Euclidean geometry

Euclidean geometry describes flat Euclidean space using lengths, angles, and rigid motions. A coordinate realization uses points $x,y\in {\mathbb{R}}^n$, Euclidean distanceand angles determined by the dot product. The coordinate axes are a convenience: translations and rotations change coordinates while preserving these measurements. This makes Euclidean geometry useful for comparing figures and converting measurements along different directions into a single distance.

In the Euclidean plane, through a point outside a line there is exactly one parallel line. This is a standard equivalent formulation of the parallel postulate. Euclid's fifth postulate in the Elements instead describes when two lines meet using the interior angles made by a crossing line.^[ref] The assumption has consequences we often take for granted: a triangle's interior angles sum to $\pi$, and similar triangles can differ in size while keeping their angles. Comparing with spherical geometry reveals that these are properties of a particular geometry.

As a computed example, the diagonal from $(0,0)$ to $(4,3)$ in a rectangle has length $\sqrt{4^2+3^2}=5$. The Pythagorean theorem explains why perpendicular contributions add as squares. If the directions cease to meet at a right angle, the law of cosines supplies the missing angle-dependent term. The beauty is that a single distance formula unites geometric constructions, coordinate calculations, and orthogonality in linear algebra.

Make the assumptions visible: in a Cartesian coordinate system with orthogonal axes and the same length unit on each axis, take $A=(0,0)$, $B=(4,0)$ and $C=(0,3)$. The vectors $B-A=(4,0)$ and $C-A=(0,3)$ have dot product zero, establishing a right angle at $A$ before using the Pythagorean theorem. A sketch that merely looks like a right triangle supplies no such guarantee. Likewise, an image stretched horizontally and vertically by different amounts can distort both lengths and angles; a displayed shape and the assumptions defining it must be kept separate.

For a less origin-dependent measurement, take $P=(1,2)$ and $Q=(5,5)$. Their displacement vector is $Q-P=(4,3)$, so their Euclidean distance is again $5$. Its Euclidean norm records the direct separation, whereas travelling first horizontally and then vertically covers length $4+3=7$. This connects measuring a diagonal to locating two positions from a Cartesian coordinate system. A translation changes where the positions are recorded without changing their separation; a rotation changes the component values without changing the Euclidean norm.

A compass-and-straightedge construction uses an unmarked straightedge to draw a Euclidean line through two already constructed points, and a compass to draw a circle with a constructed centre and radius. New points arise from intersections of these lines and circles. The allowed operations matter: an accurate numerical drawing need not supply such a construction.

Consider the constructible angle whose cosine is $5/6$ and sine is $\sqrt{11}/6$. If one fifth of this angle were constructible, its cosine $c$ would satisfy the displayed polynomial, by the identity $\cos 5\theta =16{\cos }^5\theta -20{\cos }^3\theta +5\cos \theta$. The identity follows by taking the real part of $(\cos \theta +i\sin \theta {)}^5$ and substituting ${\sin }^2\theta =1-{\cos }^2\theta$.

The polynomial is irreducible by Eisenstein's criterion at five, so $c$ has degree five over the rational numbers. The initial angle field $\mathbb{Q}(\sqrt{11})$ already has degree two, and adjoining any finite collection of constructible numbers gives total degree over the rational numbers a power of two. Such a field cannot contain the degree-five element $c$, by the tower law for field extensions. Hence this constructible angle cannot be divided into five equal angles by straightedge and compass construction. Some particular angles can be quintisected; the obstruction disproves a construction valid for all angles.

Doubling the volume of a unit cube requires constructing its new side length $\sqrt[3]{2}$. Its minimal polynomial $X^3-2$ is irreducible by Eisenstein's criterion at two, so this length has degree three. Every constructible number belongs to a finite tower of quadratic extensions, whose total degree is a power of two. The tower law for field extensions would force the degree three to divide that power of two, a contradiction. Thus no such straightedge and compass construction exists.

A constructible polygon is a polygon whose vertices can be obtained by a compass-and-straightedge construction from the specified initial data. The classical question for a regular polygon asks which numbers of sides permit an exact construction from a given side or circumscribed circle.

A constructible regular polygon is a regular polygon admitting a compass-and-straightedge construction. The Gauss-Wantzel theorem classifies its possible numbers of sides.

