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Let $P\ne 0$ in the Poincare disc model be at hyperbolic distance $\rho >0$ from zero. The hyperbolic line through $P$ perpendicular to the radial line $OP$ is represented by a Euclidean circle with radius $\mathrm{csch}\rho$ and center at Euclidean distance $\coth \rho$ from zero, on the ray through $P$.

Write $r=|P|=\tanh (\rho /2)$. A circle with center at distance $c$ on that ray is orthogonal to the unit circle when its squared radius is $c^2-1$. Passing through $P$ at a perpendicular tangent requires $c-r$ to equal that radius. Solving givesAt $P=0$, one must first choose a radial direction; the perpendicular hyperbolic line is then a diameter. The finite-circle formulas are replaced by this diameter limit as $\rho \to 0$.

In the Poincare disc model, the geodesics are diameters and interior arcs of circles orthogonal to the unit circle. To prove radial minimality, let a piecewise smooth path start at zero and end at a point of Euclidean radius $r_1<1$. Polar coordinates giveA constant-angle path with monotone radius attains the bound. Thus the radial segment minimizes hyperbolic length, and $d(0,z)=2\mathrm{artanh}|z|$.

A Poincare disc automorphism moves any interior point to zero and carries diameters to circles or lines orthogonal to the boundary. Transporting the radial minimizing paths gives all the stated hyperbolic geodesics.

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 participating in the growth and collisional evolution of a planetary system. Its collisions can build larger bodies or fragment existing ones; the result depends on relative velocity, material strength and self-gravity. Gravitational encounters can increase random motion, while gas drag and dynamical friction can reduce it.

The category marks a physical role rather than one sharply defined diameter. A population can feed planetary embryos during planet formation or act as the parent bodies of a later debris disk. The same measured total mass can have very different collision behaviour when distributed among many small bodies or a few large ones.

In a homogeneous lossless dielectric, let $\mathbf{D}=\mathbf{\epsilon }\mathbf{E}$ with a real symmetric permittivity tensor, and $\mathbf{B}=\mu \mathbf{H}$ with scalar positive magnetic permeability. A source-free plane electromagnetic wave with angular frequency $\omega \ne 0$, real wavevector $\mathbf{k}$ and electric field amplitude $\mathbf{e}$ satisfiesThe second equation follows from Gauss's law. It makes the electric displacement field transverse; the electric field itself need not be transverse in an anisotropic medium.

If $\mathbf{e}$ is an eigenvector of the permittivity tensor with nonzero eigenvalue ${\epsilon }_e$, then ${\epsilon }_e\mathbf{k}\cdot \mathbf{e}=0$, so $\mathbf{e}\perp \mathbf{k}$. The first equation reduces to the dispersion relation $|\mathbf{k}{|}^2={\omega }^2\mu {\epsilon }_e$. For a propagating solution the time-averaged Poynting vector is $⟨\mathbf{S}⟩=|\mathbf{e}{|}^2\mathbf{k}/(2\omega \mu )$ and is parallel to the wavevector.

A general anisotropic eigenmode need not have its electric field along a principal dielectric direction. Its energy flux can then depart from the wavevector, producing spatial walk-off. The nonzero-${\epsilon }_e$ assumption matters: at a zero permittivity eigenvalue, Gauss's law alone no longer forces $\mathbf{k}\cdot \mathbf{e}=0$.^[ref]

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 $[K:\mathbb{Q}]$.

If a prime ideal $\mathfrak{p}$ of $K$ lies over the prime number $p$ with ramification index of a prime ideal $e_{\mathfrak{p}}$, its normalized absolute value isIt extends the usual p-adic absolute value on $\mathbb{Q}$. Its local degree is $[K_{\mathfrak{p}}:{\mathbb{Q}}_p]=e_{\mathfrak{p}}{f}_{\mathfrak{p}}$, where $f_{\mathfrak{p}}$ is the residue-field degree.

A real embedding $K\hookrightarrow \mathbb{R}$ gives a real Archimedean place of local degree $1$. A conjugate pair of complex embeddings $K\hookrightarrow \mathbb{C}$ gives a complex Archimedean place of local degree $2$.

Reflection symmetry $x\mapsto -x$ forces the reduced vector field to be odd in $x$, naturally producing paired nonzero equilibrium branches.

The normal form $\dot{x}=\mu x+x^3$ has a stable trivial equilibrium for $\mu <0$ and two unstable nonzero equilibria for $\mu <0$. They collide with the trivial branch at $\mu =0$, after which that branch is unstable.

The photon quadrupole is the leading anisotropy that sources linear polarization through Thomson scattering.