Poncelet's Porism is a well-known and much-beloved result in plane geometry, discovered by Jean-Victor Poncelet in 1813. Here is the statement:

**Theorem [Poncelet's Porism]**
*
Let $C$, $D$ be non-singular conics in general position in the projective plane,
and $n \ge 3$ a natural number. If there exists an $n$-sided polygon,
all of whose vertices lie on $C$ and all of whose edges are tangent to
$D$, then there exists such a polygon with a vertex at any given point of
$C$.
*

If this is the first you have heard of Poncelet's Porism, pause for thought: it is not obvious.

Poncelet's Porism has a lot going for it. It turns out to be an archetypal result demonstrating the power of "modern" algebraic geometry, it admits numerous fruitful generalisations, and it has connections to many beautiful subjects including elliptic curves, modular forms, dynamical systems, the Painlevé equations, and even instantons. But there is another reason for its popularity: it asserts the existence of some very appealing pictures and animations.

On the left hand side, the following two conics are plotted: $$ \left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1, \qquad x^2 + y^2 = 1, $$ together with the results of Poncelet iteration (the red lines). There are controls for adjusting the values of $a$, $b$ and the number of edges, as well as other elements whose functions are hopefully self-explanatory. We defer an explanation of the curves on the right until we discuss Cayley's results.

For anyone interested, the source for the above is available at Microsoft's code-hosting hub here.

Note that an instance of the porism in this real plane has two natural topological invariants:

- the vertex count, $n$,
- the number of times the polygon winds around the circle, $w$.

Composing, we have an endomorphism: $$ T = i_D \circ i_C : X \to X. $$ Poncelet's Porism is the statement that $T^n$ has a fixed point iff it is the identity map.

With this setup, we need only chain together the following basic facts from general theory:

- X is a smooth and has genus one (i.e., topologically a torus).
- After choosing a point to act as identity, every smooth curve of genus one becomes an elliptic curve, isomorphic to a quotient of $\mathbb{C}$ by a maximal-rank lattice.
- The only involutions admitting a fixed point on an elliptic curve are of the form $p \mapsto q - p$ for some point $q$.

Note that if $O\in X$ is the identity, this proof gives us a new interpretation of $n$: it is the order of the torsion point $T(O)$. We will depend upon this interpretation when discussing Cayley's results below.

If you didn't know the results 1–3 already, you might not be all that happy with this proof. The good news is, we will give another more elementary proof, due to Jacobi, below. In fact it's really the same proof in different language.

Poncelet's Porism concerns only incidence and tangency, and so is projectively invariant. Since the group of projective transformations of the plane is eight-dimensional, and the space of conics is five-dimensional, the space of pairs of conics up to projective equivalence is two-dimensional (5 + 5 - 8).

There are at least three natural ways to parameterise these two dimensions:

- A circle and a concentric, axis-aligned ellipse. This yields local coordinates such as the semi-minor, semi-major radii of the ellipse dividing by the radius of the circle.
- A pair of non-concentric circles. This yields local coordinates such as the radii of the two circles divided by the distance between their centres.
- The two cross ratios of the four intersection points of the conics: one cross ratio on each conic.

Following Schoenberg, we will present Jacobi's argument in terms of the parametrisation (i) even though Jacobi used (ii). Note that (i) exists as a consequence of the well-known result in linear algebra that for any pair of quadratic forms, we may always find a basis in which one is a diagonal matrix of $\pm 1$s, and the other is diagonal. We will discuss the case over $\mathbb{R}$ with $C$ and $D$ represented by the matrices: $$ \left[\begin{array}{ccc} 1/a^2 & 0 & 0\\ 0 & 1/b^2 & 0\\ 0 & 0 & -1/r^2 \end{array}\right] \quad\mbox{and}\quad \left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -1 \end{array}\right] \quad\mbox{respectively.}\quad $$

These matrices are of course only defined up to scale which allows us to assume $r=1$. Note that even after this we have still not used all the symmetry of projective invariance: we have freedom to permute the elements of our basis. Thus the quotient by full projective invariance is naturally (an open set of) the weighted projective space $\mathbb{P}(1, 2, 3) \simeq \mathbb{P}^2/S_3$ where the eigenvalues $1/a^2$, $1/b^2$, $-1/r^2$ are homogeneous coordinates on the $\mathbb{P}^2$.

