how do we know when to homogenise an equation???

-thnx in advance

Dear student,

The importance of homogeneity is the scale invariance of the functions. Which implies that the graphs of the functions will be scale invariant. Indeed, imagine a homogeneous function is used to define a geometrical object implicitly:

meaning all points with coordinates (x1,…,xn  that satisfy this equation will belong to the geometrical figure defined by f. If f is homogeneous, it immediately follows that any multiple of these coordinates also satisfies the equations. In other words, any point that satisfies the equation immediately implies the entire ray going through that point and the origin of the space belong to the geometrical object.

Homogenizing an implicit polynomial equation means adding an extra variable z and multiply any term by zk with k such that the resulting polynomial is homogeneous. Of course, since any z-multiple of the polynomial will also be homogeneous, you choose the resulting homogeneous polynomial with smallest possible degree.