Maths AI terms in Geometry and trigonometry

An angle greater than 0° and less than 90°.

The square matrix of a graph with $n$ vertices in which the entry in row $i$, column $j$ is the number of edges from vertex $i$ to vertex $j$.

Two edges that share a common vertex.

In a right-angled triangle, the side next to a given acute angle that is not the hypotenuse.

Two vertices that are joined by at least one edge.

A transformation of the form ${\mathit{r}}^{\mathrm{\prime }}=A\mathit{r}+\mathit{b}$, where $A$ is a $2\times 2$ matrix and $\mathit{b}$ is a translation vector; it applies the matrix transformation and then the translation.

A finite sequence of precisely stated steps that solves a problem or carries out a task, and that can be followed mechanically.

A pair of equal angles formed on opposite sides of a line that crosses two parallel lines, each lying between the parallels.

The situation in which two sides of a triangle and an angle that is not between them are given, and the data fit two different triangles, one with an acute angle and one with the obtuse supplement of that angle.

The angle between a line and its projection onto the plane, measured at the point where the line meets the plane.

The acute angle (between 0° and 90°) formed by two lines, found from the angle between their direction vectors; if the angle between the direction vectors is obtuse, the angle between the lines is its supplement $180\mathrm{°}-\theta$.

The angle $\theta$, with $0\mathrm{°}\le \theta \le 180\mathrm{°}$, between two non-zero vectors drawn from the same starting point; it satisfies $\cos \theta =\frac{\mathit{v}\cdot \mathit{w}}{\mathrm{\mid }\mathit{v}\mathrm{\mid }\mathrm{\mid }\mathit{w}\mathrm{\mid }}$.

The GDC setting that fixes whether angles are entered and given in degrees or in radians.

The angle measured downwards from the horizontal to the line of sight of an object below the observer.

The angle measured upwards from the horizontal to the line of sight of an object above the observer.

The angle that an object's diameter subtends at the eye of an observer, which for a distant object is approximately its diameter divided by its distance, in radians.

The angle through which an object rotates about a fixed point, usually measured in radians.

The angle turned through per unit time by a rotating object, usually measured in radians per second (rad s⁻¹) and written $\omega$.

The part of an annulus between two radii; for central angle $\theta$ degrees, outer radius $R$ and inner radius $r$ its area is $\frac{\theta }{360}\times \pi (R^2-r^2)$.

The region between two concentric circles; for outer radius $R$ and inner radius $r$ its area is $\pi R^2-\pi r^2=\pi (R^2-r^2)$.

A part of the circumference of a circle between two points on the circle.

The distance along an arc of a circle; for radius $r$ and central angle $\theta$ degrees it is $l=\frac{\theta }{360}\times 2\pi r$.

A unit of angle equal to $\frac{1}{3600}$ of a degree, written ″; a milliarcsecond is one thousandth of an arcsecond.

The area of any triangle equals $\frac{1}{2}ab\sin C$, half the product of two sides multiplied by the sine of the angle between them.

The number by which a transformation multiplies every area; for the transformation ${\mathit{r}}^{\mathrm{\prime }}=A\mathit{r}+\mathit{b}$ it equals $\mathrm{\mid }\det A\mathrm{\mid }$.

The bearing of the starting point from the destination, which differs from the original bearing by 180°.

The vectors $\mathit{i}$, $\mathit{j}$ and $\mathit{k}$ of length 1 in the positive directions of the $x$-, $y$- and $z$-axes; every vector can be written as a sum of multiples of them.

The direction of one point from another, given as the angle measured clockwise from north, written with three digits from 000° to 360°.

A line, ray or segment that divides a line segment or an angle into two equal parts.

The volume of liquid or other material that a hollow container can hold, often measured in litres or millilitres.

An equation connecting $x$ and $y$ only, found by eliminating $t$ from the parametric equations $x=x(t)$ and $y=y(t)$ of a moving object, which describes the curve the object travels along.

The region of a Voronoi diagram that contains one site and every point that is closer to that site than to any other site.

The angle between two radii of a circle, with its vertex at the centre; it is the angle that an arc or chord subtends at the centre.

The fixed point from which distances are multiplied by the scale factor in an enlargement; it is the only point that an enlargement with scale factor $k\mathrm{\ne }1$ leaves unmoved.