Note that the same $S_3$ indeterminacy is also visible in parameterisation (iii). A cross ratio depends on the ordering of the four points; without an ordering we only have an orbit of the anharmonic group (abstractly $S_3$) rather that a single value. These cross ratios will naturally show up again below when we discuss the natural map to the pencil of conics.

Now suppose $\phi_0, \phi_1, \ldots, \phi_n$ is a sequence of angles associated with the Poncelet construction, and that the $\phi_i$ keep track of how many times we have wound round the circle by adding multiples of $2\pi$ if necessary. The $\phi_i$ correspond to a Poncelet $n$-gon iff: $$ \phi_n = \phi_0 + 2\pi w, $$ for some natural number $w$. Since the integrand is positive, this is true iff $$ \int_{\phi_0}^{\phi_n}\limits\frac{d\psi}{\sqrt{1-m\sin^2\psi}} = \int_{\phi_0}^{\phi_0 + 2\pi w}\limits\frac{d\psi}{\sqrt{1-m\sin^2\psi}}, $$ and this in turn is true iff: $$ n\omega = \omega(\phi_0) + \cdots + \omega(\phi_{n-1}) = 4wK(m), $$ where: $$ K(m) = \int_0^{\pi/2}\limits\frac{d\psi}{\sqrt{1-m\sin^2\psi}} = \frac{1}{4}\int_0^{2\pi}\limits\frac{d\psi}{\sqrt{1-m\sin^2\psi}}, $$ is the complete elliptic integral of the first kind. We are done: the above shows that the condition for the Poncelet construction to close is that: $$ \omega/K(m) \in \mathbb{Q}, $$ which does not depend on $\phi_0$.

We claimed above that Griffiths and Harris's proof was really the same as Jacobi's. The reason is that both are really just applications of the Abel-Jacobi map. Indeed, of the three "basic facts from general theory" to which we appealed when giving the Griffiths and Harris proof, number 1 is easy and 3 is an easy consequence of 2, but 2 is a deep result. It is a consequence of the fact that the Abel-Jacobi map is an isomorphism for genus-one curves. (In general the Abel-Jaocbi map is a sort-of linearisation map, but genus one curves are already linear.)

More concretely, the point is that the key quantity: $$ \frac{d\psi}{\sqrt{1-m\sin^2\psi}}, $$ which shows up in Jacobi's proof is the so-called "invariant differential" (the unique, up to scale, holomorphic 1-form) on the elliptic curve, and the map: $$ \phi \mapsto \int_{\phi_0}^{\phi}\limits\frac{d\psi}{\sqrt{1-m\sin^2\psi}}, $$ is the real component of the Abel-Jacobi map, which gives an isomorphism between the identity component of the elliptic curve and the circle: $$ \mathbb{R}/4K(m)\mathbb{Z}. $$ For clarity, note that this elliptic curve has two connected components over $\mathbb{R}$. Set-wise it is the space of pairs: $$ X = \{(p, l) \in C\times D^* ~|~ p \in l\}. $$ When working over $\mathbb{R}$ with an ellipse and circle as above, the two connected components of $X$ are distinguished according to whether $p$ is at the clockwise or anti-clockwise end of $l$ (relative to the point of tangency with the circle).

