reverse translation from step 1

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Rotation about Arbitrary Axis
y
y
l
b
a
T(-p)
x
x
Rz(a)
z
z
l
y
y
l
q
Ry(b)
Ry(-b)
Rx(Q)
x
z
l
-b
x
z
y
-a
Rz(-a)
T(p)
x
z
l
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Viewing

• Setting up an Arbitrary View, given
• Look from Point
• Look at point
• The up direction
• From these, compute VRC

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Viewing

• To display the scene, you need
• View plane
• Usually perpendicular to the view direction
• Window in the view plane
• Only object that project to the window will be
  mapped to the viewport
• View Volume
• Front clipping plane
• Back clipping plane
• Projection type
• Parallel
• Perspective

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Parallel Projection
View Reference Point
Front Clipping Plane
Back Clipping Plane
Window
-zv
View Volume
xv
Window
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Perspective Projection
Center of Projection
Front Clipping Plane
Back Clipping Plane
Window
-zv
View Volume
xv
d
(xcp, ycp, zcp)
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Lighting

• For realistic lighting effects, the following are
  generally included
• Ambient illumination
• background lighting, no light source, just simply
  there
• Diffuse reflection
• light reflects equally in all directions
• Specular reflection
• light seen depends on the view direction
• Refracted (transmitted) light
• for transparent objects

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Illumination Model

• One illumination model, that works well, and is
  not too computationally expensive is

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Shading

• Define the objects in the scene, then shade the
  surfaces of the objects
• Typically want the intensity to vary across the
  polygon
• i.e., a blue polygon should be dark blue in one
  corner then gradually change to light blue in the
  other corner
• Used to give the appearance of curvature to an
  object that has been defined as a polygonal set.
• Different types of shading have the quality vs.
  time tradeoff

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Shading

• Constant Shading
• A single intensity is used to shade an entire
  polygon
• Not too bad if
• Light source is at infinity (i.e., (N L) is
  constant across the polygon)
• Viewer is at infinity (i.e., (N V) is constant
  across the polygon)
• The polygon represents the actual surface, rather
  than an approximation to a curved surface
• Gouraud Shading
• Compute illumination model at each vertex
• Interpolate intensities between vertices
• Phong Shading
• Compute normals at each vertex
• Interpolate normals between vertices
• Computer illumination model at each point

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Shading

• Gouraud Shading
• Interpolate intensities over each polygon surface
• Compute illumination model at each vertex
• Interpolate intensities between vertices
• Compute intensities for points 1, 2, and 3
• Interpolate intensity between points 1 and 2 to
  get intensity for point 4
• Interpolate intensity between points 3 and 1 to
  get intensity for point 5
• Interpolate intensity between points 4 and 5 to
  get intensity for point 6

2
3
6
4
5
1
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Shading

• Phong Shading
• Interpolate intensities over each polygon surface
• Compute normals at each vertex
• Interpolate normals between vertices
• Compute illumination model at each point
• Compute normals for points 1, 2, and 3
• Interpolate between normals at 1 and 2 to get
  normal for point 4
• Interpolate between normals at 3 and 1 to get
  normal for point 5
• Interpolate between normals at 4 and 5 to get
  normal for point 6

2
3
6
4
5
1
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Linear Interpolation

• The equation to interpolate a value i between
  two other values a and b is

dist(a,i)
(
)

-
-
V
V
V
V
i
a
a
b
dist(b,a)