Geodesic, the Glossary
In geometry, a geodesic is a curve representing in some sense the shortest path (arc) between two points in a surface, or more generally in a Riemannian manifold.[1]
Table of Contents
113 relations: Acceleration (differential geometry), Action (physics), Affine connection, Antipodal point, Arc length, Brachistochrone curve, Calculus of variations, Cambridge University Press, Capacity of a set, Cauchy–Schwarz inequality, Christoffel symbols, Classical mechanics, Closed geodesic, Communications of the ACM, Complete manifold, Connection (mathematics), Critical point (mathematics), Curve, Differentiable manifold, Differential geometry of surfaces, Digital dentistry, Distance, Distance (graph theory), Double tangent bundle, Earth, Ehresmann connection, Einstein notation, Equation, Euclidean geometry, Euler–Lagrange equation, Exponential map (Riemannian geometry), Fiber bundle, Finsler manifold, First variation, Flow (mathematics), Free fall, Free particle, General relativity, Geodesic circle, Geodesic curvature, Geodesic dome, Geodesics as Hamiltonian flows, Geodesics in general relativity, Geodesics on an ellipsoid, Geodesy, Geodetic airframe, Geometry, Graph (discrete mathematics), Graph theory, Great circle, ... Expand index (63 more) »
- Geodesic (mathematics)
Acceleration (differential geometry)
In mathematics and physics, acceleration is the rate of change of velocity of a curve with respect to a given linear connection. Geodesic and acceleration (differential geometry) are differential geometry.
See Geodesic and Acceleration (differential geometry)
Action (physics)
In physics, action is a scalar quantity that describes how the balance of kinetic versus potential energy of a physical system changes with trajectory.
See Geodesic and Action (physics)
Affine connection
In differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. Geodesic and affine connection are differential geometry.
See Geodesic and Affine connection
Antipodal point
In mathematics, two points of a sphere (or n-sphere, including a circle) are called antipodal or diametrically opposite if they are the endpoints of a diameter, a straight line segment between two points on a sphere and passing through its center.
See Geodesic and Antipodal point
Arc length
Arc length is the distance between two points along a section of a curve.
Brachistochrone curve
In physics and mathematics, a brachistochrone curve, or curve of fastest descent, is the one lying on the plane between a point A and a lower point B, where B is not directly below A, on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time.
See Geodesic and Brachistochrone curve
Calculus of variations
The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.
See Geodesic and Calculus of variations
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge.
See Geodesic and Cambridge University Press
Capacity of a set
In mathematics, the capacity of a set in Euclidean space is a measure of the "size" of that set.
See Geodesic and Capacity of a set
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the inner product between two vectors in an inner product space in terms of the product of the vector norms.
See Geodesic and Cauchy–Schwarz inequality
Christoffel symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection.
See Geodesic and Christoffel symbols
Classical mechanics
Classical mechanics is a physical theory describing the motion of objects such as projectiles, parts of machinery, spacecraft, planets, stars, and galaxies.
See Geodesic and Classical mechanics
Closed geodesic
In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. Geodesic and closed geodesic are differential geometry and geodesic (mathematics).
See Geodesic and Closed geodesic
Communications of the ACM
Communications of the ACM is the monthly journal of the Association for Computing Machinery (ACM).
See Geodesic and Communications of the ACM
Complete manifold
In mathematics, a complete manifold (or geodesically complete manifold) is a (pseudo-) Riemannian manifold for which, starting at any point, there are straight paths extending infinitely in all directions. Geodesic and complete manifold are differential geometry and geodesic (mathematics).
See Geodesic and Complete manifold
Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as tangent vectors or tensors in the tangent space, along a curve or family of curves in a parallel and consistent manner. Geodesic and connection (mathematics) are differential geometry.
See Geodesic and Connection (mathematics)
Critical point (mathematics)
In mathematics, a critical point is the argument of a function where the function derivative is zero (or undefined, as specified below).
See Geodesic and Critical point (mathematics)
Curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus.
See Geodesic and Differentiable manifold
Differential geometry of surfaces
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Geodesic and differential geometry of surfaces are differential geometry.
See Geodesic and Differential geometry of surfaces
Digital dentistry
Digital dentistry refers to the use of dental technologies or devices that incorporates digital or computer-controlled components to carry out dental procedures rather than using mechanical or electrical tools.
See Geodesic and Digital dentistry
Distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are.
Distance (graph theory)
In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them.
