The Central Idea
A geodesic is a path that is intrinsically straight with respect to the geometry of the space it lies in.
In ordinary Euclidean geometry, this idea gives the familiar straight line. In curved geometry, the same idea can produce a path that appears curved when viewed from an outside space.
The key shift is this:
straightness is determined by the geometry of the space, not by the appearance of the path in an external picture.
From Straight Lines to Geodesics
In the Euclidean plane, the shortest path between two distinct points is the line segment joining them.
If the points are
\[ P=(x_1,y_1), \qquad Q=(x_2,y_2), \]then their Euclidean distance is
\[ d(P,Q)=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. \]The straight segment from \(P\) to \(Q\) realizes this distance. It is a Euclidean geodesic.
Geodesics Are an Intrinsic Idea
Geometry can be studied intrinsically, using measurements available within the space itself.
Imagine a traveler who is restricted to the surface of Earth. The traveler measures distances along Earth's surface and does not travel through the planet's interior.
The relevant geometry is therefore surface geometry. The shortest surface routes follow great-circle arcs, even though those arcs appear curved from a viewpoint in three-dimensional space.
Local Straightness
A geodesic is best understood first as a locally straight path.
If we examine a sufficiently small piece of a geodesic, the path cannot be shortened by making a tiny nearby adjustment that keeps its endpoints fixed.
This local condition generalizes the ordinary idea that a straight line does not turn left or right.
Local Does Not Always Mean Global
A crucial distinction is that a geodesic need not be the globally shortest route between every pair of its points.
A geodesic is fundamentally a local notion. A distance-minimizing geodesic additionally realizes the shortest distance between its endpoints.
These ideas often agree over short distances, but they can separate over larger distances.
The Sphere Shows the Difference
On a sphere, complete geodesics are great circles.
Suppose two non-antipodal points lie on the same great circle. Two arcs of that great circle connect them:
- a shorter arc;
- a longer arc.
Both arcs lie on a geodesic, but only the shorter arc is the globally shortest route between the two points.
Geodesics in the Euclidean Plane
In the Euclidean plane, geodesics are ordinary straight lines.
A convenient parametric form is
\[ \mathbf{r}(t)=\mathbf{p}+t\mathbf{v}, \]where \(\mathbf{p}\) is a fixed point and \(\mathbf{v}\neq\mathbf{0}\) is a fixed direction vector.
Since \(\mathbf{v}\) is constant, the direction does not change as the path is followed.
Zero Acceleration Along a Euclidean Geodesic
If a Euclidean geodesic is parameterized at constant speed by
\[ \mathbf{r}(t)=\mathbf{p}+t\mathbf{v}, \]then
\[ \mathbf{r}'(t)=\mathbf{v} \]and
\[ \mathbf{r}''(t)=\mathbf{0}. \]In flat space, no acceleration is needed to remain on a straight path.
Geodesics on a Sphere
On a sphere, the geodesics are the great circles.
A great circle is formed by intersecting the sphere with a plane that passes through the sphere's center.
Examples include the equator and every complete meridian circle.
Why Great Circles Are Special
A small circle such as a circle of latitude away from the equator may look smooth and regular, but it continually bends within the surface.
A great circle, by contrast, follows the locally straightest available direction on the sphere.
This is why long-distance navigation on an approximately spherical Earth is organized around great-circle routes rather than constant-latitude routes.
Great-Circle Distance
Let a sphere have radius \(R\). Suppose two points subtend a central angle \(\theta\), measured in radians, with
\[ 0\leq\theta\leq\pi. \]The length of the shorter great-circle arc is
\[ \boxed{d=R\theta}. \]This is the spherical geodesic distance between the two points.
Example: Distance Along a Sphere
Suppose a sphere has radius
\[ R=6 \]and two points are separated by a central angle
\[ \theta=\frac{\pi}{3}. \]Their geodesic distance is
\[ d=R\theta =6\left(\frac{\pi}{3}\right) =\boxed{2\pi}. \]Antipodal Points
Two points on a sphere are antipodal if they lie directly opposite one another.
Their central angle is
\[ \theta=\pi, \]so their spherical distance is
\[ d=\pi R. \]Unlike nearby points, antipodal points are joined by infinitely many distance-minimizing great-circle semicircles.
Uniqueness Can Fail
In the Euclidean plane, two distinct points determine exactly one straight line and one shortest line segment.
On curved spaces, shortest geodesics need not always be unique.
The antipodal example on the sphere is an important warning: geometric properties that seem automatic in flat space can depend on the curvature and global structure of the space.
Geodesics in Hyperbolic Geometry
Hyperbolic geometry also has geodesics, but their appearance depends on the model used to represent the hyperbolic plane.
In the Poincaré disk model, complete hyperbolic geodesics appear as:
- Euclidean diameters of the disk;
- Euclidean circular arcs meeting the boundary circle at right angles.
They may look curved on the page while remaining intrinsically straight in hyperbolic geometry.
