Euclidean geometry

Euclidean geometry studies distances, angles, and rigid figures in flat space.

An elliptic paraboloid is a smooth bowl-shaped quadratic surface with elliptical horizontal sections. In the circular case $A=B$, it is a surface of revolution. For $z=x^2+y^2$, polar coordinates give the parametrization $(r\cos \theta ,r\sin \theta ,r^2)$ and upward vector area element $(-2r^2\cos \theta ,-2r^2\sin \theta ,r)drd\theta$. A band between two positive heights has two boundary circles and no cap.

The union of the two infinite lateral conical sheets meeting at a common vertex is a double circular cone. The displayed example has axis $z$ and semi-angle $\pi /4$. It is smooth away from the vertex, where its defining gradient is nonzero. A right circular cone truncated on one side is a related bounded solid; this surface includes both axial directions and has no base disks.

Placement of geometric objects without interior overlap. A packing fraction measures how much area or volume they occupy. Circle packing and sphere packing are examples.

Arrangement of spheres without interior overlap. The occupied volume fraction is its packing fraction; it depends on the spatial arrangement and on the radii.

Equal spheres of radius $r$ occupy the sites of a cubic lattice of spacing $2r$. A cell contains one sphere, so its packing fraction is $(4\pi r^3/3)/(2r{)}^3=\pi /6$. This is an ideal arranged deposit rather than a claim about all settling suspensions.

Arrangement of circles in a plane without interior overlap. The covered area fraction is its packing fraction. Equal-radius square packing has fraction $\pi /4$.

Place equal numbers of radii $r_1,r_2$ on alternating sites of a square nearest-neighbour lattice of spacing $d$. Nonoverlap requires $d\ge \max (r_1+r_2,\sqrt{2}{r}_1,\sqrt{2}{r}_2)$. A two-circle cell has area $2d^2$, giving the displayed packing fraction. Its maximum within this geometry occurs at large-to-small radius ratio $1+\sqrt{2}$ and is $\pi (1-1/\sqrt{2})$; smaller interstitial circles leave unused space, while larger ones force the large-circle lattice apart.

Occupied area or volume divided by the corresponding total area or volume of a geometric packing. For a periodic packing, it is the total object area or volume within a repeat cell divided by the cell measure. It is distinct from the number fraction of objects of a given size.

A Euclidean ray from the origin with direction $v\ne 0$ is the geometric half-line $\{tv:t\ge 0\}$. Points in the strictly positive quadrant lie on rays with positive slope.

If finite point sets $P,Q$ lie on two distinct Euclidean rays in a plane, the map $(p,q)\mapsto p+q$ is injective because the two direction vectors are linearly independent. Thus $|P+Q|=|P||Q|$. If all points are nonzero and the rays have positive slope, every sum lies in the open planar sector between those rays.

The open planar sector between two distinct Euclidean rays of positive slope consists of nonzero points whose directions lie strictly between their directions. Sectors between consecutive members of an ordered collection of Euclidean rays are disjoint.

A hyperbolic paraboloid is a saddle-shaped quadric such as $z=xy$. Its graph parametrization $(x,y,xy)$ has Gaussian curvature $-1/(1+x^2+y^2{)}^2$ and is regular everywhere. It is not a hyperboloid: their quadratic equations have different forms.

A right circular cone has an axis through its apex and the centre of its circular base. With the axis chosen as the $z$ axis, its solid interior is $0\le z\le h$, $x^2+y^2\le (z\tan \alpha {)}^2$, where $0<\alpha <\pi /2$ is the semi-angle. Its closed boundary includes both the lateral surface and the base Euclidean disk.

A straight line in Euclidean space is the set $L=\{x_0+tv:t\in \mathbb{R}\}$, for a fixed point $x_0$ and nonzero direction vector $v$. Two distinct points determine one such line.