See Geodesic and Distance (graph theory)
Double tangent bundle
In mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle of the total space TM of the tangent bundle of a smooth manifold M. A note on notation: in this article, we denote projection maps by their domains, e.g., πTTM: TTM → TM. Geodesic and double tangent bundle are differential geometry.
See Geodesic and Double tangent bundle
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life.
Ehresmann connection
In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of the notion of a connection, which makes sense on any smooth fiber bundle.
See Geodesic and Ehresmann connection
Einstein notation
In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity.
See Geodesic and Einstein notation
Equation
In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign.
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements.
See Geodesic and Euclidean geometry
Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional.
See Geodesic and Euler–Lagrange equation
Exponential map (Riemannian geometry)
In Riemannian geometry, an exponential map is a map from a subset of a tangent space TpM of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself. Geodesic and exponential map (Riemannian geometry) are differential geometry.
See Geodesic and Exponential map (Riemannian geometry)
Fiber bundle
In mathematics, and particularly topology, a fiber bundle (''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure.
Finsler manifold
In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold where a (possibly asymmetric) Minkowski norm is provided on each tangent space, that enables one to define the length of any smooth curve as Finsler manifolds are more general than Riemannian manifolds since the tangent norms need not be induced by inner products. Geodesic and Finsler manifold are differential geometry.
See Geodesic and Finsler manifold
First variation
In applied mathematics and the calculus of variations, the first variation of a functional J(y) is defined as the linear functional \delta J(y) mapping the function h to where y and h are functions, and ε is a scalar.
See Geodesic and First variation
Flow (mathematics)
In mathematics, a flow formalizes the idea of the motion of particles in a fluid.
See Geodesic and Flow (mathematics)
Free fall
In classical mechanics, free fall is any motion of a body where gravity is the only force acting upon it.
Free particle
In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies.
See Geodesic and Free particle
General relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. Geodesic and general relativity are differential geometry.
See Geodesic and General relativity
Geodesic circle
A geodesic circle is either "the locus on a surface at a constant geodesic distance from a fixed point" or a curve of constant geodesic curvature. Geodesic and geodesic circle are geodesic (mathematics).
See Geodesic and Geodesic circle
Geodesic curvature
In Riemannian geometry, the geodesic curvature k_g of a curve \gamma measures how far the curve is from being a geodesic. Geodesic and geodesic curvature are geodesic (mathematics).
See Geodesic and Geodesic curvature
Geodesic dome
A geodesic dome is a hemispherical thin-shell structure (lattice-shell) based on a geodesic polyhedron.
See Geodesic and Geodesic dome
Geodesics as Hamiltonian flows
In mathematics, the geodesic equations are second-order non-linear differential equations, and are commonly presented in the form of Euler–Lagrange equations of motion. Geodesic and geodesics as Hamiltonian flows are geodesic (mathematics).
See Geodesic and Geodesics as Hamiltonian flows
Geodesics in general relativity
In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Geodesic and geodesics in general relativity are geodesic (mathematics).
See Geodesic and Geodesics in general relativity
Geodesics on an ellipsoid
The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. Geodesic and geodesics on an ellipsoid are differential geometry and geodesic (mathematics).
See Geodesic and Geodesics on an ellipsoid
Geodesy
Geodesy or geodetics is the science of measuring and representing the geometry, gravity, and spatial orientation of the Earth in temporally varying 3D.
Geodetic airframe
A geodetic airframe is a type of construction for the airframes of aircraft developed by British aeronautical engineer Barnes Wallis in the 1930s (who sometimes spelt it "geodesic").
See Geodesic and Geodetic airframe
Geometry
Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.
Graph (discrete mathematics)
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related".
See Geodesic and Graph (discrete mathematics)
Graph theory
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.
Great circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Great-circle distance
The great-circle distance, orthodromic distance, or spherical distance is the distance between two points on a sphere, measured along the great-circle arc between them.
See Geodesic and Great-circle distance
Group action
In mathematics, many sets of transformations form a group under function composition; for example, the rotations around a point in the plane.
Hamilton–Jacobi equation
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics.
See Geodesic and Hamilton–Jacobi equation
Hamiltonian mechanics
In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833.
See Geodesic and Hamiltonian mechanics
Hamiltonian vector field
In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian.
See Geodesic and Hamiltonian vector field
Infimum and supremum
In mathematics, the infimum (abbreviated inf;: infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists.
See Geodesic and Infimum and supremum
Interval (mathematics)
In mathematics, a (real) interval is the set of all real numbers lying between two fixed endpoints with no "gaps".