The fixed point about which a rotation turns the plane; it is the only point that a rotation through an angle that is not a multiple of 360° leaves unmoved.

The problem of finding the route of least total weight that travels along every edge of a connected weighted graph at least once and returns to its starting vertex.

A straight line segment joining two points on a circle.

A trail that starts and finishes at the same vertex, so it has no repeated edges.

Motion along a circle; for uniform circular motion about the origin, the position is $\mathit{r}=\left(\begin{array}{c}R\cos (\omega t)\\ R\sin (\omega t)\end{array}\right)$, the distance from the centre is always $R$, and the speed is constant.

The point that is equidistant from the three vertices of a triangle; it is where the perpendicular bisectors of the three sides meet.

The circle that passes through all three vertices of a triangle; its centre is the circumcentre and its radius is the distance from the circumcentre to any vertex.

The distance around a circle, equal to $2\pi r$ for a circle of radius $r$.

An instrument for measuring an angle of elevation or depression, by sighting an object along it and reading the angle from the horizontal.

A walk that starts and finishes at the same vertex.

The moment at which the distance between two moving objects, or between a moving object and a fixed point, is smallest, together with that smallest distance.

Two angles between a pair of parallel lines on the same side of a line crossing them; they add up to 180°.

Lines that have every point in common, so that two different equations describe the same line.

Lying on the same straight line; points $A$, $B$ and $C$ are collinear when $\overrightarrow{AC}$ is a scalar multiple of $\overrightarrow{AB}$.

The event of two moving objects being at the same position at the same time, so that ${\mathit{r}}_A(t)={\mathit{r}}_B(t)$ for one value of $t$ that satisfies every component.

A vector written as a vertical list of its components in brackets, with the $x$-component at the top, then the $y$-component and, in three dimensions, the $z$-component.

A simple graph in which every pair of distinct vertices is joined by exactly one edge; the complete graph on $n$ vertices is written $K_n$.

The signed amount by which a vector moves in the direction of one of the coordinate axes; a vector in two dimensions has an $x$-component and a $y$-component, and one in three dimensions also has a $z$-component.

A solid formed by joining two or more simpler solids, or by removing one solid from another.

The flat shape that folds to make the curved surface of a cone, a sector whose radius is the cone's slant height and whose arc length equals the circumference of the cone's base.

A graph in which every vertex can be reached from every other vertex by travelling along a sequence of edges.

Motion in which the velocity vector does not change, so the object travels in a straight line at constant speed and its position at time $t$ is $\mathit{r}={\mathit{r}}_0+t\mathit{v}$.

The function whose value $\cos \theta$ is the $x$-coordinate of the point where the arm of the angle $\theta$, in standard position, meets the unit circle.

For an acute angle $\theta$ in a right-angled triangle, $\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}$.

In any triangle $ABC$, $c^2=a^2+b^2-2ab\cos C$, which links the three sides with one angle; rearranged, $\cos C=\frac{a^2+b^2-c^2}{2ab}$.

The area of the curved part of the surface of a solid such as a cylinder, cone or sphere, not including any flat faces.

A walk that starts and finishes at the same vertex and repeats no other vertex, so it also repeats no edge.

A solid with two identical, parallel circular ends joined by a curved surface, so that every cross-section parallel to the ends is the same circle.

In PageRank, the probability $d$ that at each step the random surfer follows a link from the current page; with probability $1-d$ the surfer instead jumps to a page chosen at random from the whole network.

A unit of angle equal to $\frac{1}{360}$ of a full turn, written with the symbol °.

The number of edge-ends that meet at the vertex, so each ordinary edge adds 1 and each loop adds 2; it is written $\mathrm{deg}(V)$.

The division of the region between a set of sites into triangles, formed by joining every two sites whose Voronoi cells share an edge.

A method for a lower bound for the travelling salesman problem that deletes one vertex and its edges, finds a minimum spanning tree of the remaining graph, and adds the weights of the two edges of least weight from the deleted vertex.

A graph in which every edge has a direction, from one vertex to another, shown by an arrow on the edge; it is also called a digraph.

