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# GRPHICS03 - Graphical Representation

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

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### GRPHICS03 - Graphical Representation

1. 1. GRAPHICAL REPRESENTATION Michael Heron
2. 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. 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. 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. 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. 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. 7. BEZIER PATCHES http://mathforum.org/sketchpad/mis html
8. 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. 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. 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. 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. 12. CSG EXAMPLES Images here show boolean operations – union, intersection, and difference. http://www.student.cs.uwaterloo.ca/~cs488/Contrib/f02/a5/djbigham/
13. 13. INTERSECTIONS  When two primitives intersect, the new object is the volume shared by both shapes
14. 14. DIFFERENCE  Difference is all the bits that don’t intersect.
15. 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. 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. 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. 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. 19. SPACE SUBDIVISION
20. 20. SPACE SUBDIVISION
21. 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. 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.