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The Poisson bracket satisfies

A hyperbolic half-turn is the unique orientation-preserving isometry of order two fixing a given point. It exchanges every pair of points lying at equal opposite distances along a geodesic through its centre.

Reflection in a hyperbolic line is the unique orientation-reversing hyperbolic isometry fixing that line pointwise. The product of reflections in intersecting lines is a rotation through twice their angle.

PSL2(R) is $S{L}_2(\mathbb{R})/\{\pm I\}$. It acts on the upper half-plane by the Möbius transformationsand is the group of orientation-preserving hyperbolic isometries.

The geodesics in the upper half-plane are vertical lines and semicircles centred on the boundary axis. Indeed, the length integrand $\sqrt{1+y^{\prime 2}}/y$ has the Beltrami identitywhich integrates to $(x-c{)}^2+y^2=R^2$. Swapping the coordinate axes gives circles centred on the vertical boundary for the metric $(d{x}^2+d{y}^2)/x^2$.

If $P$ is at hyperbolic distance $\rho$ from the origin, the Poincare-disc geodesic through $P$ perpendicular to $OP$ is represented by a Euclidean circle of radius $\mathrm{csch}\rho$ whose centre is at Euclidean distance $\coth \rho$ from the origin.

The geodesics are diameters and arcs of circles orthogonal to the unit circle. A radial segment is length minimizing becausewith equality for constant angle and monotone radius. Hence the distance from the origin to $z$ is $2\mathrm{artanh}|z|$.

For a piecewise smooth curve $z(t)$ in the Poincare disc, its hyperbolic length is

An isometry sends the centers of a collinear family of equal hyperbolic discs to points on one hyperbolic geodesic. In the upper-half-plane model no geodesic contains three distinct points on a horizontal Euclidean line, since geodesics are vertical lines or semicircles orthogonal to the boundary.

In curvature minus one, a hyperbolic disc of radius $R$ has areaIn the Poincare disc this follows by integrating $4r(1-r^2{)}^{-2}drd\theta$ up to Euclidean radius $\tanh (R/2)$.

For a small body with orbital elements $a,e,I$ relative to a planet on a circular orbit of semi-major axis $a_p$, the Tisserand parameter is approximatelyIt is derived from the Jacobi constant and is approximately conserved across separated weak encounters in the circular restricted three-body problem.

A planetesimal is a solid body in a young or evolved planetary system whose collisions and gravitational interactions help build or erode larger bodies.

For $D_i={\epsilon }_{ij}{E}_j$, $B=\mu H$, and a plane wave with electric polarization $e$, the source-free Maxwell equations implyIf $e$ is an eigenvector of $\epsilon$, then $e\perp k$, the dispersion relation is $|k{|}^2={\omega }^2\mu {\epsilon }_e$, and the Poynting vector is parallel to $k$.

Without free surface charge or current, the tangential components of $E$ and $H$ and the normal components of $D$ and $B$ are continuous across an interface.

At the surface of a perfect conductor, the tangential electric field and normal magnetic field vanish. An incident plane electromagnetic wave is therefore accompanied by a reflected wave whose tangential electric field cancels the incident field at the surface.

A semicubical parabola is a cusped cubic curve affinely equivalent to $y^2=x^3$.

For a smooth plane curve of degree $d\ge 4$, adjunction givesIts canonical map is induced by the $(d-3)$rd Veronese map and is therefore an embedding.

For a finite connected graph with at least one vertex, embedded as a plane graph, let $n$ count vertices, $m$ count edges, and $f$ count faces of a plane graph, including the outer face. ThenThe hypotheses refer to an actual embedding; regions in a crossing-filled graph drawing do not supply its $f$.^[ref]

In the plane graph $K_4$ pictured above, three bounded faces of a plane graph and the outer face give $4-6+4=2$. A square cycle graph gives $4-4+2=2$, while the graph path $A-B-C$ gives $3-2+1=2$. Changing a crossing into a new vertex changes the graph: the square-and-diagonals drawing then has five vertices, eight edges, and five faces of a plane graph, giving $5-8+5=2$.

A proof outline deletes edges on graph cycles. Each deletion preserves the connected graph and merges two faces of a plane graph, leaving $n-m+f$ unchanged. Eventually a tree remains, with $m=n-1$ and $f=1$. Adding the point at infinity identifies the plane picture with a sphere; the value two agrees with its Euler characteristic and the cell-count formula for Euler characteristic.

A planar dual graph has one vertex in each face of a fixed planar embedding and one dual edge crossing each primal edge.

This is the planar-embedding case of a dual graph.

For a place $v$ of a number field $K$, the local degree is $d_v=[K_v:{\mathbb{Q}}_v]$. Thus $d_v=1$ at a real place, $d_v=2$ at a complex place, and $d_v=e_v{f}_v$ at a finite place over $p$. For each rational place, the local degrees above it sum to $[K:\mathbb{Q}]$.