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.

Let us consider two real numbers, *a* and *w*, and compute the
value of *a/w*. Let us hold the value of *a* fixed and vary the
value of *w*. As *w* getting smaller, the value of *a/w* is
getting larger. If *w* approaches zero, *a/w* approaches to
infinity! Thus, to capture the concept of infinity, we use two numbers
*a* and *w* to represent a value *v*, *v=a/w*.
If *w* is not zero, the value is exactly *a/w*. Otherwise, we
identify the infinite value with (*a*,0). Therefore, the concept of
infinity can be represented with a number pair like (*a*, *w*) or as
a quotient *a/w*.

Let us apply this to the *xy*-coordinate plane. If we replace *x*
and *y* with *x/w* and *y/w*, a function
*f*(*x*,*y*)=0 becomes
*f*(*x/w*,*y/w*)=0. If function
*f*(*x*,*y*) = 0 is a polynomial, multiplying it with
*w ^{n}* will clear all denominators, where

For example, suppose we have a line *Ax + By + C* = 0. Replacing
*x* and *y* with *x/w* and *y/w* yields
*A(x/w) + B(y/w) + C* = 0. Multiplying by *w* changes it to

Ax + By + Cw= 0.

Let the given equation be a second degree polynomial
*Ax ^{2} + 2Bxy + Cy^{2} + 2Dx + 2Ey + F* = 0.
After replacing

If you look at these two polynomials carefully, you will see that the degrees of all terms are equal. In the case of a line, termsAx= 0^{2}+ 2Bxy + Cy^{2}+ 2Dxw + 2Eyw + Fw^{2}

Given a polynomial of degree *n*, after introducing *w*,
all terms are of degree *n*. Consequently, these polynomials are called
*homogeneous* polynomials and the coordinates (*x*,*y*,*w*)
the *homogeneous coordinates*.

Given a degree *n* polynomial in a homogeneous coordinate system,
dividing the polynomial with *w*^{n} and replacing *x/w*,
*y/w* with *x* and *y*, respectively, will convert the
polynomial back to a conventional one. For example, if the given degree 3
homogeneous polynomial is the following:

the result isx^{3}+ 3xy^{2}- 5y^{2}w+ 10w^{3}= 0

x^{3}+ 3xy^{2}- 5y^{2}+ 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.

Conversely, what is the homogeneous coordinates of a point (*x*,*y*)
in the *xy*-plane? It is simply (*x*,*y*,1)!
That is, let the *w* component be 1. In fact, this is only part of the
story, because the answer is * not* unique. The homogeneous
coordinates of a point (

For example, a point (4,2,3) in space is convert to (4Converting from a homogeneous coordinates to a conventional one isunique; but, converting a conventional coordinates to a homogeneous one isnot.

Let us take a look at an example. Let (3,5) be a point in the *xy*-plane.
Consider (3/*w*,5/*w*). If *w* is not zero, this point lies
on the line *y* = (5/3) *x*. Or, if you like the vector form,
(3/*w*,5/*w*) is a point on the line **O** + (1/*w*)**d**,
where the base point **O** is the coordinate origin and **d** is the
direction vector <3,5>. Therefore, as *w* approaches zero, the
point moves to infinity on the line. This is why we say (*x*,*y*,0)
is the ideal point or the point at infinity
__ in the direction of (x,y)__.

The story is the same for points in space, where
(*x*,*y*,*z*,0) is the ideal point or point at infinity in the
direction of (*x*,*y*,*z*).

The concept of homogeneous coordinates and points at infinity in certain direction will become very important when we discuss representations of curves and surface.

This transformation treats a two-dimensional homogeneous point as a point
in three-dimensional space and projects (from the coordinate origin) this
three-dimensional point to the plane *w*=1. Therefore, as a
homogeneous point moves on a curve defined by homogeneous polynomial
*f*(*x*,*y*,*w*)=0, its corresponding point moves in
three-dimensional space, which, in turn, is projected to the plane *w*=1.
Of course, (*x/w*,*y/w*) moves on a curve in plane *w*=1.

The above figure also shows clearly that while the conversion from the
conventional Euclidean coordinates to homogeneous coordinates is unique,
the opposite direction is not because all points on the line joining the
origin and (*x*,*y*,*w*) will be projected to
(*x/w*,*y/w*,1). This is also an important concept to be used
in later lectures.