Jacobi's proof essentially includes answers to the above questions. We know that a necessary and sufficient condition for a pair of conics to admit a Poncelet $n$-gon is: $$ \frac{\omega}{4K(m)} = \frac{w}{n}, $$ where $w$ is the number of times the $n$-gon wraps round, $K$ is the complete elliptic ingetral of the first kind, and: $$ m = \frac{\alpha - \beta}{\alpha - 1}, $$ where for convenience we have introduced $\alpha = a^2/r^2$, $\beta = b^2/r^2$. The only thing we don't yet know is $\omega$. However an easy exercise in elementary geometry shows that if $\phi_0 = 0$ then: $$ \cos\phi_1 = u, $$ where: $$ u = \frac{\alpha + \beta - \alpha\beta}{\alpha - \beta + \alpha\beta}. $$ Then since $\omega$ is given in terms of the incomplete elliptic integral: $$ \omega = F(\phi_1, m) = \int_{0}^{\phi_1}\limits\frac{d\psi}{\sqrt{1-m\sin^2\psi}} $$ we have $\phi_1 = {\rm am}(\omega, m)$, where ${\rm am}$ is the Jacobi amplitude function, and so: $$ {\rm cn} (\omega, m) = u, $$ where ${\rm cn} = \cos \circ {\rm am}$ is one of the Jacobi elliptic functions. Since ${\rm cn}$ has period $4K(m)$, this determines $\omega$ as: $$ \omega = cn^{-1}\left(\frac{\alpha + \beta - \alpha\beta}{\alpha - \beta + \alpha\beta}, m\right) $$

As an example, suppose we wish to determine whether a pair of conics admit an inscribed quadrilateral, winding around once. By the above, this occurs iff $\omega = K(m)$, or equivalently $\phi_1 = \pi/2$ which means $u = 0$, i.e., $$ \alpha + \beta = \alpha\beta, $$ which is the same as: $$ 1/a^2 + 1/b^2 = 1/r^2. $$ Although it was easy to determine this condition for the existence of a quadrilateral using the expression for $\omega$ above, in general it is difficult to determine the equations for existence of Poncelet $n$-gons this way. Fortunately there is another much easier method, due to Cayley.

**Proposition**
*
Let $C$, $D$ be a general pair of conics in the projective plane,
together with a distinguished point $P \in C\cap D$. Let $K$
be the pencil of conics containing $C$, $D$ and let $X$ be the
associated incidence curve
(introduced above)
together with its natural endomorphism $T$.
Then we have a natural double cover:
$$
\pi_P : X \to K,
$$
whose ramification locus in $X$ naturally corresponds with $C\cap D$, and
whose branch locus in $K$ is the four points that are three singular
conics in $K$, together with $C \in K$. Furthermore, if $O \in X$ is the
ramification point over the branch point $C \in K$, then:
$$
\pi_P(T(O)) = D.
$$
*

**Proof** (sketch) Firstly, given a non-singular conic $C$ with distinguished point
$P \in C$, let $L_P$ be the set of lines through $P$, and let:
$$
\begin{align}
\rho_P : C &\to L_P,\\
Q &\mapsto \mbox{line $PQ$ if $P \ne Q$, otherwise the tangent at $P$},
\end{align}
$$
be the natural isomorphism.

Secondly, given a general pair of conics $C$, $D$, together with a distinguished point $P \in C\cap D$, let $K$ denote the pencil of conics containing $C$, $D$. Because of generality, $C_1^*\cap C_2^*$ is not incident with $C_1\cap C_2$ for any $C_1, C_2 \in K$ and so we have a well-defined natural isomorphism: $$ \begin{align} \tau_P : L_P &\to K,\\ l &\mapsto \mbox{unique conic in $K$ with tangent $l$ at $P$.} \end{align} $$ As zero set of the determinant, the space of singular conics is a cubic hypersurface in the space of all conics and so meets $K$ in three points. Under $\tau_P$, these three singular conics in $K$ naturally correspond to the lines joining $P$ to the three points of $C\cap D \setminus \{P\}$ and of course $C$ itself, as point in $K$, correponds to the tangent to $C$ at $P$.

Thirdly, recall the incidence curve $X$ defined above, together with its natural involutions $i_C$, $i_D$ and their composition $t = i_D\circ i_C$. Let: $$ \gamma : X \to C, $$ be the natural double cover (i.e., the quotient by $i_D$) and note that its branch locus is $C\cap D$. The map we seek is: $$ \pi_P = \tau_P \circ \rho_P \circ \gamma $$ and is trivially verified to have the stated properties. ∎

As an aside, note that the above is related to the "cross ratio coordinates" mentioned above as parametrisation (iii). Indeed if we choose an ordering of the three singular conics in $K$ then we have a unique isomorphism $K \simeq \mathbb{P}^1$ under which the singular conics correspond to $0, 1, \infty$. This allows us to regard $C, D \in K$ as elements of $\mathbb{P}^1 \setminus \{0, 1, \infty\}$, i.e., as two numbers (differing from 0, 1). This connects with the natural coordinates (iii) and some related systems.

