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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

 > with(algcurves):
 > f:=y^2-x^3;
 ${f}{≔}{-}{{x}}^{{3}}{+}{{y}}^{{2}}$ (1)
 > homogeneous(f,x,y,z);
 ${-}{{x}}^{{3}}{+}{{y}}^{{2}}{}{z}$ (2)

Now one can convert back as follows:

 > subs(z=1,(2));
 ${-}{{x}}^{{3}}{+}{{y}}^{{2}}$ (3)