The Euclidean disk of radius $R$ is $D_R=\{x\in {\mathbb{R}}^2:|x|\le R\}$. Its area is $\pi R^2$.

A polyhedron is a three-dimensional solid assembled from polygonal faces meeting along edges and vertices.

A tetrahedron is the three-dimensional simplex, with four vertices, six edges, and four triangular faces.

For a positively oriented basis $\mathbf{a},\mathbf{b},\mathbf{c}$, put $\tau =\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})>0$. The center of an interior sphere at distance $r$ from each of the three tetrahedron face planes through the origin has the displayed expression. The three inward unit vectors are the normalized cross products $\mathbf{b}\times \mathbf{c}$, $\mathbf{c}\times \mathbf{a}$, $\mathbf{a}\times \mathbf{b}$. Taking inner products with these vectors solves three independent signed-distance equations. Reversing some signs instead produces centers on exterior sides. Existence of a sphere contained in the full tetrahedron also requires clearance from the fourth face.

Two edges of a tetrahedron are opposite when they share no vertex. The six edges split into three opposite pairs.

A Platonic solid is a convex polyhedron whose faces are congruent regular polygons and whose vertex figures are all alike.

A Platonic solid with six square faces, twelve edges and eight vertices. Its rotational symmetry group of a cube has order 24; including orientation-reversing isometries gives the order-48 symmetry group of a cube.

The regular dodecahedron is the Platonic solid with 12 regular pentagonal faces, 20 vertices, and 30 edges.

An Archimedean solid is a convex uniform polyhedron with regular polygonal faces and equivalent vertices, allowing more than one type of face.

The icosidodecahedron has 20 triangular and 12 pentagonal faces, with the faces alternating around each of its 30 vertices. It can be obtained by cutting a dodecahedron through the midpoints of its edges.

For a right triangle with perpendicular side lengths $a,b$ and hypotenuse $c$, the Pythagorean theorem states $a^2+b^2=c^2$.

A Pythagorean triple consists of three positive integers satisfying $x^2+y^2=z^2$. It is primitive when their greatest common divisor is one. Reducing their square numbers modulo $3$ shows that at least one leg is divisible by $3$; reducing modulo $4$ shows that both legs cannot be odd.

Given sites in a Euclidean space, the Voronoi cell of a site is the set of points at least as close to it as to every other site. The Voronoi diagram is the subdivision formed by all of these cells.

A sphere is the set of points at one fixed positive Euclidean distance from a center.

The Cartesian product $S^p\times S^q$ has one cell in each of dimensions $0,p,q,p+q$, counting the two middle cells separately when $p=q$. For positive $p,q$, its integral cohomology ring has generators $a,b$ of degrees $p,q$, with $a^2=b^2=0$ and $ba=(-1{)}^{pq}ab$. The class $ab$ is a product orientation class. This follows from the Künneth theorem.

For positive $p,q$, quotient $S^p\times S^q$ by $(x,y)\mapsto (-x,-y)$. This is a free double covering space action. Its closed manifold quotient is orientable exactly when $p+q$ is even, because the product mapping degree is $(-1{)}^{p+q+2}$. If $p,q\ge 2$, the product is simply connected and the quotient has fundamental group $\mathbb{Z}/2$. For $p=q=2$, the quotient has integral cohomology groups $\mathbb{Z}$ in degrees $0,4$, $\mathbb{Z}/2$ in degrees $2,3$, and zero elsewhere. Every positive-degree cup product is zero. These conclusions follow from Poincare duality, the Euler characteristic under a finite covering, and the universal coefficient theorem for cohomology.

Surface area is the two-dimensional measure induced on the unit sphere in three dimensions. It is invariant under orthogonal rotations. It is not the restriction of ambient three-dimensional Lebesgue measure, under which the sphere has measure zero. Normalizing by the sphere area gives a probability measure.

On a sphere centred at the origin, an antipodal pair is the unordered pair $\{x,-x\}$. The antipodal map exchanges its two points.