For $N\ge 3$, a regular polygon with $N$ sides is a constructible regular polygon exactly when $N=2^a{p}_1\cdots p_r$, where $a$ is a nonnegative integer and the $p_j$ are distinct Fermat primes; the product may be empty. Thus seventeen and fifteen qualify, while nine does not. Carl Friedrich Gauss established sufficiency, and Pierre Wantzel proved necessity in 1837.^[ref]

Intersection of $o+\lambda r$ with the plane $n\cdot (p-a)=0$, when $n\cdot r\ne 0$. For a ray, require $\lambda \ge 0$. A zero denominator requires separate parallel or coplanar handling.

A ray-plane intersection followed by an inside-triangle test, conveniently using nonnegative barycentric coordinates. Oriented surfaces may additionally require a front-facing hit through back-face culling.

Points are coplanar when they belong to a common plane. For three vectors based at the origin of three-dimensional Euclidean space, this means their span has dimension at most two. Equivalently their scalar triple product, and hence the determinant with those vectors as columns, is zero. This includes dependent or zero vectors.

Geometry in three-dimensional Euclidean space, including volume and surface area.

Removing a central cylindrical hole from a sphere leaves a solid whose volume depends only on its remaining height $h$. With $a=h/2$, sphere radius $R$ and hole radius $r_0$, the rim gives $R^2-r_0^2=a^2$. Every horizontal slice is an annulus of area $\pi (a^2-z^2)$, so integrating from $-a$ to $a$ gives $V=4\pi a^3/3=\pi h^3/6$. This slice equality explains why changing $R$ while retaining $h$ leaves the volume unchanged.

The common region inside equal-radius circular cylinders whose axes meet orthogonally. Two cylinders give a bicylinder; three give a tricylinder.

The solid intersection of two equal-radius infinite circular cylinders whose axes meet at a right angle. For radius $a$, the cross section at height $z$ along the common perpendicular coordinate is a square of area $4(a^2-z^2)$, so its volume is $16a^3/3$.

The common intersection of three equal-radius circular cylinders with mutually perpendicular axes. The displayed inequalities describe a centered example of radius $a$. It is a Steinmetz solid; its surface area is given by the surface area of a tricylinder formula.

For three radius-$a$ circular cylinders with mutually perpendicular axes through one point, the boundary of their common intersection has three congruent cylindrical surface pieces. On $x^2+y^2=a^2$, the height restriction is $|z|\le a\min (|\sin \theta |,|\cos \theta |)$. Eight congruent sectors give surface area $8a^2(2-\sqrt{2})$ per piece. Pairwise intersections are curves of zero surface area, so the three pieces add.

The area of a planar section through a body. A homogeneous diagonal deformation with transverse stretch ratios ${\lambda }_1$ and ${\lambda }_2$ changes the area of a transverse section from ${\mathcal{A}}_0$ to $\mathcal{A}={\lambda }_1{\lambda }_2{\mathcal{A}}_0$. Cauchy stress tensor components use current cross-sectional area, whereas nominal tractions use reference cross-sectional area.

A line for which reflection leaves a figure unchanged.

The dihedral angle measures the angle between two intersecting planes. Molecular torsion angles use an oriented four-atom convention to specify a related signed angle.

For four consecutively bonded atoms A–B–C–D, the torsion angle is the signed dihedral between planes A–B–C and B–C–D. State the viewing and sign convention when comparing projections.

The ratio between corresponding lengths of two geometrically similar figures. Multiplying every length by $t>0$ multiplies areas by $t^2$ and volumes by $t^3$. For integer-sided triangles, choosing a positive integer scale factor preserves integer side lengths.

Two intersecting straight lines are perpendicular if their angle is $90^{\circ }$. Their direction vectors have zero dot product.

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.

For a coaxial cylinder standing on the base of a right circular cone of height $h>0$ and semi-vertical angle $0<\alpha <\pi /2$, similar triangles give radius $r=(h-x)\tan \alpha$ at cylinder height $x$. Its volume is $V=\pi x(h-x{)}^2{\tan }^2\alpha$. Differentiation gives $V^{\prime }=\pi {\tan }^2\alpha (h-x)(h-3x)$, so the unique positive-volume maximum has $x=h/3$ and occupies $4/9$ of the cone's volume. Its curved surface area is instead $A=2\pi x(h-x)\tan \alpha$, maximized at $x=h/2$. Changing the optimized quantity changes the optimal shape, even though the geometric constraint is identical.

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 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.

