Homogeneous Cartesian Coordinates/Examples/Arbitrary Example 1

Examples of Homogeneous Cartesian Coordinates

Consider the polynomial equation $\map P {x, y}$:

$2 x^2 + x + 7 = y$

This can be expressed in homogeneous Cartesian coordinates $\map P {X, Y, Z}$ as:

$2 X^2 + X Z + 7 Z^2 = Y Z$

$\ds 2 x^2 + x + 7$

$=$

$\ds 7$

$\ds \leadsto \ \ $

$\ds 2 \paren {\dfrac X Z}^2 + \dfrac X Z + 7$

$=$

$\ds \dfrac Y Z$

substituting $x \gets \dfrac X Z$, $y \gets \dfrac Y Z$

$\ds \leadsto \ \ $

$\ds 2 X^2 + X Z + 7 Z^2$

$=$

$\ds Y Z$

multiplying both sides by $Z^2$

We confirm this is a homogeneous equation:

$\ds 2 \paren {k X}^2 + \paren {k X} \paren {k Z} + 7 \paren {k Z}^2$

$=$

$\ds \paren {k Y} \paren {k Z}$

$\ds \leadsto \ \ $

$\ds k^2 \paren {2 X^2 + X Z + 7 Z^2}$

$=$

$\ds k^2 Y Z$

and homogeneity has been confirmed.

$\blacksquare$