Primitive Instancing

Primitive Instancing

The concept of primitive instancing is to construct more complex objects from simpler primitive shapes. This concept is utilized in many commercial CAD solid modeling systems. Blocks, cylinders, spheres, and cones are some of the typical primitive solids that these systems offer. For fairly simple objects, it can be easy to model their shapes using combinations of these primitives. However, for the components with complex shapes, it can be difficult or very tedious to describe their shapes starting with primitive solids.

To implement the primitive instancing concept, it is first necessary to identify the primitives of interest. For this example, I will use cubes, cones, and squares. Then, it is necessary to be able to scale the primitives to generate shapes of the appropriate sizes and proportions. It is also necessary to position and orient the scaled shapes.

Let's investigate the cube first. For our purposes, we will be using MATLAB to model our objects, and MATLAB's graphics capabilities are limited to wireframes (at present). So, it is only necessary to model end-points of line segments that will be displayed. For a cube, I will assume that one face is centered in the XZ plane, a unit cube is most convenient, and the cube extends along the Y axis. This is shown in the figure below. In MATLAB code, the cube vertices are:

Cube = [-0.5, -0.5, -0.5, -0.5, -0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, -0.5, -0.5, 0.5, 0.5, -0.5; BR> 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1;
-0.5, 0.5, 0.5, -0.5, -0.5, -0.5, 0.5, 0.5, -0.5, -0.5, 0.5, 0.5, 0.5, 0.5, -0.5, -0.5;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1];

To position, orient, and scale the cube using relative transformations, the following equation is used. Note that I am using the Roll-Pitch-Yaw convention for rotations.

InstantiatedCube = Translate(tx,ty,tz) * Rotate(theta,phi,psi) * Scale(sx,sy,sz) * Cube

For a cone, I assume that the base of the cone (circle) lies in the XY plane, with a radius of 1 unit, and four line segments connect the circle to the cone's apex, 1 unit along the Z axis.

For a square, I center a unit square in the XY plane as follows:

square = [-0.5, -0.5, 0.5, 0.5, -0.5;
-0.5, 0.5, 0.5, -0.5, -0.5;
0, 0, 0, 0, 0;
1, 1, 1, 1, 1];