A line segment with a direction from its starting point to its end point, drawn as an arrow, used to represent a vector; the segment from $A$ to $B$ is written $\overrightarrow{AB}$.

A non-zero vector parallel to a line, which gives the line's direction; any non-zero scalar multiple of it is also a direction vector of the same line.

The vector $\overrightarrow{AB}$ from a point $A$ to a point $B$, which describes the change in position from $A$ to $B$; it equals $\mathit{b}-\mathit{a}$, the position vector of the end point minus that of the starting point.

A connection between two vertices of a graph, drawn as a line (straight or curved) joining them.

A boundary between two neighbouring cells of a Voronoi diagram; it is a line segment or ray lying on the perpendicular bisector of the two sites whose cells it separates.

A transformation that maps every point $P$ to the point $P^{\mathrm{\prime }}$ on the line through a fixed centre $C$ and $P$ with $\overrightarrow{C{P}^{\mathrm{\prime }}}=k\overrightarrow{CP}$, where $k$ is the scale factor; it changes the size of a shape but not its shape.

Vectors that have the same magnitude and the same direction; in component form, vectors whose corresponding components are all equal.

A boundary on which every point is the same distance from the nearest points of two coastlines or regions; between two single points it is their perpendicular bisector.

At the same distance from two or more given points or objects.

The length of the straight line segment joining two points, calculated in two dimensions as $\sqrt{(x_1-x_2{)}^2+(y_1-y_2{)}^2}$.

A circuit that uses every edge of a graph exactly once, starting and finishing at the same vertex.

A connected graph that has an Eulerian circuit, which happens exactly when every vertex has even degree.

A trail that uses every edge of a graph exactly once.

A vertex whose degree is an even number, including 0.

A shape with detailed structure at every scale, usually generated by iteration, in which parts of the shape repeat the pattern of the whole.

The part of a cone or pyramid that remains when the top is cut off by a plane parallel to the base.

A set of vertices together with a set of edges, where each edge joins two vertices (or one vertex to itself).

The branch of mathematics that studies graphs, structures made of objects (vertices) and connections between pairs of them (edges).

An algorithm that makes the choice that looks best at each step, without reconsidering earlier choices.

A cycle that visits every vertex of a graph exactly once and returns to the starting vertex.

A path that visits every vertex of a graph exactly once.

Half of a sphere, formed by cutting the sphere with a plane through its centre; a solid hemisphere has a curved surface and a flat circular face.

A level line through the observer's position, from which angles of elevation and depression are measured.

The side of a right-angled triangle opposite the right angle; it is always the longest side.

The number of edges of a directed graph that end at (point into) a vertex.

The angle between two given sides of a triangle, at the vertex where they meet.

A method of constructing a Voronoi diagram by adding the sites one at a time, updating the existing diagram after each new site is added.

The position vector ${\mathit{r}}_0$ of a moving object at time $t=0$.

Two lines that have exactly one point in common, their point of intersection.

A function, ${\sin }^{-1}$, ${\cos }^{-1}$ or ${\tan }^{-1}$, that takes the value of a trigonometric ratio and gives the angle with that ratio.

A finite set of transformations, usually affine, that are applied repeatedly, either all together to a shape or one at a time chosen at random to a point, to generate a fractal.

The process of applying the same rule repeatedly, each time to the result of the previous application.

The branch of mechanics that describes the motion of objects (their position, velocity and acceleration) without considering the forces that cause the motion.

A method for finding a minimum spanning tree by repeatedly choosing the edge of least weight that does not form a cycle with the edges already chosen, until $n-1$ edges have been chosen.

The largest circle, centred inside a given region, that has no site inside it; its centre solves the toxic waste dump problem and its radius is the distance from that centre to the nearest site.

The number of edges in the walk, counting each repeated edge every time it is used.

The straight line from an observer's eye to the object being observed.

The part of a straight line between two endpoints, written $[AB]$ for the segment with endpoints $A$ and $B$; its length is written $AB$.

The set of all points that satisfy a given condition.

An edge that joins a vertex to itself.

The length of a vector, written $\mathrm{\mid }\mathit{v}\mathrm{\mid }$; for $\mathit{v}=v_1\mathit{i}+v_2\mathit{j}+v_3\mathit{k}$ it equals $\sqrt{v_1^2+v_2^2+v_3^2}$.

