projective coordinates
the projective plane can be thought of as the euclidean plane with additional points added, which are called points at infinity, and are considered to lie on a new line, the line at infinity.
and 
in projective coordinates is found from their vector cross product.
a line is turned into a vector from its general form equation, where we turn 
into 
, this allows us to have the same format for both lines and points. however, transformation matrices are applied differently to lines from points, for a point 
, we simply use matrix multiplication:
but for a line 
, it is just a little more involved computationally:
for 2d transformation matrices, with the simple trick of adding a third coordinate and requiring that it always be 1, we could unify rotation, scaling, and all the other linear transformations with translations, to get the class of affine transformations.
we consider transformations on the form:
if we examine the special case where the upper-left corner is a 2x2 identity matrix, we get:
as long as we pay attention only to the 
and 
coordinates, this looks like a translation.
we can regard points of the plane as being ellements of
whose third coordinate is 1, and vectors as being elements of whose third coordinate is 0. with this convention, its clear that the difference of points is a vector, the sum of a vector and a point is a point, and combinations yield a point if and only if the sum of the coefficients is 1 (because the third coordinate of the result will be exactly the sum of the coefficients; for the sum to be a point, this third coordinate is required to be 1).
[cite:;taken from @graphics_hughes_2014 chapter 10.4 translation]
the line passing through two points we consider transformations on the form:
if we examine the special case where the upper-left corner is a 2x2 identity matrix, we get:as long as we pay attention only to the and coordinates, this looks like a translation.we can regard points of the plane as being ellements of
whose third coordinate is 1, and vectors as being elements of whose third coordinate is 0. with this convention, its clear that the difference of points is a vector, the sum of a vector and a point is a point, and combinations yield a point if and only if the sum of the coefficients is 1 (because the third coordinate of the result will be exactly the sum of the coefficients; for the sum to be a point, this third coordinate is required to be 1).[cite:;taken from @graphics_hughes_2014 chapter 10.4 translation]
and in projective coordinates is found from their vector cross product.
a line is turned into a vector from its general form equation, where we turn into , this allows us to have the same format for both lines and points. however, transformation matrices are applied differently to lines from points, for a point , we simply use matrix multiplication:
but for a line , it is just a little more involved computationally: