The center of a circle or sphere is the unique point $\mathbf{c}$ at the fixed positive radius $R$ from every point of that circle or sphere. In Cartesian coordinates, its equation is $|\mathbf{x}-\mathbf{c}{|}^2=R^2$. In the complex plane, a circle has equation $|z-c{|}^2=R^2$; completing the square identifies $c$ as its center. For example, $|z+b{|}^2=R^2$ has center $-b$.

A quadric surface in three-dimensional Euclidean space is the zero set of a real polynomial of degree two. Its quadratic part can be written with a symmetric matrix $A$, since the antisymmetric matrix part contributes zero to $x^{T}Ax$. An orthogonal diagonalization and a translation expose its geometry: examples include an ellipsoid, an elliptic cone, and a hyperboloid. This affine geometric description allows singular surfaces such as a cone's vertex.

Points are collinear when they lie on a single line. Two vectors are collinear when their linear span has dimension at most one. For a nonzero vector $v$, this means that the other vector is $\lambda v$ for some scalar $\lambda$. This algebraic convention includes the zero vector; the directional use of “parallel” sometimes assumes both vectors are nonzero.

A geometric cone joins a fixed point to points of a plane curve by straight lines. A circular cone uses a circle as that curve; its solid version includes the enclosed circular cross-sections. For a right circular cone of height $h$ and base radius $R$, the cross-section radius varies linearly along the axis, and its volume is $\pi R^{2}h/3$ by integration. An elliptic cone allows elliptical cross-sections. The construction differs from the abstract topological cone, which does not specify lengths or angles.

A system of equations requires one choice of unknowns to satisfy several equations at once. Its solution set is the intersection of the solution sets of the individual equations. For example, $x^2+y^2=1$ and $y=(x-1{)}^2$ ask for intersections of a unit circle and a parabola. A linear system is the special case in which every equation is linear in the unknowns.

An implicit equation specifies a relation between variables without isolating a designated output as an explicit function of the inputs. For example, $x^2+y^2=1$ specifies a circle. On $-1<x<1$, it determines the two real branches $y=\pm \sqrt{1-x^2}$, each an implicit function when one branch is selected. The implicit function theorem gives local conditions for a unique differentiable branch; implicit differentiation computes its derivative without first solving explicitly.

Excitation energy is an energy level's energy above a specified ground state. Emitted photon energies measure differences between levels, with a small recoil correction when the emitter has finite mass.