The longer of the two arcs joining two points on a circle; its central angle is greater than 180°.

The point on a line segment that is the same distance from both endpoints; for endpoints $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ it is $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2})$.

A spanning tree of a weighted graph whose total weight is as small as possible.

The shorter of the two arcs joining two points on a circle; its central angle is less than 180°.

The fixed line in which a reflection takes place; every point on it is unmoved, and each other point is mapped to the point the same distance away on the other side, along a perpendicular to the line.

Two or more edges that join the same pair of vertices.

A method that builds a Hamiltonian cycle by starting at a given vertex, repeatedly travelling to the nearest vertex not yet visited, and finally returning to the start; the weight of this cycle is an upper bound for the travelling salesman problem.

Estimating the value of a quantity at a point by the value measured at the nearest site, so that every point in a Voronoi cell is given the value of that cell's site.

The vector $-\mathit{v}$ with the same magnitude as $\mathit{v}$ and the opposite direction; its components are the components of $\mathit{v}$ with their signs changed.

A two-dimensional shape that can be folded along its edges to form the surface of a three-dimensional solid.

A triangle in which none of the three angles is 90°.

A non-zero vector perpendicular to a plane, or to every vector lying in that plane; $\mathit{v}\times \mathit{w}$ is a normal to the plane containing $\mathit{v}$ and $\mathit{w}$, and dividing it by its magnitude gives a unit normal vector $\mathit{n}$.

Dividing a non-zero vector by its magnitude to give the unit vector in the same direction, $\frac{\mathit{v}}{\mathrm{\mid }\mathit{v}\mathrm{\mid }}$.

Describes a problem that is at least as hard as every problem in a large class of problems whose solutions can be checked quickly; no known algorithm solves every instance of an NP-hard problem in a time that grows only as a power of the problem's size.

The point, line or shape to which a transformation is applied, before it is moved.

An angle greater than 90° and less than 180°.

A vertex whose degree is an odd number.

In a right-angled triangle, the side across from a given acute angle, which does not form either arm of that angle.

The direction, clockwise or anticlockwise, in which the labelled vertices of a shape occur; a transformation with a positive determinant preserves it and one with a negative determinant reverses it.

The number of edges of a directed graph that start at (point out of) a vertex.

A method of ranking the pages of a network of hyperlinks by the long-run proportion of time a random surfer, following links at random (with occasional random jumps), spends on each page.

A way of splitting the odd vertices of a graph into pairs, so that the shortest path between each pair can be repeated in the Chinese postman problem.

The apparent change in the direction of an object when it is viewed from two different positions; for a star, the parallax angle is half the shift seen from opposite sides of Earth's orbit.

Non-zero vectors $\mathit{v}$ and $\mathit{w}$ such that $\mathit{w}=k\mathit{v}$ for some scalar $k$; they have the same direction when $k>0$ and opposite directions when $k<0$.

The separate equations $x=x_0+\lambda l$, $y=y_0+\lambda m$, $z=z_0+\lambda n$ that give each coordinate of a point on a line in terms of the parameter $\lambda$.

A walk in which no vertex is repeated, and therefore no edge is repeated.

The total length of the boundary of a sector, equal to its arc length plus two radii, $P=\frac{\theta }{360}\times 2\pi r+2r$.

A function whose values repeat at regular intervals, so that $f(x+p)=f(x)$ for every $x$ and some fixed positive number $p$.

The line that passes through the midpoint of a line segment and is perpendicular to it; it is the set of all points that are the same distance from the two endpoints of the segment.

The length of the part of a vector $\mathit{a}$ acting at right angles to a vector $\mathit{b}$, in the plane containing both, given by $\frac{\mathrm{\mid }\mathit{a}\times \mathit{b}\mathrm{\mid }}{\mathrm{\mid }\mathit{b}\mathrm{\mid }}=\mathrm{\mid }\mathit{a}\mathrm{\mid }\sin \theta$.

The shortest distance from a point $P$ to a line, measured along the line through $P$ that meets it at right angles; for a line through $A$ with direction $\mathit{b}$, it equals $\frac{\mathrm{\mid }\overrightarrow{AP}\times \mathit{b}\mathrm{\mid }}{\mathrm{\mid }\mathit{b}\mathrm{\mid }}$.

