Mumbai university > mechanical engineering > sem 7 > cad/cam/cae

Marks : 5M

Difficulty : Medium

Homogeneous coordinates, introduced by August Ferdinand Mobius, make calculations of graphics and geometry possible in projective space. Homogeneous coordinates are a way of representing N-dimensional coordinates with N+1 numbers. 

To make 2D Homogeneous coordinates, we simply add an additional variable, w, into existing coordinates. Therefore, a point in Cartesian coordinates, (X, Y) becomes (x, y, w) in Homogeneous coordinates. And X and Y in Cartesian are re-expressed with x, y and w in Homogeneous as; X = x/w , Y = y/w 

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.

For example, if the given degree 3 homogeneous polynomial is the following:

x³ + 3xy² - 5y^2w + 10w³ = 0

the result is

x³ + 3xy² - 5y^2w + 10 = 0

This works for three-dimension as well. One can replace a point (x, y, z) with (x/w, y/w, z/w) and multiply the result by w raised to certain power.

The resulting polynomial is a homogeneous one. Converting a degree n homogeneous polynomial in x, y, z and w back to the conventional form is exactly identical to the two-variable case.