3D Computer Graphics

1
3D Computer Graphics

• An oh so brief introduction

2
What you need to know

• Theres really only a couple of things
• Coordinate systems
• Matrices
• 3D ? 2D transformation

3
Homogeneous Coordinates

• A computational convenience

4
Coordinate systems

• To represent a location (point) in 3D space one
  needs 3 numbers
• (X, Y, Z)
• Each value specifies a distance along the
  respective coordinate axis
• The resultant location (point) is the sum of the
  axis unit vectors multiplied by the values

5
Manipulating points

• As we will see soon, manipulations of points in 3
  space are generally performed using matrix
  notation
• But, as it turns out, this is not readily done
  using points represented by 3 values
• Something better is needed

6
Homogeneous coordinates

• Homogeneous coordinates were introduced as a
  means of facilitating matrix-based
  transformations applied to points
• It is a 4D representation of a 3D point
• (X, Y, Z) ? (X, Y, Z, 1.0)
• (X/W, Y/W, Z/W) ? (X, Y, Z, W)
• Thats really all you need to know about
  homogeneous coordinates

7
Matrix Operations

• Uses of Matrix Multiplication

8
Translation

• To move a point (X, Y, Z) by amounts (x, y, z)

9
Scale

• To scale a point (X, Y, Z) by sizes (x, y, z)

10
Rotate X

• To rotate a point (X, Y, Z) about the X axis by
  an angle T

11
Rotate Y

• To rotate a point (X, Y, Z) about the Y axis by
  an angle T

12
Rotate Z

• To rotate a point (X, Y, Z) about the Z axis by
  an angle T

13
Shear X

• To shear a point (X, Y, Z) int the X direction
  by an angle T

14
Shear Y

• To shear a point (X, Y, Z) in the Y direction
  by an angle T

15
Shear Z

• To shear a point (X, Y, Z) in the Z direction
  by an angle T

16
Combining matrices
17
Problems

• Rotation based on matrix operations potentially
  suffers some afflictions
• Difficult to interpolate between rotations when
  you want to create a smooth sequence
• Gimbal lock when one of the three axes rotates
  to align with another essentially rendering it
  redundant (reduces the number of degrees of
  freedom)
• Non-linear speed of rotation objects dont
  rotate smoothly with constant velocity
• These afflictions are due to the use of Euler
  angles and trigonometric functions that dont
  always behave well (sign changes at quadrant
  changes, asymptotic behavior)

18
Quaterions

• Another method for performing rotations
• Based on complex arithmetic (complex numbers
  not complicated numbers)
• Straight forward conversion from Euler (matrix
  based) operations to Quaternions
• The underlying concepts are nasty
• The implementation is easy
• Just a bunch of multiplications and additions
• Handles the constant velocity rotation issue
• SLERP (Spherical LinEaR interPolation)
• Ken Shoemake is credited for coming up with the
  approach

19
3D ? 2D Transformation
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Projections Orthographic

• Projectors are perpendicular to the projection
  plane
• Project plane is parallel to one of the principal
  faces

21
Projections Axonometric

• Projectors are perpendicular to the projection
  plane
• Project plane has any desired orientation with
  respect to the object faces

22
Projections Oblique

• Projectors are arbitrary with respect to the
  projection plane

23
Projections Perspective

• Projectors converge at the center of projection

24
Projections

• Each has advantages and disadvantages dealing
  with
• Retention of angles between lines
• Retention of distances between points
• Visibility of surfaces
• Realization via camera models
• Realistic synthesis of scenes

25
Triangulation

• Because you asked

26
Triangulation

• Problem given a set of points, find a set of
  triangles that connects those points in a mesh
• Solution computational geometry provides us with
  the Voronoi Diagram and (its dual) the Delaunay
  Triangulation

27
Delaunay Triangulation

• For a given set of points
• Find a set of edges satisfying an "empty circle"
  property
• for each edge we can find a circle containing the
  edge's endpoints but not containing any other
  points
• Deals with

28
Voronoi Diagram

• For a given set of points
• Every point in the region around a point is
  closer to that point than to any of the other
  point

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Inscribe and Circumscribe

• Inscribe
• Given a triangle find a circle that fits inside
  it
• Circumscribe
• Given a triangle find a circle that passes
  through all three vertices

30
Delaunay, circles, Voronoi
31
Summary

• This is all stuff that is fundamental to computer
  graphics (with the possible exception of
  triangulation thats computational geometry)
• Typical 1 semester undergraduate course spends
  most of its time on these topics
• Lots of good books on this material

32
Bibliography

• Interactive Computer Graphics 4th edition
• Angel
• Addison-Wesley
• Computer Graphics Principles and Practice 2nd
  edition
• Foley, van Dam, Feiner, Hughes
• Addison-Wesley
• Java 2D Graphics
• Knudsen
• OReilly
• Computer Graphics for Java Programmers
• Ammeraal
• Wiley

33
Bibliography

• Essential Mathematics for Games Interactive
  Applications
• van Verth, Bishop
• Morgan Kaufmann
• Visualizing Quaternions
• Hanson
• Morgan Kaufmann
• 3D Game Engine Design
• Eberly
• Morgan Kaufmann

34
Bibliography

• Game Physics
• Eberly
• Morgan Kaufmann
• Killer Game Programming in Java
• Davison
• OReilly
• Developing Games in Java
• Brackeen, Barker, Vanhelsuwe
• New Riders
• Software Engineering and Computer Games
• Rucker
• Addison-Wesley