Algebraic curves are often represented using a polynomial in two variables, or as a homogeneous polynomial in three variables. This command converts a polynomial to the homogeneous representation.

If f is a polynomial in x and y of degree $d$, and f can be written as $\sum _{i,j}{c}_{i,j}{x}^i{y}^j$ for some coefficients $c_{i,j}$, then the output of this command is $\sum _{i,j}{c}_{i,j}{x}^i{y}^j{z}^{d-i-j}$.

$\mathrm{with}\left(\mathrm{algcurves}\right)\:$

$f≔y^2-x^3$

$f≔-x^3+y^2$

$\mathrm{homogeneous}\left(f\,x\,y\,z\right)$

$-x^3+y^{2}z$

$\mathrm{subs}\left(z=1\,\right)$

$-x^3+y^2$