CAD/CAM Machine Drawing and Computer Graphics

LESSON 11. Homogeneous coordinates, transformations, planners and space curves design

11.2.Homogenous Coordinates

These are a system of coordinates used in projective geometry like Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix. Homogeneous coordinates are everywhere in computer graphics because they allow common operations such as translation, rotation, scaling and perspective projection to be implemented as matrix operations. One of the many purposes of using homogeneous coordinates is to capture the concept of infinity. In the Euclidean coordinate system, infinity is something that does not exist. Mathematicians have discovered that many geometric concepts and computations can be greatly simplified if the concept of infinity is used. This will become very clear when we move to curves and surfaces design. Without the use of homogeneous coordinates system, it would be difficult to design certain classes of very useful curves and surfaces in computer graphics and computer-aided design.

11.3.Transformations

Transformations are used to position objects, to shape objects to change viewing positions. There are different types of transformations namely, Geometric Transformations which includes,  Translation, Rotation and scaling. Linear transformation which preserves parallel lines, including Non-uniform scales and shears or skews, Projection which preserves lines which includes Perspective projection and Parallel projection and finally  Non-linear transformation where lines become curves including Twists, bends, warps and morphs. The transformations are done by using matrices to change the homogenous coordinates.

11.4.Planners

It is planning for the application of computers for the design and manufacturing operations. The path of movement of a particular component may also be planned by using algorithm. The motion of robot can be formulated using planners.

11.5.Space curve design

Computer graphics depends on parametric forms to describe curves and surfaces. Normally a curve is presented as a graph of a function y = f(x). As x is varied, y = f(x) is computed by the function f, and the pair of coordinates (x, y) sweeps out the curve. This is called the  explicit form of the curve. An explicit curve cannot have infinite slope. The derivative f' (x) is not defined parallel to the y axis. Hence there are points on the curve that cannot be defined. Any transformation, such as rotation or shear, may cause an explicit curve to violate the coordinates.  So a parametric form is used. A parametric curve that lies in a plane is defined by two functions, x(t) and y(t), which use the independent parameter t. x(t) and y(t) are coordinate functions, since their values represent the coordinates of points on the curve. As t varies, the coordinates (x(t), y(t)) sweep out the curve. If we consider the two functions: x(t) = sin(t), y(t) = cos(t). As t varies from zero to 2p, a circle is swept out by (x(t), y(t)).