Constructive Solid-Geometry Methods

Constructive models represent a solid as a combination of primitive solids. (CSG)

Sweep Representation

Specifying a 2D shape and a sweep that moves the shape through a region of space.

We perform a sweep by moving the shape along a path. At intervals along this path, we replicate the shape and draw a set of connectiong line in the direction of the sweep to obtain the wireframe reprensentation.

Example of torus designed using a rotational sweep. The periodic spline cross section is rotated about an axis of rotation specified in the plane of the cross section.

Implicit/mathematical definition. Difficult to implement.

Constructive Solid Geometry (CSG)

1. Definition

2. Implementation with ray casting

Ray Casting is often used to implement CSG operation when objects are described with boundary representation. We fire a ray from the plane xy (which represent the screen). The surface limits for the composite object are determnined bu the specified set operation (see Table).

Surface Limit
A, D
C, B
Difference (Obj2 - Obj1)
B, D

  This method can also be used for physical simulation.


The use of graphical methods to aid in analysis, often of datasets so large or complex they can not be conseptualized otherwise. The analysed data can be scalars, vectors, tensors, or a combination of these types.

An example of visualization. Proteins are too complex to describe verbally

Scalar Fields

A topographic map of Marfa, Texas. Topographical maps are the most common utilization of isolinear plots.

Volume Rendering

Project the data into 2D using ray casting, but the 2D value can be some function of the values along that ray.


  The Visualization Toolkit is a cool program for doing some of these applications.

Spatial Data Structures

We will use heirarchical tree structures to store object information.



Let's start with 2D.




We will apply quadtrees to storing planar polygons. Our attribute will be whether current quadrant is filled by the interior of a polygon:

The succesive subdivisions can be reprensented as a tree with heterogeneous quadrants as nodes and homogeneous quadrants as leaves.

Here, we are grouping by color:

F: Full ; E: Empty; P: Partially Full

For an area containing 2^n by 2^n pixels, a quadtree representation contains at most n levels.


Octrees, like Quadtrees, use a node structure to store the volume elements:


In this figure, the octree has been refined twice. First, the root is refined into eight cells, each representing an octant of the root's domain. Then, one of the root's children is refined yet again.


Here is an example of a fully subdivided torus:

BSP Trees

BSP Trees for Polygons We can represent a polygon with a BSP tree. Let each face of the polygon coincide with a partitioning plane of the tree.

Now, each node of the tree encodes a face of the polygon. Each leaf of the tree is either inside or outside of the polygon.

It is easy to find out if a given point is inside the polygon; put it in the tree and see if it ends up in a + or - leaf (also have case of point exactly on a plane)