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
Constructive Solid Geometry
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.
By manipulating the control points, we can change
the nature of the curve.
Governed by a mathematical formula.
Bezier curves are used for:
Modelling smooth curves
In a way that cannot be done with a polymesh
Curves appear smooth at all scales
Used to define smooth, realistic paths for movement.
Related to the idea of a Bezier curve is a Bezier
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
The define the topology of the shape through mathematical
modelling of attraction
Bezier patches offer several advantages
Integrity of representation
Continuity across boundaries
Awareness of neighbours
Can measure volumes and surfaces
A single patch can represent many polygonal equivalents.
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
Non-Uniform Rational B-Splines
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
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
CONSTRUCTIVE SOLID GEOMETRY
Complex shapes modeled by using simple shapes.
To these shapes we add or subtract other simple
Combination of shapes using boolean operators
and set theory allows for new and interesting
Unions, Intersections, etc
Images here show boolean operations – union, intersection, and
When two primitives intersect, the new object is the
volume shared by both shapes
Difference is all the bits that don’t intersect.
A union is both shapes together.
A merge, in other words.
The new object can be treated as a single new
And be transformed as a holistic unit.
Sometimes need to represent 3D space as voxels.
3D space broken up into three dimensional arrays
Each voxel labeled according to its object
Voxel arrays are:
Costly in terms of memory
Entire object space must be separated down into cubic
Labelled according to object occupancy
Frequently used in:
Game representations of terrain
Process of space subdivision common to 2D and
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.
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
Easy to navigate and store
Offers greater efficiency of representation.
Advancement over simple voxel array.
Various ways to represent 3D graphics beyond
Each technique has its real life applications.
The nature of the intended destination will dictate the