Hello Everyone,

i am reading a book on the Mathematics involved in Graphics. On homogeneous coordinates, this is what i read:

Basically, homogeneous coordinates define a point in a plane using three

coordinates instead of two. Initially, Pl¨ucker located a homogeneous point

relative to the sides of a triangle, but later revised his notation to the one

employed in contemporary mathematics and computer graphics. This states

that for a point P with coordinates (x, y) there exists a homogeneous point

(x, y, t) such that X = x/t and Y = y/t. For example, the point (3, 4) has

homogeneous coordinates (6, 8, 2), because 3 = 6/2 and 4 = 8/2. But the

homogeneous point (6, 8, 2) is not unique to (3, 4); (12, 16, 4), (15, 20, 5) and

(300, 400, 100) are all possible homogeneous coordinates for (3, 4).

The reason why this coordinate system is called ‘homogeneous’ is because

it is possible to transform functions such as f (x, y) into the form f (x/t, y/t)

without disturbing the degree of the curve. To the non-mathematician this

may not seem anything to get excited about, but in the field of projective

geometry it is a very powerful concept.

Well i have seen its use quite a lot in Graphics, but i have never quite understood its essence. Could anyone please elaborate why it so useful and its advantages, with a clear example? i read some links online, but im still not very clear about it.

Well using [x, y, 1] instead of [x, y] does provide a facility for all transformations.So, i understood that part. But why do we always use a “1” there?

Thanks.