A corner of a polygon or polyhedron. In a polygon, it is the shared endpoint of consecutive polygon edges.

A bounding box is a rectangle, or its higher-dimensional counterpart, containing a specified set. For finitely many points, the smallest axis-aligned box is ${\prod }_j[{\min }_i{x}_{ij},{\max }_i{x}_{ij}]$. For $(1,4),(3,-1),(2,2)$ this is $[1,3]\times [-1,4]$. Every coordinate extreme forces the corresponding box endpoint, proving minimality for the chosen axes. Allowing a rotation changes the optimization problem. Disjoint bounding boxes certify disjoint enclosed objects and accelerate geometric searches.

Two smooth curves have tangency at a shared point when their tangent lines agree. A tangent to a circle is perpendicular to its radius at the contact point: at $(r,0)$ on $x^2+y^2=r^2$ the tangent is $x=r$. For smooth surfaces the analogous condition uses tangent planes. Tangency is local and does not preclude crossing: $y=0$ and $y=x^3$ have the same tangent line at zero but pass through one another there.

The surface area of an object divided by its volume.

A measure of separation between locations or objects. In a metric space, distance is specified by a metric satisfying positivity, symmetry and the triangle inequality. The familiar straight-line separation in Euclidean space is Euclidean distance; other metrics can assign different distances to the same pair of points.

For a positive refractive index $\mu (r)$ in cylindrical coordinates, a light ray with constant radius $R>0$ and varying axial coordinate has the form $\theta (z)={\theta }_0+\omega z$ when it exists. Fermat principle makes $L=\mu (r)\sqrt{1+r^{\prime 2}+r^2{\theta }^{\prime 2}}$ stationary. Conservation of the momentum conjugate to the cyclic angle makes ${\theta }^{\prime }$ constant along a constant-radius path. The radial Euler-Lagrange equation is thengiving the displayed formula. A nonzero real helix requires $0<-R{\mu }^{\prime }(R)/\mu (R)<1$. If ${\mu }^{\prime }(R)=0$, $\omega =0$ gives a straight axial light ray; a generic radial profile need not support a constant-radius helix. Its helical pitch is $2\pi /|\omega |$. This ray model is useful for a graded-index optical fiber when the refractive index varies smoothly and geometrical optics applies; it does not by itself impose wave-mode quantization or boundary confinement.

For a light ray parametrized by arc length $s$, with unit tangent vector $t$ in a smooth positive refractive index $\mu$, the optical momentum is $p=\mu t$. This is the momentum for optical path-length normalization. The Euler-Lagrange equation for Fermat principle, $\int \mu (r)|r^{\prime }(\tau )|d\tau$, gives $d(\mu t)/ds=\nabla \mu$. Therefore an index independent of $z$ conserves $p_z=\mu dz/ds$. In coordinates $r(z),\theta (z)$ this is $\mu /\sqrt{1+r^{\prime 2}+r^2{\theta }^{\prime 2}}$, the first integral supplied by the Beltrami identity for several dependent variables.

For the refractive index $\mu =1/y$ on the upper half-plane, Fermat principle and the Beltrami identity give $1/[y\sqrt{1+(dy/dx{)}^2}]=1/R$. The nonvertical rays are arcs of circles centered on $y=0$; vertical lines are the remaining rays. These are also geodesics of the Poincaré half-plane model. A ray entering the strip $0<y\le 1$ toward decreasing $y$ at angle $\theta$ to the vertical has limiting horizontal displacement $\tan (\theta /2)$, at most one. The line $y=0$ is approached only after infinite optical path length because the refractive index diverges there.

A stereoisomer distinguished by the spatial arrangement of connected groups, such as cis versus trans ligands.

For ideal $\mathrm{M}(\mathrm{AA}{)}_2{\mathrm{a}}_2$ with equivalent ends of each chelate, the monodentate donors can be trans or cis. The cis form has two mirror-related configurations; the trans form is achiral when the chelate backbones introduce no additional stereochemistry.

In an octahedral facial isomer, three selected like donors occupy one triangular face and are pairwise cis. Unsymmetrical bidentate ligands can give facial and meridional isomers, each with distinct optical configurations.

In an octahedral meridional isomer, three selected like donors lie in a plane through the metal and include one trans pair. For three unsymmetrical bidentate ligands, this changes both steric contacts and the identity of opposite donor pairs.

Global navigation satellite systems can measure station position time series in a stated terrestrial reference frame. Vertical and horizontal secular or transient motions require treatment of station offsets, atmospheric delays, loading and reference-frame uncertainty.

A chromosomal inversion reverses the orientation of a chromosome segment relative to its flanking sequence. Breakage and rejoining can change gene regulation at the boundaries even when the segment retains the same genes. In an individual carrying different orientations, crossing over within the inverted region can produce unbalanced products, reducing recovery of recombinant descendants. An inversion therefore affects inheritance as well as the visual order of chromosomal markers. ^[ref]

Variation among genomes in whether a specified gene or genomic segment is present.

Gene annotation identifies the structure and attributes of a gene in a reference sequence. It records features such as exons, splice forms and coding regions, often combining RNA evidence with sequence comparison. A supported transcript structure does not automatically establish a known biochemical function for its product. Distinguishing structural evidence from functional interpretation prevents a prediction from being mistaken for a fully characterized gene. ^[ref]

A mitochondrial genome is the genetic material carried in a mitochondrion. In comparing sampled mitochondrial genomes, the age of their ancestral split is the relevant branch-time calibration; it need not equal the time of separation of the species carrying them.