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algcurves

 homogeneous
 Make a polynomial in two variables homogeneous in three variables.

 Calling Sequence homogeneous(f, x, y, z)

Parameters

 f - polynomial in x and y x, y, z - variables

Description

 • 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 $\sum _{i,j}{c}_{i,j}{x}^{i}{y}^{j}{z}^{d-i-j}$.

Examples

 > $\mathrm{with}\left(\mathrm{algcurves}\right):$
 > $f≔{y}^{2}-{x}^{3}$
 ${f}{≔}{-}{{x}}^{{3}}{+}{{y}}^{{2}}$ (1)
 > $\mathrm{homogeneous}\left(f,x,y,z\right)$
 ${-}{{x}}^{{3}}{+}{{y}}^{{2}}{}{z}$ (2)

Now one can convert back as follows:

 > $\mathrm{subs}\left(z=1,\right)$
 ${-}{{x}}^{{3}}{+}{{y}}^{{2}}$ (3)