Derivation of Dual Curve for Parametric Equations

Question

I am studying projective geometry and am stuck on understanding the following:

For a parametric curve

$$ x = x(t), y = y(t) $$

the dual curve is given by

$$ X=\frac{-y′}{xy′-yx'}, Y=\frac{x′}{xy′−yx′} $$

This is explained on Wikipedia, in Chapter 1, Section 2 of Discriminants, Resultants, and Multidimensional Determinants by Gelfand, Kapranov, and Zelevinsky (referred by @Jan-Magnus-Økland; see this question), and in this document.

I feel like something so simple and fundamental is escaping me...how was this derived? Is Cramer's Rule involved?

Answer 1

Let

$$ Xx + Yy + 1 = 0 \tag{1} $$

be the equation of a line in line coordinates. A point $(x)$ and line $[X]$ (in the notation of Coxeter) are said to be incident in projective geometry if their inner product is zero:

$$ \{xX\} = 0. $$

The tangent to our parametric equation at point $t_0$ is given by $<x'(t_0),y'(t_0)>$. Hence, the point $(x')$ is incident on our line if

$$ \{x'X\} = 0. $$

That is,

$$ Xx' + Yy' = 0 \tag{2} $$

Thus, we have two equations in two unknowns ($(1)$ and $(2)$), which via Cramer's rule yield

$$ X=\frac{-y'}{xy'-yx'}, Y=\frac{x'}{xy'-yx'}. $$

Q.E.D.