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?