**Theorem**
*
Let $C$, $D$ be $3\times 3$ matrices defining quadratic
forms which cut out a general pair of conics in the projective plane.
Let:
$$
\sqrt{\det(tC + D)} = a_0 + a_1 t + a_2 t^2 + \cdots
$$
be the taylor series of a branch of the square root of the cubic
$\det(tC + D)$. Then there exists a Poncelet polygon,
inscribed in $C$ and circumscribed about $D$, with vertex count
dividing $n$ iff the determinant of an associated $\left[\frac{n-1}{2}\right]$-dimensional
Hankel matrix
vanishes, specifically iff:
$$
\left|\begin{array}{cccc}
a_2 & \cdots & a_{p+1}\\
\cdot & & \cdot\\
\cdot & & \cdot\\
a_{p+1} & \cdots & a_{2p}
\end{array}\right| = 0,~\mbox{if $n = 2p+1$,}
\qquad \mbox{or:}\qquad
\left|\begin{array}{cccc}
a_3 & \cdots & a_{p+1}\\
\cdot & & \cdot\\
\cdot & & \cdot\\
a_{p+1} & \cdots & a_{2p-1}
\end{array}\right| = 0,~\mbox{if $n = 2p$}.
$$
*

The choice of matrices $C$, $D$ for our conics (which by abuse of notation we continue to denote $C$, $D$) determines a natural affine coordinate $t$ on $K$, the pencil of conics containing $C$, $D$: $$ t \mapsto \mbox{conic cut out by $tC + D$}, $$ and so provides an isomorphism $K \simeq \mathbb{P}^1$, under which $C$, $D$ correspond to $t = \infty, 0$ respectively, and under which the singular conics correspond to the roots of $\det(tC + D)$.

If we pick a point $P \in C\cap D$ and, in the notation of the proposition above, take $O \in X$ as identity, rendering $X$ an elliptic curve, then by Griffiths and Harris's proof, we know that $C$, $D$ admit a Poncelet $n$-gon iff $T(O)$ has order $n$. Cayley's result is thus equivalent to the following:

**Theorem [restatement of Cayley's result]**
*
Let $X \to \mathbb{P}^1$ be a double covering by an elliptic curve $X$,
under which the identity is sent to the branch point $\infty$, and the
remaining branch points are the values
$t_1, t_2, t_3 \in \mathbb{P}^1 - \{0,\infty\}$.
Then the points of $X$ over $t=0$
have order dividing $n$ iff the Cayley-type Hankel matrix,
associated to the taylor series:
$$
\sqrt{(t - t_1)(t - t_2)(t - t_3)} = a_0 + a_1 t + a_2 t^2 + \cdots
$$
has vanishing determinant.
*

By the above, when the identity is taken to be a point of $C\cap D$ (together with the tangent to $D$ at that point) our incidence curve $X$ is naturally isomorphic to the elliptic curve with Weierstrass equation: $$ s^2 = \det(tC + D). $$ Since the invariant differential of an elliptic curve in this form is $dt/s$, the invariant differential of $X$ (with this choice of identity) is: $$ \frac{dt}{\sqrt{\det(tC + D)}}. $$ We should be able to use this together with Jacobi's closure condition given above to obtain a proof of Cayley's result but we prefer a slick proof due to Atiyah, presented by Hitchin in [14].

**Proof** (sketch) Recall that a point $x$ on an elliptic curve has
order dividing $n$
iff there exists a meromorphic function with a zero of order $n$ at $x$,
a pole of order $n$ at the identity and no other zeros or poles. We will
show that the vanishing of the Cayley-type Hankel determinant is sufficient
for the existence of such a function and leave necessity as an exercise.