Two non-zero vectors at right angles to each other; $\mathit{v}$ and $\mathit{w}$ are perpendicular if and only if $\mathit{v}\cdot \mathit{w}=0$.

The vector $\overrightarrow{OA}$ from the origin $O$ to a point $A$, usually written $\mathit{a}$; its components are the coordinates of $A$.

A method for finding a minimum spanning tree that starts at a chosen vertex and repeatedly adds the edge of least weight joining a vertex already in the tree to a vertex not yet in the tree, until every vertex is included.

The single value returned by an inverse trigonometric function: for ${\sin }^{-1}$ and ${\tan }^{-1}$ an angle from $-\frac{\pi }{2}$ to $\frac{\pi }{2}$, and for ${\cos }^{-1}$ an angle from 0 to $\pi$.

A solid with two identical, parallel polygonal ends joined by rectangular faces, so that every cross-section parallel to the ends is the same shape and size.

The motion of an object moving freely under gravity alone, in which the horizontal component of velocity is constant and the vertical component decreases at the constant rate $g$, so that the path is a parabola.

For a line segment, the segment in the plane that joins the feet of the perpendiculars dropped from the segment's endpoints to the plane.

In a right-angled triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides, $c^2=a^2+b^2$.

The trigonometric identity ${\cos }^2\theta +{\sin }^2\theta =1$, true for every angle $\theta$.

One of the four regions into which the $x$- and $y$-axes divide the coordinate plane, numbered 1 to 4 anticlockwise starting from the region where both coordinates are positive.

The angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle.

A process in which a walker moves from vertex to vertex along edges, at each step choosing one of the edges leaving the current vertex at random, all such edges being equally likely.

The part of a line that starts at a point and continues without end in one direction; with the line $\mathit{r}=\mathit{a}+\lambda \mathit{b}$, it is the set of points with $\lambda \ge 0$.

The position of an object B from the viewpoint of an object A, given by the vector $\overrightarrow{AB}={\mathit{r}}_B-{\mathit{r}}_A$; when both objects move, it is a function of time.

The velocity of an object B from the viewpoint of an object A, given by ${\mathit{v}}_B-{\mathit{v}}_A$; it is the rate of change of the relative position ${\mathit{r}}_B-{\mathit{r}}_A$.

Multiplying a vector by a positive scalar so that it has a required magnitude while keeping its direction; the vector of magnitude $L$ in the direction of $\mathit{v}$ is $\frac{L}{\mathrm{\mid }\mathit{v}\mathrm{\mid }}\mathit{v}$.

The single vector equal to the sum of two or more vectors, which has the same effect as all of them acting together.

A unit of rotation rate equal to the number of complete turns made in one minute.

A solid with a circular base and a curved surface that rises to a point (the apex) directly above the centre of the base.

A solid with a polygonal base and triangular faces meeting at a point (the apex) that lies directly above the centre of the base.

The rule fixing the direction of $\mathit{v}\times \mathit{w}$; if the fingers of the right hand curl from $\mathit{v}$ towards $\mathit{w}$ through the angle between them, the thumb points in the direction of $\mathit{v}\times \mathit{w}$.

A transformation that turns every point through the same angle, in a stated direction (clockwise or anticlockwise), about a fixed centre, keeping its distance from the centre the same.

A drawing made using only an unmarked straight edge and a pair of compasses, in which every length and point comes from circles and lines rather than from measurement.

A quantity that is fully described by its size (a single number with a unit) and has no direction.

The length of the part of a vector $\mathit{a}$ that acts in the direction of a vector $\mathit{b}$, given by $\frac{\mathit{a}\cdot \mathit{b}}{\mathrm{\mid }\mathit{b}\mathrm{\mid }}=\mathrm{\mid }\mathit{a}\mathrm{\mid }\cos \theta$, where $\theta$ is the angle between them; it is negative when $\theta$ is obtuse.

The number $\mathit{v}\cdot \mathit{w}=v_1{w}_1+v_2{w}_2+v_3{w}_3$ formed from two vectors $\mathit{v}$ and $\mathit{w}$, which also equals $\mathrm{\mid }\mathit{v}\mathrm{\mid }\mathrm{\mid }\mathit{w}\mathrm{\mid }\cos \theta$, where $\theta$ is the angle between them.

