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The "dancing metric" is a pseudo-riemannian metric $\pmb{g}$ of signature$(2,2)$ on the space $M^4$ of non-incident point-line pairs in the realprojective plane $\mathbb{RP}^2$. The null-curves of $(M^4,\pmb{g})$ are givenby the "dancing condition": the point is moving towards a point on the line,about which the line is turning. We establish a dictionary between classicalprojective geometry (incidence, cross ratio, projective duality, projectiveinvariants of plane curves...) and pseudo-riemannian 4-dimensional conformalgeometry (null-curves and geodesics, parallel transport, self-dual null2-planes, the Weyl curvature,...). There is also an unexpected bonus: byapplying a twistor construction to $(M^4,\pmb{g})$, a $\mathrm G_2$-symmetryemerges, hidden deep in classical projective geometry. To uncover thissymmetry, one needs to refine the "dancing condition" by a higher-ordercondition, expressed in terms of the osculating conic along a plane curve. Theoutcome is a correspondence between curves in the projective plane and itsdual, a projective geometry analog of the more familiar "rolling withoutslipping and twisting" for a pair of riemannian surfaces.

The dancing metric, $\mathrm G_2$-symmetry and projective rolling