COMPUTER GRAPHICS & MULTIMEDIA
Chapter Seven
Representing 3D objects
1
3D OBJECT REPRESENTATION
Graphics scenes can contain many different kinds of
objects like trees, flowers, clouds, rocks, water etc.
Classification
Boundary representation ( b-reps)
Space partitioning representation
1. Boundary representation ( b-reps)
It describes a three dimensional object as a set of
surfaces that separate the object interior from the
environment.
Examples are polygon facets and spline patches.
CONT…
2. Space-partitioning representation
It describes the interior properties, by partitioning the
spatial region containing an object into a set of small,
non-overlapping, contiguous solids(usually cubes).
Eg: octree representation.
POLYGON SURFACES
Many graphics systems store all object descriptions
as sets of surface polygons.
This simplifies and speeds up the surface rendering
and display of objects, since all surfaces are
described with linear equations.
Polygon descriptions are often referred to as
“standard graphics objects”.
Generally polygon surfaces are specified using;
1. Polygon table
2. Plane equations
3. Polygon meshes.
1. POLYGON TABLES
The specification of polygon surfaces using vertex
coordinates and other attributes:
Geometric data table: vertices, edges, and polygon
surfaces.
Attribute table: e.g. Degree of transparency and surface
reflectivity etc.
Geometric data table:
Every vertex is listed as an endpoint for at least 2 edges
Every edge is part of at least one polygon
Every polygon is closed
Each polygon has at least one shared edge.
WORKOUT
2. PLANE EQUATIONS
Used to determine the spatial orientation of the
individual surface component of the object.
The equation for a plane surface can be expressed in
the form
Ax + By + Cz + D=0
where (x, y, z) is any point on the plane, and the
coefficients A , B , C , and D are constants.
Let (x1, y1 , z1 ,), (x2 , y2 , z2 ), and (x3 , y3 , z3 ) be
three successive polygon vertices of the polygon.
Using Cramer's Rule
Ax1 + By1+ Cz1 + D= 0,
Ax2 + By2+ Cz2 + D= 0,
Ax3 + By3+ Cz3 + D= 0
CONT…
A = y1 (z2 –z3 ) + y2(z3 –z1 ) + y3 (z1 –z2 )
B = z1 (x2 -x3 ) + z2 (x3 -x1 ) + z3 (x1 -x2 )
C = x1 (y2 –y3 ) + x2 (y3 –y1 ) + x3 (y1 -y2 )
D = -x1 (y2 z3 -y3 z2 ) - x2 (y3 z1 -y1 z3 ) - x3 (y1 z2 -y2 z1)
CONT…
Inside outside tests of the surface:
A x + B y + C z + D < 0, point (X,Y,Z) is inside the surface
A x + B y + C z + D > 0, point (X,Y,Z) is outside the surface
3. POLYGON MESHES
Is a collection of vertices, edges and faces that defines the
shape of a polyhedral object in 3D computer graphics.
Polyhedron is a connected mesh of simple planar polygons
that enclose a finite amount of space
Common types of polygon meshes are triangle strip and
quadrilateral mesh.
. Triangular Mesh : produces n – 2 connected triangles,
given the coordinates for n vertices.
. Quadrilateral mesh : generates a mesh of (n-1) by (m-1)
quadrilaterals, given the coordinates for an n by m array of vertices.
CONT.
Representation polygon meshes
P = { (X 1 ,Y 1 ,Z 1 ), (X 2 ,Y 2 ,Z 2 ),………,( X
n ,Y n ,Z n )}
Each polygon represent by a list of vertex of
coordinate
P1= { V1,V2,V3}
P2={ V2,V3,V5}
P3={V3,V4,V5}
CURVED SURFACES
1. Regular curved surfaces can be generated as
Quadric surfaces, eg. Sphere, ellipsoid,
super quadrics, eg. Super ellipsoids
Eg, for ellipsoid simple parametric equations:
where s1, rx,ry, and rx are constants. By varying the values
of φ
and θ, points on the surface can be computed.
2. IRREGULAR SURFACES(BLOBBY OBJECTS)
Shapes showing a certain degree of fluidity.
Nonrigid object. That is things, like cloth, rubber, liquids,
water droplets, etc.
Models
Use a combination of gaussian density functions (gaussian
bumps)
Meta-ball technique is to describe the object as being
made of density functions much like balls. Advantage here
is that the density function falls of in a finite interval.
3. SPLINE REPRESENTATIONS
A spline is a flexible strip used to produce a smooth
curve through a designated set of points.
Used to design and control the shape of complex
curves and shapes
a sequence of points and a curve is constructed whose shape
closely follows this sequence . The point are called control
point.
CONT.
A curve is actually passes through each
control point is called interpolating curve
A curve that passes near to the control point
but not necessarily through them is called an
approximating curve.
SWEEP REPRESENTATIONS
Sweep representations mean sweeping a 2D
surface in 3D space to create an object.
However, the objects created by this method
are usually converted into polygon meshes
and/or parametric surfaces before storing.
Constructive solid-geometry(CSG) methods
Combines the volumes occupied by overlapping 3D
objects using set operations: - union - intersection –
difference
VOXEL REPRESENTATION
In voxel representation, an object is decomposed into identical
cells arranged in a fixed regular grid.
These cells are called voxels (volume elements), in analogy to
pixels.
Eg. A cylinder can be represented as
follows by voxels. A '1' may represent
inside the cylinder while a '0' may
represent outside of the cylinder.
Alternatively we may use 8 bits to
represent the transparency value. If a
voxel has a value of '0', it is a fully-
transparent cell. If a voxel has a value
of '255', it is a non-transparent cell.
OCTREES
Octrees are hierarchical tree structures that describe
each region of 3D space as nodes.
When compared with the basic voxel representation,
octrees reduce storage requirements for 3D objects.
It also provides a convenient representation for
storing information about object interiors.
For
octree representation of 3D data, the
numbering method is as follows:
FRACTALS
Fractal objects refer to those objects which are self-
similar at all resolutions.
Generated recursively by applying the same
transformation function to an object, eg. Scale down +
rotate + translate.
For example, a Fractal Snowflake:
THANK YOU FOR YOUR ATTENTION
24
