2.3 The homogenous representation of a point ...

It is useful to represent both points and vectors using the same set of basic underlying coordinate frame. (See figure 2.4).

Now, let "v" be a vector represented by: v=v1 a + v2 b+ v3 c, or v= (v1, v2, v3), where v1, v2 and v3 are the components in the underlying coordinate system. On the other hand, to represent a point P, we view its location as an offset from the origin "O" by a certain amount: we represent the vector "P-O" by finding three numbers (p1,p2,p3) such that P-O= p1 a + p2 b + p3 c and then equivalently, write P itself as: P =O + p1 a + p2 b + p3 c.

The representation of P is not just a triple, but a triple along with an origin. P is "at" a location that is offset from the origin by p1 a + p2 b + p3 c. This means that the vector v=v1 a + v2 b+ v3 c needs the four coefficients (v1, v2, v3, 0), whereas the point P= O + p1 a + p2 b + p3 c needs the four coefficients (p1, p2, p3, 1).

2.3 The homogenous representation of a point ...

The fourth component designates whether the object does or doesn't include the "O". We can formally write any "v" and "P" using matrix multiplication as:

The equations above are examples of the homogeneous representation of vectors and points.

The use of homogeneous coordinates is one of the hallmarks of computer graphics, as it both helps to keep straight the distinction between points and vectors and provides a compact notation when one works with affine transformations as we will see later.

2.3 The homogenous representation of a point ...