A projective quadric is the zero set of a nonzero homogeneous quadratic polynomial in projective space. Over a field of characteristic different from two, represent it by a symmetric matrix $B$, up to a nonzero scalar. The gradient at a representative $v$ is $2Bv$, so the singular locus is the projectivization of $\ker B$. Thus the quadric is nonsingular exactly when $B$ is invertible. In the complex projective plane, ranks three, two, and one give respectively a smooth projective conic, two distinct projective lines, and a double projective line.

The equation of a quadric surface after orthogonal diagonalization of its symmetric matrix and appropriate translation. The spectral theorem for real symmetric matrices removes mixed quadratic terms, and completing squares removes linear terms in directions with nonzero eigenvalues. Linear terms in zero-eigenvalue directions can remain and generate elliptic paraboloids or hyperbolic paraboloids. The signs, zero coefficients and constant distinguish ordinary surfaces, degenerate conics in plane sections, and empty real loci.

A quadric surface obtained by extending a parabola along a fixed direction outside its plane. For example, $x=c-y^2/(4c)$ with arbitrary $z$ is a parabolic cylinder parallel to the $z$ axis. Each cross-section of fixed $z$ has vertex $(c,0)$ and focus $(0,0)$ in its $xy$ plane. It occurs as a coordinate surface in parabolic cylindrical coordinates.

For a real symmetric matrix $A$, every stationary squared distance $x^{T}x$ on $x^{T}Ax=1$ occurs in a positive-eigenvalue eigenspace. The Lagrange multiplier equation is $x=\Lambda Ax$; multiplying by $x^T$ gives $\Lambda =|x{|}^2>0$, hence $Ax=|x{|}^{-2}x$. If $A$ is a positive-definite matrix, the nearest and farthest squared distances are $1/{\lambda }_{\max }$ and $1/{\lambda }_{\min }$. With a repeated eigenvalue, the stationary set may contain an entire sphere within its eigenspace. For an indefinite or singular form the surface may be unbounded, and a stationary point need not be a global extremum.

Let $A$ be a real positive-definite matrix and let the columns of $U$ be an orthonormal basis of a two-dimensional linear subspace in three-dimensional Euclidean space. Points in that central plane are $x=Uq$, with $|x{|}^2=|q{|}^2$. Restricting the ellipsoid $x^{T}Ax=1$ gives $q^T(U^{T}AU)q=1$. The compressed matrix $U^{T}AU$ is a positive-definite matrix, and its extreme eigenvalues give the distance formulas in the title by the Rayleigh quotient. The two extremal axes are the images under $U$ of the corresponding eigenvectors. This restriction reduces a problem with two Lagrange multipliers to a two-dimensional quadratic form.

For $a,b>0$, this quadric surface is independent of its third coordinate and is formed by translating a hyperbola along that direction. Its two connected components correspond to $u\ge a$ and $u\le -a$. In elliptic cylindrical coordinates, fixing a generic angular coordinate selects a half-branch of such a cylinder because the signs of both planar coordinates are fixed by that angular value.

For $a,b>0$, an elliptic cylinder is a quadric surface with the displayed equation in suitable Cartesian coordinates, independent of the third coordinate. It is formed by translating an ellipse parallel to a line perpendicular to its plane. The generator direction is the zero-eigenvalue direction of its quadratic part, while the transverse semiaxes are $a,b$. It is circular if $a=b$.

After an orthogonal diagonalization, the sign of the third coefficient classifies the displayed quadric surface. If $C>0$, it is an ellipsoid with semiaxes $A^{-1/2},B^{-1/2},C^{-1/2}$. If $C=0$, it is an elliptic cylinder parallel to the $w$ axis, with elliptical transverse section. If $C<0$, it is a one-sheet hyperboloid with axis $w$ and waist ellipse $A{u}^2+B{v}^2=1$. The continuous limit $C\to 0$ sends the associated axial length scale to infinity. Repeated positive eigenvalues make the corresponding principal-axis choices nonunique within their eigenspace, without changing the surface.

An elliptic cone is a real quadric surface consisting of the two nappes $u=\pm \sqrt{(b{v}^2+c{w}^2)/a}$, joined at the vertex. Its intersections with planes $u=u_0\ne 0$ are ellipses. It is the zero level of an indefinite quadratic form of signature $(1,2)$. Changing the level to a positive value produces a two-sheeted hyperboloid; changing it to a negative value produces a one-sheet hyperboloid, when the positive coefficient is assigned to the axial coordinate $u$.