For distinct points and $0<\lambda <1$, the distance-ratio locus is a sphere with center $(\mathbf{a}-{\lambda }^2\mathbf{b})/(1-{\lambda }^2)$ and radius $\lambda |\mathbf{a}-\mathbf{b}|/(1-{\lambda }^2)$. Completing the square proves this formula. It encloses $\mathbf{a}$ but excludes $\mathbf{b}$, because their distances to the center are $\lambda R$ and $R/\lambda$. It is the zero-potential locus of two opposite-sign point charges with magnitude ratio $\lambda$. Equal magnitudes instead give a plane.

A hemisphere is either half of a sphere cut by a plane through its center. Its curved surface has area $2\pi r^2$. Closing that surface with its equatorial disk encloses a half-ball of volume $2\pi r^3/3$. The curved surface and enclosed solid have distinct roles in surface-cost and volume calculations.

For a nonzero vector $u$, the open hemisphere of $S^d$ with direction $u$ is $\{x\in S^d:u\cdot x>0\}$. Its boundary is excluded, so opposite open hemispheres are disjoint.

For integers $n=2k+d$ with $k\ge 1$ and $d\ge 0$, there are $n$ labelled points of $S^d$ such that every open hemisphere contains at least $k$ of them. The normalized vectors $(-1{)}^i(1,t_i,\dots ,t_i^d)$ for increasing $t_i$ give a signed moment curve construction. A nonzero polynomial of polynomial degree at most $d$ cannot have enough sign changes for fewer than $k$ positive alternating sample values. Zero sample values can be perturbed negatively by a Lagrange interpolation polynomial. Labels allow repeated positions when $d=0$.

A great circle is the intersection of a sphere with a plane through its centre. Its shorter arcs are geodesics of the round sphere; celestial meridians and the equator are examples.

The gnomonic projection intersects rays from a sphere’s centre with a tangent plane. If $\mathbf{b}$ is its unit normal and $\mathbf{e},\mathbf{n}$ are tangent basis vectors, a direction $\mathbf{s}$ maps to $f(\mathbf{s}\cdot \mathbf{e},\mathbf{s}\cdot \mathbf{n})/(\mathbf{s}\cdot \mathbf{b})$. It maps great circle arcs to straight lines and is singular at the tangent-plane horizon.

The unit sphere $S^n\subseteq {\mathbb{R}}^{n+1}$ is the regular level set $\Vert x{\Vert }^2=1$. Its tangent space is

The two-sphere is the two-dimensional sphere embedded in three-dimensional Euclidean space.

The three-sphere is the unit sphere in four-dimensional Euclidean space:Identifying ${\mathbb{R}}^4$ with ${\mathbb{C}}^2$ realizes it as $\{(z_1,z_2):|z_1{|}^2+|z_2{|}^2=1\}$.

A solid angle is the area cut out on a unit two-sphere by a cone of directions. Its SI unit is the steradian, and a full sphere subtends $4\pi$ steradians.

The normalized angular average is over the unit sphere, with solid angle measure $d\Omega$. Rotational symmetry gives $⟨e^i{⟩}_{\Omega }=0$, $⟨e^i{e}^j{⟩}_{\Omega }={\delta }^{ij}/3$, and $⟨e^i{e}^j{e}^k{e}^{\ell }{⟩}_{\Omega }=({\delta }^{ij}{\delta }^{k\ell }+{\delta }^{ik}{\delta }^{j\ell }+{\delta }^{i\ell }{\delta }^{jk})/15$. The constants follow by contracting indices and using $e^i{e}_i=1$. These identities extract photon angular temperature moments. For normalized spherical harmonics, $\int Y_{\ell m}^{\ast }{Y}_{{\ell }^{\prime }{m}^{\prime }}d\Omega ={\delta }_{\ell {\ell }^{\prime }}{\delta }_{m{m}^{\prime }}$; their angular average instead has an extra factor $1/(4\pi )$.

