Page 83 - DCAP504_Computer Graphics

Computer Graphics

When this concept is applied to the xy-coordinate plane, the x and y is replaced with x/z and y/z
respectively, i.e. a function f(x, y) =0 becomes f(x/z, y/z) =0.
Consider a line represented in an equation form, Ax + By + C =0. Replace x and y with x/z and y/z
respectively. The equation now becomes,
A(x/z) + B(y/z) + C =0
Multiply the above equation by z, then it becomes,
Ax + By + Cz =0.
This is for first order polynomial equation of a line.

The second order polynomial equation of a line is given as,
Ax + 2Bxy + Cy + 2-Dx + 2Ey + F = 0
2
2
In the above equation replace x and y with x/z and y/z respectively and multiply it with z , then the
2
equation becomes,
Ax + 2Bxy + Cy + 2-Dxz + 2Eyw + Fz = 0
2
2
2
From the above two polynomial equations, it is clear that the degree of all terms are equal, i.e. in a first
degree polynomial the terms x, y, and z are of degree one. And in second degree polynomial all terms
x , y , xy, xz, yz and z are of degree two. Hence, we observe that after introducing z in a polynomial of
2
2
2
degree n, the terms of the polynomial still hold the degree n. Such polynomials are known as
homogeneous polynomials and [x y z] are the homogeneous coordinates.
Linear transformations such as translation, scaling, rotation, and shearing can be easily described using
homogeneous coordinate system. The point coordinate (x, y) in two-dimensions is represented by a
vector of three numbers [a b c]. The three components of the vector are interpreted as coordinates in a
three-dimensional space. Therefore, a point coordinate (x, y) in two-dimensions is transformed into a
homogeneous coordinate system by using a non-zero number z. The homogeneous representation of (x,
y) is [zx zy z].
Here, the number z is known as the homogeneous coordinate or scale factor.

The point coordinate values can be obtained from the homogeneous representation by
dividing the elements of homogeneous representation by the homogeneous coordinate z.

1. Point coordinates to Homogeneous representation:
(x, y)  [zx zy z]
2. Homogeneous representation to Point coordinates:
[a b c]  (a/c, b/c)

Consider a point coordinate (2, 3). This point coordinate is transformed to
homogeneous representation as [2 3 1]. Here, the value of homogeneous
coordinate, ‘z’ is equal to ‘1’. If the value of z=2, then the homogeneous
representation is [4 6 2].
The transformation such as scaling and rotation can be done by multiplying the point coordinate matrix
with the scaling and rotation matrix respectively, and translation transformation is obtained by adding
the translation matrix with the point coordinate matrix.
This is a major limitation, since different types of matrix operations have to be performed for different
transformations, i.e., if you need to perform transition, scaling, and rotation, you have to first apply
transition transformation to the object, followed by scaling transformation and then rotation
transformation.

76 LOVELY PROFESSIONAL UNIVERSITY