The region of a circle bounded by two radii and the arc between them.

The area of a sector of a circle; for radius $r$ and central angle $\theta$ degrees it is $A=\frac{\theta }{360}\times \pi r^2$.

The region of a circle bounded by a chord and the arc between the chord's ends.

The property of a shape that contains smaller copies of itself, each the image of the whole shape under a transformation.

A connected graph that has an Eulerian trail but no Eulerian circuit, which happens exactly when it has exactly two vertices of odd degree.

A number system with base 60, in which each unit is divided into 60 smaller units.

The path of least total weight between two given vertices of a weighted graph.

The fractal obtained from a solid triangle by repeatedly removing the middle triangle joining the midpoints of the sides of every remaining triangle; it is made of three copies of itself, each enlarged with scale factor $\frac{1}{2}$.

Solids with the same shape, in which every length of one is $k$ times the corresponding length of the other, so that corresponding areas are in the ratio $k^2$ and volumes in the ratio $k^3$.

A graph with no loops and no multiple edges, so each pair of distinct vertices is joined by at most one edge.

The function whose value $\sin \theta$ is the $y$-coordinate of the point where the arm of the angle $\theta$, in standard position, meets the unit circle.

For an acute angle $\theta$ in a right-angled triangle, $\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}$.

In any triangle $ABC$, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$, where each side is divided by the sine of the angle opposite it.

One of the given points around which a Voronoi diagram is built, such as a town, a shop, a phone mast or a weather station.

Lines in three dimensions that are not parallel and do not intersect.

The distance from the apex of a cone or pyramid to the edge of its base, measured along the sloping surface; for a pyramid it is the height of a triangular face, measured from the midpoint of a base edge.

A memory aid for the right-angled ratios: Sine = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse, Tangent = Opposite ÷ Adjacent.

Finding all the unknown sides and angles of a triangle from the ones that are given.

A line segment joining two vertices of a solid that do not lie on the same face, passing through the inside of the solid; in a cuboid it joins opposite corners.

A subgraph of a connected graph that is a tree and contains every vertex of the graph.

The solid made of all the points within a fixed distance $r$ (the radius) of a fixed point (the centre); its surface is the set of points exactly $r$ from the centre.

A figure on the surface of a sphere bounded by three arcs of great circles; its angles add up to more than 180°.

An angle drawn with its vertex at the origin, its first arm along the positive $x$-axis, and measured anticlockwise to its second arm; a negative angle is measured clockwise.

A directed graph in which, for every ordered pair of vertices, there is a sequence of edges, each followed in its own direction, leading from the first vertex to the second.

A graph whose vertices all belong to a given graph and whose edges all belong to that graph, each joining two of the chosen vertices.

A line segment or arc subtends an angle at a point when lines drawn from its two ends to the point form that angle.

Two angles whose sum is 180° ($\pi$ radians); supplementary angles have equal sines and cosines of opposite sign.

The total area of all the faces and surfaces of a solid, measured in square units.

A square matrix that is unchanged when reflected in its leading diagonal, so the entry in row $i$, column $j$ equals the entry in row $j$, column $i$ for every $i$ and $j$.

A table giving, for every pair of vertices in a graph, the weight of the shortest path between them.

The function defined by $\tan \theta =\frac{\sin \theta }{\cos \theta }$ for every angle $\theta$ with $\cos \theta \mathrm{\ne }0$.

For an acute angle $\theta$ in a right-angled triangle, $\tan \theta =\frac{\text{opposite}}{\text{adjacent}}$.

An ordered triple $(x,y,z)$ that gives the position of a point in space as its distances along three mutually perpendicular axes that meet at the origin $O(0,0,0)$.

Replacing $t$ by $t-a$ in an equation of motion, which describes the same motion started $a$ units of time later; an object that starts from ${\mathit{r}}_0$ at time $t=a$ with constant velocity $\mathit{v}$ has position $\mathit{r}={\mathit{r}}_0+(t-a)\mathit{v}$ for $t\ge a$.