To avoid case analysis, assume $n = 2p + 1$ is odd; the argument for even $n$ requires only trivial modifications. Represent $X$ by the plane curve with equation: $$ s^2 = (t-t_1)(t-t_2)(t-t_3), $$ represent finite points on $X$ using coordinates $(t, s)$, let $O$ be the identity, and let $a_0 = \sqrt{-t_1 t_2 t_3}$ be a choice of square root. Consider the two natural meromorphic functions $s$, $t$ on $X$ and note that:

- $s$ has a pole of order 3 at $O$, and simple zeros at the three points $(t_i, 0)$,
- $t$ has a pole of order 2 at $O$, and simple zeros at the two points $(0, \pm a_0)$,

With the setup in place, we're ready for Atiyah's trick. Consider the following $p$ functions: $$ f_i = t^{p-i}(s - a_0 - a_1t - a_2t^2 - \cdots - a_i t^i) $$ for $i = 1, \ldots, p$. Note that $f_1$ has a pole of order $2p+1$ at $O$ and that $f_i$ has a pole of order $2p$ at $O$ for $i = 2, \ldots, p$.

Now, in a neighbourhood of $(0, a_0)$ we have: $$ s = a_0 + a_1 t + a_2 t^2 + \cdots, $$ and so: $$ \begin{array}{cccccccc} f_1 & = & a_{2}t^{p+1} & + & a_{3}t^{p+2} & + ~\cdots~ + & a_{p+1}t^{2p} & + ~\cdots,\\ f_2 & = & a_{3}t^{p+1} & + & a_{4}t^{p+2} & + ~\cdots~ + & a_{p+2}t^{2p} & + ~\cdots,\\ &\cdots\\ f_p & = & a_{p+1}t^{p+1} & + & a_{p+2}t^{p+2} & + ~\cdots~ + & a_{2p}t^{2p} & + ~\cdots.\\ \end{array} $$ Thus, if the relevant determinant vanishes, we can find scalars $\lambda_i$, $i=1, \ldots, p$ not all zero such that: $$ f = \lambda_1 f_1 + \cdots + \lambda_p f_p $$ has a local expansion with no terms of degree less than $t^{2p+1}$, i.e., a zero of order at least $2p+1$ at $(0, a_0)$. Since $f$ has a unique pole of order at most $2p+1$, it must in fact have a unique zero, and pole, both of order exactly $2p+1$, as required. ∎

We can finally explain the curves on the right hand side of the app above. These are the curves corresponding Poncelet configurations for triangles, quadrilaterals, pentagons etc. plotted in the weighted projective space $\mathbb{P}(1, 2, 3)$ of Poncelet configurations with local coordinates: $$ \begin{align} x &= e_2 / e_1^2,\\ y &= e_3 / e_1^3, \end{align} $$ where $e_i$ are the elementary symmetric functions in $1/a^2$, $1/b^2$, $1/r^2$, i.e.,: $$ -\det(tC + D) = e_3 t^3 + e_2 t^2 + e_1 t + 1. $$

Thus, in these coordinates, the series for Cayley's result is: $$ \sqrt{yt^3 + xt^2 + t + 1} = 1 + \frac{1}{2}t + \frac{1}{8}(-1 + 4x)t^2 + \frac{1}{16}(1 - 4x + 8y)t^3 + \frac{1}{128}(-5 + 24x - 16x^2 - 32y)t^4 + \cdots $$ and the curves are: $$ \begin{array}{ll} \mbox{triangles} & 1 - 4x = 0,\\ \mbox{quadrilaterals} & 1 - 4x + 8y = 0,\\ \mbox{pentagons} & 1 - 12 x + 48 x^2 - 64 x^3 - 32 y + 128 x y - 256 y^2 = 0,\\ \mbox{hexagons} & 3 - 20 x + 16 x^2 + 64 x^3 + 96 y - 384 x y + 512 y^2 = 0,\\ \end{array} $$ etc.

One last thing though! I believe the paper [16] of Kerawala potentially contains some interesting results but I have been unable to obtain a copy. I would be very grateful to anyone who might be able to help.

References to Bertrand's work are conspicuously absent. This is because, though I could locate the works of Bertrand that Schoenberg mentions, I could not find any mention of Poncelet's porism within them! In particular I could not find anything relevant at the page locations provided by Schoenberg. ¯\_(ツ)_/¯

Note that there are numerous beautiful expository articles on Poncelet's Porism, its history, and its generalisations. (I have yet to read most of these myself.)