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 $-b$.

For two parallel lines ${\mathbf{a}}_1+s\mathbf{v}$ and ${\mathbf{a}}_2+t\mathbf{v}$ in three-dimensional Euclidean space, with $\mathbf{v}\ne 0$ and $\mathbf{d}={\mathbf{a}}_2-{\mathbf{a}}_1$, their shortest distance isEvery joining displacement is $\mathbf{d}+(t-s)\mathbf{v}$. Its component perpendicular to $\mathbf{v}$ is fixed, and its parallel component can be cancelled by a suitable $t-s$. The Pythagorean theorem proves that this perpendicular component has the shortest Euclidean norm. The cross product expression follows from $\Vert \mathbf{d}\times \mathbf{v}\Vert =\Vert \mathbf{d}\Vert \Vert \mathbf{v}\Vert \sin \theta$. For coincident lines the distance is zero. The perpendicular-component formula also applies in any finite-dimensional Euclidean space, where a three-dimensional cross product need not be available.

Two equally spaced families of perpendicular lines partitioning the Euclidean plane into congruent squares. A general affine map can turn this grid into parallelograms, or collapse directions if its linear part of an affine map is not injective.

In the Euclidean plane, precisely one line through a point outside a given line is parallel to it. This formulation is equivalent to Euclid's fifth postulate within the usual axioms of plane geometry.

Figures are congruent when a rigid motion maps one onto the other, preserving all Euclidean distances.

This surface consists of rays from a vertex $\mathbf{a}$ whose directions make a fixed angle $\beta$ with a unit axis vector $\hat{\mathbf{n}}$. The displayed equation follows from the dot product formula for that angle. For $0<\beta <\pi$, $\beta \ne \pi /2$, it gives one circular nappe; $\beta =\pi /2$ gives a plane, and $\beta =0,\pi$ gives a degenerate ray. Squaring the equation produces both nappes, so the unsquared sign distinguishes them. Generalized angular momentum in a radial monopole field confines particle directions to such a surface.

The Euclidean group $E(n)=O(n)⋉{\mathbb{R}}^n$ is the group of all isometries $x\mapsto Ax+b$ of Euclidean space, with $A$ an orthogonal matrix. Its orientation-preserving subgroup is $SE(n)=SO(n)⋉{\mathbb{R}}^n$, the semidirect product of the special orthogonal group with translations. The Euclidean motion of Euclidean three-space and the proper Euclidean motion of Euclidean three-space are the respective three-dimensional instances.

The group of Euclidean isometries preserving orientation of ${\mathbb{R}}^n$. Its elements have the form $x\mapsto Ax+b$, with $A$ in the special orthogonal group and $b\in {\mathbb{R}}^n$. The function composition law is $(A,b)(B,c)=(AB,Ac+b)$, the identity element is $(I,0)$, and the inverse element is $(A^{-1},-A^{-1}b)$. These formulas prove closure and inverses; function composition supplies associativity. For $n=2$, $A$ is a planar rotation.

A geometric locus is the set of points satisfying a specified geometric or algebraic condition in an ambient space. For example, points at a fixed positive distance $r$ from $p$ form a circle in the plane and a sphere in three dimensions. In the complex plane, $|z-z_0|=r$ has Cartesian equation $(x-\mathrm{Re}{z}_0{)}^2+(y-\mathrm{Im}{z}_0{)}^2=r^2$, proving the circle description. A locus can also be a union, such as $x(y-1)=0$, which consists of two lines; domain restrictions in the original equation must be retained when simplifying it.

A hypercube is the Cartesian product of $n$ equal-length closed intervals in Euclidean space. The centered cube $[-1,1{]}^n$ has $2^n$ vertices, each with coordinates in $\{-1,1\}$. Antipodal vertices determine $2^{n-1}$ body diagonals.

Two sign-vector vertices of a centered hypercube differing in $h$ coordinates have dot product $n-2h$. Thus their body diagonals can be orthogonal only in even dimension. A collection of mutually orthogonal nonzero diagonals has at most $n$ members; achieving $n$ is equivalent to the existence of a Hadamard matrix of order $n$. For unoriented lines the angle is determined by $|n-2h|/n$, while oriented vertex vectors retain the sign of their dot product.

The arithmetic mean of a finite nonempty collection of vertices, treating each vertex as an equal point mass. For a polygon, this need not equal the uniform-area centroid of its interior. A translation or linear map commutes with this average; consequently rotating all vertices about their vertex centroid leaves that centroid fixed.