A triangle is the convex hull of three noncollinear points.

For a triangle with side lengths $a,b,c$ and angle $C$ opposite $c$, the law of cosines states

An acute triangle has all three interior angles strictly smaller than a right angle.

An equilateral triangle has three equal side lengths.

A right triangle has one interior angle equal to a right angle.

A trapezium is a quadrilateral with at least one pair of parallel opposite sides.

A triangular prism is the orthogonal Cartesian product of a triangle with a line segment.

The Borsuk conjecture asserted that every bounded subset of ${\mathbb{R}}^d$ with positive diameter can be partitioned into $d+1$ subsets of strictly smaller diameter. It is false in sufficiently high dimension.

Kahn and Kalai represented uniform subsets by quadratic sign vectors so that diameter pairs correspond to one forbidden intersection size. The Frankl-Wilson theorem bounds each smaller-diameter part, forcing exponentially many parts while the ambient dimension grows only quadratically.

The unit ball in ${\mathbb{R}}^n$ can be covered by at most $9^n$ balls of radius $1/4$. A maximal $1/4$-separated set supplies the centres, and disjoint radius-$1/8$ balls around them fit inside the ball of radius $9/8$.

The open unit disk in the Euclidean plane is

Every bounded planar domain with a sufficiently regular boundary of length $L$ and area $A$ satisfiesEquality holds exactly for circular domains.

Parametrize a positively oriented closed plane curve at constant arc length speed on $[0,2\pi ]$, and let $c_n$ be its complex-position Fourier coefficients. Green's theorem and Parseval's identity give $A=\pi {\sum }_{n}n|c_n{|}^2$ and $L^2/(4{\pi }^2)={\sum }_n{n}^2|c_n{|}^2$. Since $n\le n^2$ for integers, the planar isoperimetric inequality follows. Equality leaves only the constant and positive first harmonic, a translated circle.

Equality in the Cauchy-Schwarz and periodic Wirtinger bounds makes the arclength boundary parametrization a first harmonic with orthogonal equal-length coefficient vectors. The boundary is therefore a circle, and the enclosed domain is a disc.

A cyclic quadrilateral has all four vertices on one circle. Among quadrilaterals with fixed cyclically ordered side lengths, a cyclic quadrilateral has maximal area.

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 parameter $e\ge 0$ in a focus-centred polar coordinates equation $r=\ell /(1-e\cos \theta )$ for a conic section. It distinguishes an ellipse ($e<1$), a parabola ($e=1$), and a hyperbola ($e>1$).

The distance from a focus to a conic section along a line perpendicular to its symmetry axis. In focus-centred polar coordinates, an ellipse may be written $r(1-e\cos \theta )=\ell$, where $e$ is its eccentricity of a conic.

A parabola is a plane curve whose points are equally distant from a focus and a directrix. In suitable coordinates it has equation $y=a{x}^2$, with $a\ne 0$. Its quadratic geometry can also describe a contour deformation through a saddle point.

A hyperbola is a conic section with two unbounded branches. In suitable coordinates its equation is $x^2/a^2-y^2/b^2=1$ with $a,b>0$. The unit hyperbola is parametrized by $x=\cosh t$, $y=\sinh t$ on its right branch, relating it to the hyperbolic functions.

An ellipse is the locus of points whose distances from two fixed foci have constant sum. If the focal distance is $d$ and the sum is $c>d$, its semimajor and semiminor axes are $c/2$ and $\sqrt{c^2-d^2}/2$.

A nondegenerate ellipse has two positive semiaxis lengths. It is a smooth closed curve rather than a point or line segment.

The focal line of an ellipse is the straight line through its two foci.

An ellipsoid is a level set $(x-c{)}^{T}A(x-c)=1$ for a real symmetric positive-definite matrix $A$. Its principal axes are the orthonormal eigenvectors of $A$, and an eigenvalue $\lambda$ gives semi-axis length $1/\sqrt{\lambda }$.

