#### Poles, polars and duality -the algebraic version

In this section, we give an algebraic treatment of these topics.

The proofs may be obtained by clicking on the link below the statement of each theorem.

A plane conic has an equation of the form
**ax**^{2}+bxy +cy^{2}+fx+gy+h=0.

In terms of homogeneous coordinates, this becomes
**ax**^{2}+bxy +cy^{2}+fxz+gyz+hz^{2}=0

which can be written as
__x__^{T}M__x__=0
where __x__=(x,y,z), and **M** is a symmetric 3x3 matrix.

For a non-degenerate conic,
**M** must be non-singular and have eigenvalues of different sign.

Note that, if a conic contains three (distinct) collinear points, then it must be degenerate.

**Definition **

If **C: **__x__^{T}M__x__=0 is a non-degenerate conic and
**U=[**__u__] is any point,

then the algebraic polar of **U** with respect to **C** is the line
__u__^{T}M__x__=0.

Note that, as **M** is non-singular, we cannot have __u__^{T}M=__0__,
so that the line always exists.

A line **L** has an equation __a__^{T}__x__=0.
Now, __u__^{T}M__x__=0 and __a__^{T}__x__=0
give the same line if and only if **[**__u__]=[M^{-1}__a__].

Thus **L** is the polar of a unique point **U=[**__u__].

**Definition **

If **C: **__x__^{T}M__x__=0 is a non-degenerate conic and
**L** is any line,

then the algebraic pole of **L** with respect to **C**
is the point **U=[**__u__] such that **L** has equation
__u__^{T}M__x__=0.

**Remark**

If **L** has equation __a__^{T}__x__=0, then, as we have seen,
the pole of **L** is **U=[M**^{-1}__a__].

**Theorem 1**

If **C: **__x__^{T}M__x__=0 is a non-degenerate conic and
**U** is any point on **C**,

then the algebraic polar of **U** with respect to **C** is the tangent to **C** at **U
**.

Proof of Theorem 1

We now show that the algebraic polar coincides with the geometrical polar.

**Theorem 2**

Suppose that **C: **__x__^{T}M__x__=0 is a non-degenerate conic and
that **VW** is a chord of **C** passing through a fixed point **U**.

Then the tangents at **V** and **W** meet on the algebraic polar of **U** with respect to **C**.

Proof of Theorem 2

It follows readily that the algebraic and geometric poles of a line coincide.

We now have the idea of duality defined in algebraic terms.

The fundamental result is that duality preserves incidence.

**Theorem 3 ****La Hire's Theorem**

Suppose that **C: **__x__^{T}M__x__=0 is a non-degenerate conic and that
**U** and **V** are any points.

Then **U** lies on the polar of **V**
if and only if **V** lies on the polar of **U**.

Proof of Theorem 3

From certain points in the plane, it is possible to draw tangents to the conic **C**.

Provided the point does not lie on the conic, there will be exactly two tangents, as the following theorem shows.

**Theorem 4**

Suppose that **C: **__x__^{T}M__x__=0 is a non-degenerate conic and that
**U** is any point.

Then the pair of tangents to **C** from **U** has equation
__u__^{T}M__u__ __x__^{T}M__x__
=__u__^{T}M__x__ __u__^{T}M__x __.

Proof of Theorem 4

Together, Theorems 1, 2 and 4 establish

**Joachimsthal's Formulae**

Suppose that **C: **__x__^{T}M__x__=0 is a non-degenerate conic. Then

- The polar of
**U=[**__u__] has equation __u__^{T}M__x__=0.
- If
**U=[**__u__] is on **C**, then the tangent to **C** at **U**
has equation __u__^{T}M__x__=0.
- The pair of tangents to
**C** from **U=[**__u__] has equation
__u__^{T}M__u__ __x__^{T}M__x__
=__u__^{T}M__x__ __u__^{T}M__x __.

Tell me about Joachimsthal

Of course, the second is really a special case of the first
(but we proved it as Theorem 1 to help with the proof of Theorem 2).

We now show that the dual of a conic with respect to a fixed conic is actaully another conic.

There is a macro to draw the dual conic in the conics toolbar.

**Theorem 5**

Suppose that **C: **__x__^{T}M__x__=0 and
**D: **__x__^{T}N__x__=0 are non-degenerate conics.

Then the dual of **D** with respect to **C** is the non-degenerate conic with equation
__x__^{T}MN^{-1}M__x__=0.

Proof of Theorem 5