For a planar region $R$ of finite positive area $A$, its first moments of area about the coordinate axes are the displayed double integrals, provided they exist. Its centroid is $(\overline{x},\overline{y})=(M_y/A,M_x/A)$. This follows by applying the definition $\overline{\mathbf{r}}=A^{-1}{\iint }_R\mathbf{r}dA$ to each component. The subscript indicates the axis about which the moment is taken: the first moment about the $x$-axis weights by $y$, not $x$.

For a nondegenerate $d$-dimensional simplex $S$ with vertices $v_0,\dots ,v_d$ and an affine function $f$, the integral equals the volume times the value at the centroid $c=(v_0+\cdots +v_d)/(d+1)$. To prove this, write $x={\sum }_i{\lambda }_i(x)v_i$ in a barycentric coordinate system. On the standard simplex, symmetry permuting the vertices gives equal integrals of all ${\lambda }_i$; since ${\sum }_i{\lambda }_i=1$, each is $\mathrm{vol}(S)/(d+1)$. An invertible affine map transfers this result to any simplex. Affineness gives $f(x)={\sum }_i{\lambda }_i(x)f(v_i)$, so linearity of integration proves the formula. For a tetrahedron the factor is one quarter. The rule is exact for affine integrands; a nonlinear integrand generally needs additional moments.

A regular star polygon $\{p/q\}$ joins every $q$-th vertex of a regular $p$-gon. When $\gcd (p,q)=1$, this traces one closed polygonal circuit.

For a simple polygon with cyclically ordered vertices of a polygon $(x_i,y_i)$, its signed area is $\frac{1}{2}{\sum }_i(x_i{y}_{i+1}-x_{i+1}{y}_i)$, with indices read cyclically. Reversing the traversal changes the sign; taking the absolute value gives the area. Each summand is a two-dimensional determinant.

Positive real number lengths $a_1,\dots ,a_k$, with $k\ge 3$, are the sides of a nondegenerate polygon exactly when $2{\max }_i{a}_i<{\sum }_i{a}_i$. Necessity follows from the triangle inequality: the longest side must be shorter than the path formed by the remaining sides.

The semiperimeter is half the perimeter. For a triangle with side lengths $a,b,c$, it is $s=(a+b+c)/2$.

The sum of the lengths of its sides of a polygon. For a triangle with side lengths $a,b,c$, the perimeter of a polygon is $a+b+c$.

A polygon with five sides and five vertices. A regular pentagon has equal sides and angles; its symmetry group is the order-ten dihedral group, consisting of five rotations and five reflections. Each reflection axis passes through a vertex and the midpoint of the opposite edge.

A polygon with six sides and six vertices. A regular hexagon has equal sides and angles, with each interior angle equal to $2\pi /3$. Its symmetry group consists of six rotations and six reflections, forming a dihedral group of order twelve.

A vertex of a polygon is an endpoint where consecutive sides of a polygon meet.

A side of a polygon is a line segment joining two consecutive vertices of a polygon in its boundary.

A diagonal of a polygon joins two vertices of a polygon that are not endpoints of the same side of a polygon. A square has two diagonals of a polygon, forming a two-element group orbit under its dihedral group of symmetries.

A Euclidean polygon with four sides and four right angles. Opposite sides are parallel and have equal lengths. A rectangle is a square exactly when its adjacent side lengths agree.

A rectangle with all four side lengths equal. It is a regular four-sided polygon. Rotations and translations preserve this definition; its sides need not be parallel to the coordinate axes.

A square frame formed by removing a concentric square hole from a larger square. If outer and inner sides are positive integers $a>b$ with the same integer parity, the border has integer thickness $t=(a-b)/2$ and contains $a^2-b^2=4t(b+t)$ unit tiles.

The unit square here is the Cartesian product $[0,1]\times [0,1]$ in the Euclidean plane, a square of side length one.

A parallelogram is the convex set with vertices $p,p+u,p+v,p+u+v$ for two linearly independent planar vectors $u,v$. Its opposite edges are parallel and its centre $p+(u+v)/2$ gives central symmetry. Its area is $|\det [u,v]|$, so an invertible linear map changes its area by the absolute value of that map's determinant. These facts make parallelograms convenient convex bodies for two-dimensional applications of the Minkowski convex body theorem.