For nonzero real $a_i$, scale the ambient coordinates by $q_i=a_i{r}_i$. On the half where $\epsilon q_i>0$, omitting $q_i$ is a manifold chart onto the open unit ball; its inverse restores $q_i=\epsilon \sqrt{1-{\sum }_{j\ne i}{q}_j^2}$. These $2(n+1)$ charts form a smooth atlas of the ellipsoid, including for negative $a_i$.

A prolate spheroid is an ellipsoid with two equal shorter semiaxes and a longer symmetry-axis semiaxis. Its meridional section is an ellipse rotated about its major axis. In the coordinates $R^2/(\lambda +\alpha )+z^2/(\lambda +\beta )=1$ with $\beta >\alpha$, a fixed regular $\lambda$ has equatorial radius $\sqrt{\lambda +\alpha }$ and polar radius $\sqrt{\lambda +\beta }$; its focal separation is independent of $\lambda$, giving a confocal family.

If $D=\mathrm{diag}(a_1,\dots ,a_{n+1})$ is invertible, $x\mapsto Dx$ sends the ellipsoid ${\sum }_i{a}_i^2{x}_i^2=1$ to the unit sphere. On $u=(v,s)\in S^n$, the two stereographic projections $v/(1-s)$ and $v/(1+s)$ omit opposite poles and have inverses $(2w/(1+|w{|}^2),\pm (|w{|}^2-1)/(1+|w{|}^2))$, with the sign chosen for the appropriate pole. Their overlap transition is $w\mapsto w/|w{|}^2$, smooth and involutive on ${\mathbb{R}}^n\setminus \{0\}$. Composing their inverses with $D^{-1}$ gives a two-chart smooth atlas of the ellipsoid. Its subspace topology is Hausdorff and has a second-countable space topology, so it is a smooth manifold. For $n=0$ the overlap is empty and each chart is a point.

The perpendicular bisector of a line segment is the line through its midpoint at a right angle to the segment.

The circumcenter of a nondegenerate triangle is the common intersection of its three perpendicular bisectors. It is equidistant from the three vertices and is the center of the triangle's circumcircle.

Skew lines are two lines in three-dimensional Euclidean space that are neither parallel nor intersecting.

For lines ${\mathbf{a}}_1+\lambda {\mathbf{u}}_1$ and ${\mathbf{a}}_2+\mu {\mathbf{u}}_2$ with ${\mathbf{u}}_1\times {\mathbf{u}}_2\ne 0$, their shortest distance is

An Apollonius circle is the locus of points whose distances from two fixed points have a prescribed positive ratio different from one. Ratio one gives the perpendicular bisector.

A right angle is an angle of $\pi /2$ radians; two nonzero vectors meet at a right angle exactly when their inner product is zero.

A rigid motion preserves Euclidean distances and therefore preserves angles, lengths, and curvatures.

An angle subtended by a diameter of a circle is right; conversely, the hypotenuse of a right triangle inscribed in a circle is a diameter.

A regular polygon has equal side lengths and equal interior angles. Its vertices on a circle are scaled and rotated roots of unity.

A regular octagon of side length s has area 2(1+sqrt(2))s^2.

A spherical cap is the part of a sphere cut off by a plane.

On the unit sphere $S^{n-1}$, let $D_{\rho }$ be a spherical cap of angular radius $0<\rho <\pi /2$. If $\sin \rho \ge 1/\sqrt{n}$, its surface area is at most $s_{n-1}{\sin }^n\rho$. This includes $\rho \ge \pi /4$ for every $n\ge 2$. The bound follows by comparing the cap-area derivative with the derivative of ${\sin }^n\rho$ and checking the two endpoint values.

A lattice polygon has all vertices in the integer lattice; triangulation shows that its area is a half-integer.