This is a course on the theoretical underpinnings of 3D Graphics in computing, suitable for students with a suitable grounding in technical computing.

1. GRAPHICAL REPRESENTATION
Michael Heron

2. INTRODUCTION
 In the last lecture we talked about the processes we
go through to build a 3D object.
 We focused primarily on the polymesh.
 In this lecture we are going to look at alternate
representations.
 Bezier curves
 Bezier Patches
 NURBS surfaces
 Constructive Solid Geometry

3. BEZIER CURVES
 A Bezier Curve (or Bezier spline) is a curve defined
through the use of four vertices.
 A start and end vertice
 Two supplementary vertices that define the curve.
 Known as the control points.
P1
P2
P3
P4

4. BEZIER CURVES
 By manipulating the control points, we can change
the nature of the curve.
 Governed by a mathematical formula.
P1
P2
P3
P4

5. BEZIER CURVES
 Bezier curves are used for:
 Modelling smooth curves
 In a way that cannot be done with a polymesh
 Curves appear smooth at all scales
 Font representation
 Vector/Outline fonts
 Animation
 Used to define smooth, realistic paths for movement.

6. BEZIER PATCHES
 Related to the idea of a Bezier curve is a Bezier
patch.
 Curves are 2D, patches are 3D
 Bezier patches are shaped nets, the appearance of
which are determined by control points.
 Control points act like gravity spots on underlying
shape.
 The define the topology of the shape through mathematical
modelling of attraction

7. BEZIER PATCHES
http://mathforum.org/sketchpad/mis
html

8. BEZIER PATCHES
 Bezier patches offer several advantages
 Integrity of representation
 Continuity across boundaries
 Awareness of neighbours
 Fluid representation
 Exact representation
 Can measure volumes and surfaces
 Economical
 A single patch can represent many polygonal equivalents.

9. BEZIER PATCHES
 However, they also have drawbacks.
 Simple patch has 16 control points.
 Rendering with a Bezier patch is expensive.
 Around 10 times slower than equivalent polymesh
 But this does not take into account the difference in time
requirements to model a polymesh correctly.
 Large number of perspectives required to render
properly.

10. NURBS
 Non-Uniform Rational B-Splines
 Yikes!
 Extension of the idea of a Bezier curve
 Defined by control points.
 Each control point has a weight
 Defines how much that control point influences the
curve.
 NURBS have knots
 Vectors that describe how the resulting curve is
influenced by the control points.
 NURBS have an order
 How closely the curve follows the lines between control
points.

11. CONSTRUCTIVE SOLID GEOMETRY
 Complex shapes modeled by using simple shapes.
 To these shapes we add or subtract other simple
shapes.
 Combination of shapes using boolean operators
and set theory allows for new and interesting
complex topographies.
 Unions, Intersections, etc

12. CSG EXAMPLES
Images here show boolean operations – union, intersection, and
difference.
http://www.student.cs.uwaterloo.ca/~cs488/Contrib/f02/a5/djbigham/

13. INTERSECTIONS
 When two primitives intersect, the new object is the
volume shared by both shapes

14. DIFFERENCE
 Difference is all the bits that don’t intersect.

15. UNION
 A union is both shapes together.
 A merge, in other words.
 The new object can be treated as a single new
shape
 And be transformed as a holistic unit.

16. VOXEL REPRESENTATION
 Sometimes need to represent 3D space as voxels.
 3D Pixels
 3D space broken up into three dimensional arrays
of voxels.
 Each voxel labeled according to its object
occupancy.

17. VOXEL REPRESENTATION
 Voxel arrays are:
 Costly in terms of memory
 Entire object space must be separated down into cubic
elements
 Labelled according to object occupancy
 Space subdivision
 Frequently used in:
 Game representations of terrain
 Medical imagery
 Ray tracing

18. SPACE SUBDIVISION
 Process of space subdivision common to 2D and
3D graphics.
 Easier to understand in 3D.
 Start with a 1x1 square representing entire image.
 Then recursively subdivide squares.
 Stop when each pixel inside a subdivided square
has a single colour.

19. SPACE SUBDIVISION

20. SPACE SUBDIVISION

21. VOLUME SUBDIVISON
 Works the same way in 3D as in 2D
 We start with cubes
 Recursive subdivide cubes
 From our cube, subdivide into eight smaller cubes.
 Can represent space/volume subdivision as a tree
data structure.
 Easy to navigate and store
 Offers greater efficiency of representation.
 Advancement over simple voxel array.

22. SUMMARY
 Various ways to represent 3D graphics beyond
polymeshes.
 Bezier Curves
 Bezier Patches
 NURBS
 CSG
 Space Subdivision
 Each technique has its real life applications.
 The nature of the intended destination will dictate the
appropriate tool.