See Geodesic and Interval (mathematics)
Intrinsic metric
In the mathematical study of metric spaces, one can consider the arclength of paths in the space.
See Geodesic and Intrinsic metric
Jacobi field
In Riemannian geometry, a Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic.
Levi-Civita connection
In Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves the (pseudo-)Riemannian metric and is torsion-free.
See Geodesic and Levi-Civita connection
Lexico
Lexico was a dictionary website that provided a collection of English and Spanish dictionaries produced by Oxford University Press (OUP), the publishing house of the University of Oxford.
Line (geometry)
In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a straightedge, a taut string, or a ray of light.
See Geodesic and Line (geometry)
Line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints.
Liouville's theorem (Hamiltonian)
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics.
See Geodesic and Liouville's theorem (Hamiltonian)
Local coordinates
Local coordinates are the ones used in a local coordinate system or a local coordinate space.
See Geodesic and Local coordinates
Local property
In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some sufficiently small or arbitrarily small neighborhoods of points).
See Geodesic and Local property
Mathematical constant
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems.
See Geodesic and Mathematical constant
McGraw Hill Education
McGraw Hill is an American publishing company for educational content, software, and services for pre-K through postgraduate education.
See Geodesic and McGraw Hill Education
Metric space
In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points.
Metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. Geodesic and metric tensor are differential geometry.
See Geodesic and Metric tensor
Michael Stevens (YouTuber)
Michael David Stevens (born January 23, 1986) is an American educator, public speaker, entertainer, and editor best known for creating and hosting the education YouTube channel Vsauce.
See Geodesic and Michael Stevens (YouTuber)
Motion planning
Motion planning, also path planning (also known as the navigation problem or the piano mover's problem) is a computational problem to find a sequence of valid configurations that moves the object from the source to destination.
See Geodesic and Motion planning
Neighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.
See Geodesic and Neighbourhood (mathematics)
Number line
In elementary mathematics, a number line is a picture of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin point representing the number zero and evenly spaced marks in either direction representing integers, imagined to extend infinitely.
Open set
In mathematics, an open set is a generalization of an open interval in the real line.
Orbit
In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point.
Ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable.
See Geodesic and Ordinary differential equation
Oxford University Press
Oxford University Press (OUP) is the publishing house of the University of Oxford.
See Geodesic and Oxford University Press
Parallel transport
In differential geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold.
See Geodesic and Parallel transport
Planetary surface
A planetary surface is where the solid or liquid material of certain types of astronomical objects contacts the atmosphere or outer space.
See Geodesic and Planetary surface
Point particle
A point particle, ideal particle or point-like particle (often spelled pointlike particle) is an idealization of particles heavily used in physics.
See Geodesic and Point particle
Poisson's equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics.
See Geodesic and Poisson's equation
Proceedings of the National Academy of Sciences of the United States of America
Proceedings of the National Academy of Sciences of the United States of America (often abbreviated PNAS or PNAS USA) is a peer-reviewed multidisciplinary scientific journal.
See Geodesic and Proceedings of the National Academy of Sciences of the United States of America
Projective connection
In differential geometry, a projective connection is a type of Cartan connection on a differentiable manifold. Geodesic and projective connection are differential geometry.
See Geodesic and Projective connection
Pseudo-Riemannian manifold
In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. Geodesic and pseudo-Riemannian manifold are differential geometry.
See Geodesic and Pseudo-Riemannian manifold
Pushforward (differential)
In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Geodesic and pushforward (differential) are differential geometry.
See Geodesic and Pushforward (differential)
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an inner product on the tangent space at each point that varies smoothly from point to point). Geodesic and Riemannian geometry are differential geometry.
See Geodesic and Riemannian geometry
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Geodesic and Riemannian manifold are differential geometry.
See Geodesic and Riemannian manifold
Rubber band
A rubber band (also known as an elastic band, gum band or lacky band) is a loop of rubber, usually ring or oval shaped, and commonly used to hold multiple objects together.
Satellite
A satellite or artificial satellite is an object, typically a spacecraft, placed into orbit around a celestial body.
Shortest path problem
In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized.
See Geodesic and Shortest path problem
SIAM Journal on Computing
The SIAM Journal on Computing is a scientific journal focusing on the mathematical and formal aspects of computer science.
See Geodesic and SIAM Journal on Computing
Skew-symmetric matrix
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative.
See Geodesic and Skew-symmetric matrix
Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number, called differentiability class, of continuous derivatives it has over its domain.
Spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum.
