Geometric equations often involve a determinant with a column of 1s. For example, the equation of a line through two points:

The equation of a circle through three points:

The equation of a general conic section through five points

The equation of a Mobius (bilinear) transformation sending *z*_{1}, *z*_{2}, and *z*_{3} to *w*_{1}, *w*_{2}, and *w*_{3}:

Why all the determinants and why all the 1s?

When you see a determinant equal to zero, you immediately think of matrix rows or columns being linearly dependent. But in the examples above it isn’t the Cartesian coordinates that are linearly dependent but **projective coordinates** that are dependent.

The 1s are in the last column, though they need not be, as a clue as to where they came from. You could permute the rows and columns any way you like and the determinant would still be zero. The 1s are in the last column because you can take Cartesian coordinates into projective coordinates by adding a 1 at the end.

This 1 is sort of a silent partner, and can be ignored much of the time. But the last projective coordinate is critical when it’s necessary to be rigorous about points at infinity. The examples above are interesting because they are an application of homogeneous coordinates when there’s no concern about points at infinity.

It’s one of those things that’s obvious once you see it. *Why didn’t I ever think of that!*

Since scale indeterminacy is not a problem for linear independence, all of the above examples should work with general projective coordinates (with an arbitrary number at the end instead of the 1). Perhaps this can be used to improve numerical accuracy in some applications.

It should also be possible to have some points at infinity and still use the equation. For example a line through a normal point and one at infinity. Just change the bottom right 1 to a 0 in the first formula.