Different versions of Duality principles

Question

1. The duality principle for projective geometry states that the roles of the words "points" and "lines" in the theorems/definitions are interchangeable. Wikipedia says that it can be explained in "a more functional approach through special mappings". Can anyone explains in plain language how these special mappings prove the duality principle in projective geometry? (I just finished my undergraduate year 1)

2. There are so many duality principles in maths, is there a single grand principle of duality in some general structure that includes all these other duality principles as its consequences?

Answer 1

Let us consider $2$ dimensional projective geometry.

With respect to a certain basis,

• A point is defined as a $3 \times 1$ matrix (column vector) with coordinates $X=\begin{pmatrix}x\\y\\t\end{pmatrix}$ (defined up to a factor)

• A line is defined as a $1 \times 3$ matrix (row vector) with coordinates $L=(u,v,w)$ (defined up to a factor).

The equation of a line is :

$$LX=0 \ \iff \ (u,v,w)\begin{pmatrix}x\\y\\t\end{pmatrix}=ux+vy+wt=0$$

Duality takes its full strength when we deal with conic curves. Just a glance at it :

Do you know that a conic curve

$$ax^2+by^2+2cxy+2dxt+2eyt+ft^2=0$$ (think to $t$ being equal to $1$) can be represented in the following way:

$$X^TCX=0 \ \iff \ (x \ \ y \ \ t)\begin{pmatrix}a&c&d\\c&b&e\\d&e&f\end{pmatrix}\begin{pmatrix}x\\y\\t\end{pmatrix}=0$$

where $C$ is the (symmetric) matrix associated with the conic curve.

The important thing to understand is that $C$ can be used as a linear operator $X \to CX$ transforming a point into a point, or $L \to LC$ transforming a line into a line, or in a mixed way and this is what is interesting here, transforming a point into a line in this way;

$$\underbrace{X}_{\text{coord. of a point}} \to \ \ \ \ (CX)^T=X^TC^T=\underbrace{X^TC}_{\text{coord. of a line}} \ \ \text{because} \ C \ \text{is symmetric.}$$

expressing the duality with respect to the conic curve. $X$ is called a "pole" and $(CX)^T$ is called the "polar line" dual of this pole (https://mathworld.wolfram.com/Polar.html) ; a particular case : when $X$ belongs to the conic curve, its polar line is the tangent to the curve at this point.

Here is the fundamental transformation, involving the "tangential equation" :

$$X^TCX=0 \ \ \iff \ \ \underbrace{(X^TC)}_LC^{-1}\underbrace{(X^TC)^T}_{L^T}=0$$

which represents in the dual space (the space of projective lines) the relationship that must exist between the coordinates of a line to be tangent to the conic curve. It has also the form of a second degree equation, but associated with the inverse of the matrix of the conic curve.