The turning effect of a force $\mathit{F}$ applied at position $\mathit{r}$ relative to a pivot, given by the vector product $\mathit{\tau }=\mathit{r}\times \mathit{F}$; its magnitude $\mathrm{\mid }\mathit{r}\mathrm{\mid }\mathrm{\mid }\mathit{F}\mathrm{\mid }\sin \theta$ is measured in newton metres (N m), and its direction, by the right-hand screw rule, is the axis about which the force turns the object.

The problem of finding the point in a given region whose distance to the nearest site is as large as possible.

A walk in which no edge is repeated; vertices may be repeated.

A $2\times 2$ matrix $A$ that defines a transformation of the plane by mapping each point with position vector $\mathit{r}$ to the point with position vector $A\mathit{r}$.

The problem of finding a route of least total weight that visits every vertex of a weighted graph and returns to the start. In the classical problem the graph is complete and the route is a Hamiltonian cycle of least weight; in the practical problem each vertex must be visited at least once.

A connected graph that contains no cycles, where a cycle is a closed route that returns to its starting vertex without repeating any edge or any other vertex.

The rule that to add two vectors, the second is drawn starting from the end of the first, and the sum is the vector from the start of the first to the end of the second.

A method of finding the positions of distant points by measuring the angles to them from the ends of a baseline of known length, and solving the resulting triangles.

An equation in which the unknown appears inside one or more trigonometric functions.

An equation involving trigonometric functions that is true for every value of the angle for which both sides are defined.

The circle of radius 1 with its centre at the origin of a coordinate grid.

The square with vertices $(0,0)$, $(1,0)$, $(1,1)$ and $(0,1)$, whose image under a transformation matrix shows the effect of the matrix, with the columns of the matrix as the images of $(1,0)$ and $(0,1)$.

A vector with magnitude 1.

A velocity whose magnitude or direction (or both) changes with time, so that the velocity vector is a function $\mathit{v}(t)$ and the path is in general curved.

A quantity that has both a size (magnitude) and a direction.

An equation $\mathit{r}=\mathit{a}+\lambda \mathit{b}$ that gives the position vector $\mathit{r}$ of every point on a line, where $\mathit{a}$ is the position vector of a known point on the line, $\mathit{b}$ is a direction vector and $\lambda$ is a real parameter.

For two vectors in three dimensions, the vector $\mathit{v}\times \mathit{w}=\left(\begin{array}{c}{v}_2{w}_3-v_3{w}_2\\ v_3{w}_1-v_1{w}_3\\ v_1{w}_2-v_2{w}_1\end{array}\right)$, which is perpendicular to both $\mathit{v}$ and $\mathit{w}$ and has magnitude $\mathrm{\mid }\mathit{v}\mathrm{\mid }\mathrm{\mid }\mathit{w}\mathrm{\mid }\sin \theta$.

The vector $\mathit{v}$ whose components are the rates of change of the components of an object's position, so that its direction is the direction of motion and its magnitude is the speed.

One of the objects of a graph, drawn as a point or a small dot; the plural is vertices.

A point where three or more edges of a Voronoi diagram meet; it is equidistant from the three or more sites whose cells meet there.

The amount of three-dimensional space occupied by a solid, measured in cubic units such as ${\text{cm}}^3$ or ${\text{m}}^3$.

A division of a plane, or of a region of it, into cells, one for each of a given set of sites, in which every point of a cell is closer to that cell's site than to any other site.

A sequence of edges in which each edge after the first starts at the vertex where the previous edge ends; vertices and edges may be repeated.

A table for a weighted graph with one row and one column for each vertex, in which each entry is the weight of the edge joining the two vertices, and a blank (or a dash) means there is no direct edge.

A graph in which each edge has a number, called its weight, attached to it, representing a quantity such as a distance, a cost or a time.

The energy transferred when a constant force $\mathit{F}$ moves an object through a displacement $\mathit{d}$, equal to the scalar product $W=\mathit{F}\cdot \mathit{d}=\mathrm{\mid }\mathit{F}\mathrm{\mid }\mathrm{\mid }\mathit{d}\mathrm{\mid }\cos \theta$; with force in newtons and displacement in metres, work is in joules (J).

The vector $\mathbf{0}$, all of whose components are 0; it has magnitude 0 and no direction.