3D Object Representations

Introduction Line and circle and polygon algorithms- represented simple and smooth object. Some Natural object are neither perfectly flat nor smoothly curved but often---- rough,Jagged. These object have different characteristic and different material So there is no one method to described such object

3D Objects How can this object be represented in a computer?

3D Objects This one? H&B Figure 10.46

3D Objects How about this one?

3D Objects This one? H&B Figure 9.9

3D objects can be represented in a variety of ways. We survey these today

Different Method Polygon and Quadric surfaces: For simple objects such as polyhedrons and ellipsoids Spline surfaces and construction: For curved surfaces for eg. Aircraft wings,gear and other engg. Structure Procedural methods: Fractals, Particle systems for eg. Mountains,clouds, clumps of grass etc..

Physically based modeling methods: Nonrigid behavior of object such as cloth. Octree Encoding: represent internal feature of object such as CT Scan detail, etc.

Representation of solid object Two type : Boundary Representations (B-reps): Space-partitioning representations :

Boundary Representations (B-reps) IT describe a 3D object as a set of surfaces that separate the object interior from the environment. eg. Polygon facets and spline patches.

Space-partitioning representations IT describes interior properties by partitioning the spatial region containing an object into a set of small,nonoverlapping,contiguous solids. Eg. Octree Representation

Polygon Surfaces The most commonly used boundary representation object is a set of polygon surfaces that enclose the object interior. It describe polygon surfaces with linear equation This method simplifies and speeds up the surface rendering and display of objects.

Polygon Tables This is the specification of polygon surfaces with a set of vertex coordinates and other attributes parameters. This information of polygon is placed in the tables and used in the subsequent processing, display and manipulation of object in scene. It organized into two groups.: 1) Geometric table 2) attribute table

1) Geometric table It contain the vertex co-ordinates and parameter to identify the spatial orientation of the polygon surface. Geometric data table: vertices, edges, and polygon surfaces

2) attribute table It contain an object includes parameters specification the degree of transparency of the object and the surface reflectivity and texture characteristics.

Three Type of Geometric table 1)Vertex table 2)Edge table 3)Polygon surface table

Vertex table Co-ordinate values fro each vertex in the object are stored in the vertex table

Edge table It contain pointer back into vertex tables to identify the vertices for each polygon edges.

Polygoan table It contain pointers back into the edge table to identify the edges for each polygon.

VERTEX TABLE V 1 : x 1, y 1, z 1 V 2 : x 2, y 2, z 2 V 3 : x 3, y 3, z 3 V 4 : x 4, y 4, z 4 V 5 : x 5, y 5, z 5 EDGE TABLE E 1 : V 1, V 2 E 2 : V 2, V 3 E 3 : V 3, V 1 E 4 : V 3, V 4 E 5 : V 4, V 5 E 6 : V 5, V 1 POLYGON TABLE P 1 : V 1, V 2, V 3 P 2 : V 1, V 3, V 4, V 5 POLYGON TABLE P 1 : E 1, E 2, E 3 P 2 : E 3, E 4, E 5, E 6 or E1E1 E2E2 E3E3 E5E5 E4E4 E6E6 V2V2 V3V3 V4V4 V5V5 V1V1 P1P1 P2P2

Cont… This geometric table is less convenient : If we use only two table vertex table and polygon table to draw the object,some edges could get drawn twice. Add extra information to table: 1) forward pointer 2) slope of each edge 3) co-ordinates extents for each polygon

Some consistency checks of the 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

Plane equation Consider a cube, each of the 6 planes has 2 sides: inside face and outside face. For each plane (in a right-handed coordinate system), if we look at its surface and take 3 points in counter-clockwise direction: (x1,y1), (x2,y2),and (x3,y3), we can compute 4 values: A,B,C,D as

Then, the plane equation at the form: Ax+By+Cz+D=0 has the property that: If we substitute any arbitrary point (x,y) into this equation, then, Ax + By + Cz + D < 0 implies that the point (x,y) is inside the surface, and Ax + By + Cz + D < 1 implies that the point (x,y) is outside the surface

Polygon meshes A polygon mesh is collection of edges,vertices and polygon connected such that each edge is shared by at most two polygons. A polygon mesh can be represented in several ways 1)Explicitly 2)Pointers to a vertex list, 3)Pointers to an edge list.

The application programmer has to apply space and time criteria to evaluate different representation and select the best suitable representation.

Operation perform on the polygon mesh Finding the vertices connected by an edge Finding the edges of the polygon Displaying the mesh Identifying error in representation

Representing Polygon Meshes

1).The explicit way: When relation among polygon,vertices and edges are explicitly defined then the time require for above operation is less,but it require more space. In this each polygon is represented by a list of vertex co-ordinates just list 3D vertices of each polygon in a certain order. The problems are, firstly it represents same vertex many times and secondly, no explicit representation of shared edges and vertices

store all vertices once into a numbered list, and represent each polygon by its vertices. It saves space (vertex only listed once) but still has no explicit representation of shared edges and vertices 2) Pointer to a vertex list:

3).Explicit edges: list all edges that belong to a polygon, and for each edge list the vertices that define it along with the polygons of which it is a member.