Sphere
A sphere (from Greek) is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. Geodesic and sphere are differential geometry.
Spherical Earth
Spherical Earth or Earth's curvature refers to the approximation of the figure of the Earth as a sphere.
See Geodesic and Spherical Earth
Spherical trigonometry
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions.
See Geodesic and Spherical trigonometry
Spray (mathematics)
In differential geometry, a spray is a vector field H on the tangent bundle TM that encodes a quasilinear second order system of ordinary differential equations on the base manifold M. Usually a spray is required to be homogeneous in the sense that its integral curves t→ΦHt(ξ)∈TM obey the rule ΦHt(λξ). Geodesic and spray (mathematics) are differential geometry.
See Geodesic and Spray (mathematics)
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
See Geodesic and Springer Science+Business Media
Sub-Riemannian manifold
In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold.
See Geodesic and Sub-Riemannian manifold
Tangent bundle
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself.
See Geodesic and Tangent bundle
Tangent space
In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. Geodesic and tangent space are differential geometry.
See Geodesic and Tangent space
Tautological one-form
In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T^Q of a manifold Q. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus providing a bridge between Lagrangian mechanics and Hamiltonian mechanics (on the manifold Q).
See Geodesic and Tautological one-form
Test particle
In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be insufficient to alter the behaviour of the rest of the system.
See Geodesic and Test particle
Torsion tensor
In differential geometry, the torsion tensor is a tensor that is associated to any affine connection. Geodesic and torsion tensor are differential geometry.
See Geodesic and Torsion tensor
Torus
In geometry, a torus (tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle.
Unit tangent bundle
In Riemannian geometry, the unit tangent bundle of a Riemannian manifold (M, g), denoted by T1M, UT(M) or simply UTM, is the unit sphere bundle for the tangent bundle T(M).
See Geodesic and Unit tangent bundle
UV mapping
UV mapping is the 3D modeling process of projecting a 3D model's surface to a 2D image for texture mapping.
Vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n.
Vertex (graph theory)
In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices).
See Geodesic and Vertex (graph theory)
Vertical and horizontal bundles
In mathematics, the vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle.
See Geodesic and Vertical and horizontal bundles
Wiley (publisher)
John Wiley & Sons, Inc., commonly known as Wiley, is an American multinational publishing company that focuses on academic publishing and instructional materials.
See Geodesic and Wiley (publisher)
See also
Geodesic (mathematics)
- Alexandrov's uniqueness theorem
- Closed geodesic
- Complete manifold
- Complex geodesic
- Geodesic
- Geodesic bicombing
- Geodesic circle
- Geodesic convexity
- Geodesic curvature
- Geodesic deviation
- Geodesic map
- Geodesics as Hamiltonian flows
- Geodesics in general relativity
- Geodesics on an ellipsoid
- Lyusternik–Fet theorem
- Prime geodesic
- The spider and the fly problem
- Theorem of the three geodesics
- Vector flow
References
[1] https://en.wikipedia.org/wiki/Geodesic
Also known as Affine parameter, Geodesic Spray, Geodesic Triangle, Geodesic equation, Geodesic flow, Geodesic length, Geodesic path, Geodesic planes, Geodesic segment, Geodesics.
, Great-circle distance, Group action, Hamilton–Jacobi equation, Hamiltonian mechanics, Hamiltonian vector field, Infimum and supremum, Interval (mathematics), Intrinsic metric, Jacobi field, Levi-Civita connection, Lexico, Line (geometry), Line segment, Liouville's theorem (Hamiltonian), Local coordinates, Local property, Mathematical constant, McGraw Hill Education, Metric space, Metric tensor, Michael Stevens (YouTuber), Motion planning, Neighbourhood (mathematics), Number line, Open set, Orbit, Ordinary differential equation, Oxford University Press, Parallel transport, Planetary surface, Point particle, Poisson's equation, Proceedings of the National Academy of Sciences of the United States of America, Projective connection, Pseudo-Riemannian manifold, Pushforward (differential), Riemannian geometry, Riemannian manifold, Rubber band, Satellite, Shortest path problem, SIAM Journal on Computing, Skew-symmetric matrix, Smoothness, Spacetime, Sphere, Spherical Earth, Spherical trigonometry, Spray (mathematics), Springer Science+Business Media, Sub-Riemannian manifold, Tangent bundle, Tangent space, Tautological one-form, Test particle, Torsion tensor, Torus, Unit tangent bundle, UV mapping, Vector field, Vertex (graph theory), Vertical and horizontal bundles, Wiley (publisher).