- [pdf] Avksentyev, E.A.
*"A universal measure for a pencil of conics and the Great Poncelet Theorem"*Sbornik: Math., 205:5, 613–632 (2014). - [pdf] Avksentyev, E.A. and Protasov, V.Yu.
*"Universal measure for Poncelet-type theorems"*Proc. Amer. Math. Soc. (to appear) (2018). - [pdf] Barth, W., and Bauer Th.
*"Poncelet theorems"*, Expo. Math., 14, 125–144 (1996). - [pdf] Barth, W., and Michel, J.
*"Modular curves and Poncelet polygons"*, Math. Ann., 295, 25–49 (1993). - [pdf] Bos, H.J.M., Kers, C., Oort, F., and Raven, D.W.
*"Poncelet's closure theorem"*, Expo. Math., 5, 289–364 (1987). - [pdf] Del Centina, A.
*"Poncelet's Porism: a long story of renewed discoveries, I"*Arch. Hist. Exact Sci., 70(1) (2014). - [pdf] Del Centina, A.
*"Poncelet's Porism: a long story of renewed discoveries, II"*Arch. Hist. Exact Sci., 70(2) (2014). - [pdf] Cayley, A.
*"Note on the Porism of the in-and-circumscribed Polygon"*, Philos. Mag., 6, 99–102, (1853). - [pdf] Cayley, A.
*"On the Porism of the in-and-circumscribed Polygon"*, Phil. Trans. Roy. Soc. London, 151, 225–239, (1861). - [pdf] Griffiths, P., and Harris, J.
*"On Cayley's explicit solution to Poncelet's Porism"*, Enseign. Math., 24, 31–40 (1978). - [pdf] Griffiths, P., and Harris, J.
*"A Poncelet theorem in space"*, Commentarii Mathematici Helvetici, 52 (1977). - [pdf] Halbeisen, L., Hungerbühler N.
*"A Simple Proof of Poncelet’s Theorem (on the occasion of its bicentennial)"*, Amer. Math. Monthly, 121, 1–14 (2014). - [pdf] Hartshorne, R.
*"Stable vector bundles and Instantons"*, Commun. Math. Phys., 59, 1–15 (1978). - [pdf] Hitchin, N.J.
*"Poncelet Polygons and the Painlevé Equations"*, Geometry and Analysis (Bombay, 1992) Tata Inst. Fund. Res., Bombay, 151–185 (1995). - [pdf] Jacobi, C.G.J.
*"Ueber die Anwendung der elliptischen Transcendenten auf ein bekanntes Problem der Elementargeometrie"*, J. Reine Angew. Math., 3, 376–387 (1828). - [help]Kerawala, S.M.
*"Poncelet Porism in Two Circles"*, Bull. Calcutta Math. Soc. 39, 85–105 (1947). - [pdf] King, J.L.
*"Three problems in Search of a Measure"*Amer. Math. Monthly, 101, 609–628 (1994). - [pdf] Poncelet, J.V.
*"Reflexions sur l'usage de l'analise algebrique dans la geometrie"*, Annales de Mathematique, 8, 143–155 (1817). - [pdf] Poncelet, J.V.
*"Traité des propriétés projectives des figures"*Paris: Gauthier-Villars (1865, first ed. 1822). - [pdf] Richelot, F.J.
*"Anwendung der der elliptischen Transcendenten auf die sphärischen Polygone, welche zugleich einem kleinen Kreise der Kugel eingeschrieben und einem andern umgeschrieben sind"*, J. Reine Angew. Math., 5, 250–267, (1830). - [pdf] Schoenberg, I.J.
*"On Jacobi-Bertrand's proof of a Theorem of Poncelet"*, in: Erdős P., Alpár L., Halász G., Sárközy A. (eds) Studies in Pure Mathematics. Birkhäuser, Basel (1983). - [pdf] Vallès, J.
*"A vector bundle proof of Poncelet's closure theorem"*, Expo. Math., 30, 399–405 (2012).