For a quadrilateral with side lengths $a,b,c,d$, semiperimeter $s$, area $K$, and opposite angles $A,C$,The second term vanishes exactly when $A+C=\pi$, which is the cyclicity condition for a nondegenerate convex quadrilateral.

The point group of a Euclidean crystallographic group is the group of its orthogonal linear parts, equivalently its quotient by the translation subgroup. It preserves the translation Euclidean lattice. In an arbitrary dimension it is a finite orthogonal group; for a plane crystallographic group this lattice preservation restricts the allowed rotation orders.

A physical property tensor must be invariant under every symmetry operation of the material's point group. For a polar rank-$r$ tensor, $T_{i_1\dots i_r}=R_{i_1{j}_1}\cdots R_{i_r{j}_r}{T}_{j_1\dots j_r}$. Taking $R=-I$ proves that an odd-rank polar tensor vanishes in a centrosymmetric material. A susceptibility of order $n$ has rank $n+1$, so local bulk even-order electric-dipole response is forbidden there. The name refers to Franz Ernst Neumann.

The spatial symmetry of a fixed molecular geometry. Its point group acts on atomic positions, orbitals and vibrational coordinates; these actions need not have the same group representation.

A linear combination of atomic displacements adapted to an irreducible representation of a molecular point group. Symmetry coordinates reduce the dynamical matrix into independent symmetry blocks; several coordinates in the same block can still mix.

The molecular point group of a centrosymmetric linear molecule. Longitudinal coordinates have $\Sigma$ symmetry; their behaviour under inversion about the inversion centre supplies the $g$ or $u$ label, and a vertical reflection supplies the sign on a $\Sigma$ representation.

The eight-operation molecular point group of an ideal square pyramid. Its polar-vector components transform as $A_1$ for the principal-axis coordinate and $E$ for the transverse pair. Symmetric quadratic functions also contain $B_1$ and $B_2$, allowing Raman spectroscopy activity beyond the infrared modes.

A transformation that preserves a specified structure, including the chemical identity of sites in a molecule. Rotations, reflections, inversion and improper rotations can occur in a molecular point group. The associated action on atomic orbitals also changes their orientations and signs, which must be included when computing a character of a representation.

The point group of orthogonal spatial transformations that preserve a fixed molecular geometry, including the positions and chemical identities of all nuclei. Changing a conformation can change the molecular point group even when the molecular connectivity is unchanged. Its group representations on atomic orbitals depend on orbital signs and rotations as well as the permutation of centres.

The molecular point group containing identity, one twofold rotation, an inversion centre and a mirror plane perpendicular to the twofold axis. Its four operations are $E,C_2,i,{\sigma }_h$. An extended anti conformation of hexa-1,5-diyne provides an example; changing the conformation can change the symmetry.

The six-operation molecular point group with a threefold proper rotation axis and three vertical mirror planes. Its rotational symmetry number is three because mirror operations do not count as proper spatial rotations.

The 24-operation molecular point group of a regular planar hexagon with equivalent sites. Its irreducible representations have inversion labels $g,u$; polar-vector components transform as $A_{2u}$ for $z$ and $E_{1u}$ for $(x,y)$.

The 24-operation molecular point group of an ideal regular tetrahedron, including improper operations. Its irreducible representations are $A_1,A_2,E,T_1,T_2$, with dimensions $1,1,2,3,3$. The polar-vector representation is $T_2$, the axial-vector representation is $T_1$, and symmetric quadratic functions transform as $A_1+E+T_2$.

The 15 Cartesian displacements of tetrahedral methane have characters $(15,0,-1,-1,3)$ on classes $(E,8C_3,3C_2,6S_4,6{\sigma }_d)$. Removing translations $T_2$ and rotations $T_1$ gives $(9,0,1,-1,3)$ and the stated decomposition. There are nine vibrational coordinates in four frequency families.

In a centrosymmetric molecule, a fundamental vibration cannot be both electric-dipole infrared active and ordinary Raman spectroscopy active. Dipole components are odd under inversion; polarizability components are even. This selection rule concerns modes, not accidental numerical equality of unrelated frequencies.

For a finite molecular point group, the multiplicity of irrep $\lambda$ in a representation with character $\chi$ is $n_{\lambda }=|G{|}^{-1}{\sum }_C|C|\chi (C){\chi }_{\lambda }(C{)}^{\ast }$. The sum is over conjugacy classes.