The Boundary of the Poincaré Disk
The boundary circle in the Poincaré disk is not part of the hyperbolic plane.
A complete geodesic approaches two ideal boundary points, but those points are infinitely far away in the hyperbolic metric.
This is another reminder that Euclidean appearance and hyperbolic distance are different structures.
Geodesics Depend on the Metric
A metric tells us how lengths and distances are measured.
If the metric changes, the paths regarded as straight or shortest may also change.
Therefore, the statement
this curve is a geodesic
is incomplete unless the geometry or metric is understood.
Length of a Curve
In the Euclidean plane, a differentiable parametric curve
\[ \mathbf{r}(t)=\bigl(x(t),y(t)\bigr), \qquad a\leq t\leq b, \]has length
\[ L=\int_a^b\sqrt{\left(\frac{dx}{dt}\right)^2+ \left(\frac{dy}{dt}\right)^2}\,dt. \]Geodesic questions generalize the same optimization idea: among nearby admissible paths, which paths make length stationary or minimal?
Length on a Curved Surface
A curved surface may be described using coordinates \((u,v)\). Its metric can have the form
\[ ds^2=E\,du^2+2F\,du\,dv+G\,dv^2. \]Along a path \(u=u(t)\), \(v=v(t)\), the length becomes
\[ L=\int \sqrt{ E\left(\frac{du}{dt}\right)^2+ 2F\frac{du}{dt}\frac{dv}{dt}+ G\left(\frac{dv}{dt}\right)^2 }\,dt. \]This formula is an advanced extension of ordinary arc length. The metric coefficients \(E\), \(F\), and \(G\) encode the local geometry of the surface.
Shortest Paths as an Optimization Problem
If two endpoints are fixed, we can compare the lengths of different paths joining them.
A distance-minimizing geodesic is a path whose length is no greater than the length of any competing path between those endpoints.
This makes geodesics an important meeting point between geometry and optimization.
Stationary Length
More generally, geodesics arise as curves for which the length is stationary under sufficiently small variations.
Stationary does not automatically mean globally minimal. It means the first change in length vanishes for small admissible perturbations.
This idea connects geodesics to calculus of variations, a subject developed more fully in advanced mathematics.
A Flat Surface Can Look Curved
Consider the lateral surface of a circular cylinder.
The cylinder looks curved when embedded in three-dimensional space, but its surface is intrinsically flat. If the cylinder is cut vertically and unrolled, the surface becomes a Euclidean rectangle without stretching.
This makes the cylinder a useful example for separating intrinsic geometry from external appearance.
Geodesics on a Cylinder
Unroll the cylinder into a plane. A geodesic on the cylinder becomes a straight line in the unrolled rectangle.
Depending on the direction of that straight line, the corresponding path on the cylinder may be:
- a vertical generator;
- a horizontal circle;
- a helix winding around the cylinder.
A helix can therefore be intrinsically straight on a cylinder even though it is visibly curved in three-dimensional space.
Unrolling Reveals the Geometry
Suppose a cylinder has circumference \(C\) and two surface points differ by vertical amount \(h\) and by horizontal unrolled displacement \(w\).
In the unrolled rectangle, the straight-line distance is
\[ d=\sqrt{w^2+h^2}. \]Rolling the rectangle back into a cylinder turns that segment into a geodesic on the surface.
Example: A Geodesic on a Cylinder
Suppose two points on an unrolled cylinder differ by
\[ w=6, \qquad h=8. \]The corresponding geodesic length is
\[ d=\sqrt{6^2+8^2} =\sqrt{100} =\boxed{10}. \]When the surface is rolled back up, the path may appear helical, but its intrinsic length remains \(10\).
Multiple Wraps on a Cylinder
Because a cylinder wraps around, different straight segments in the unrolled plane can connect one point to different copies of the other point.
If the circumference is \(C\), horizontal displacements may differ by
\[ w+kC, \qquad k\in\mathbb{Z}. \]The corresponding candidate lengths are
\[ d_k=\sqrt{(w+kC)^2+h^2}. \]The shortest surface route is found by choosing the integer \(k\) that minimizes this quantity.
Geodesics on Other Surfaces
On a general curved surface, geodesics need not have a simple elementary description.
Their behavior depends on the local metric and curvature of the surface. Some surfaces have geodesics that close into loops, while others have geodesics that continue indefinitely without repeating.
The same general principle remains: the path follows intrinsic straightness according to the geometry of the surface.
Geodesic Curvature
A curve lying on a surface can bend in two conceptually different ways:
- because the surface itself bends in the surrounding space;
- because the curve turns within the surface.
Geodesic curvature measures the second kind of bending. A geodesic has geodesic curvature
\[ \boxed{k_g=0}. \]Thus a geodesic does not turn intrinsically within the surface.
Why a Great Circle Can Curve in Space
A great circle on a sphere clearly curves when viewed in three-dimensional Euclidean space.
That external curvature does not contradict its geodesic status. Its intrinsic geodesic curvature is zero.
The sphere itself forces the path to curve through the surrounding space. Within the surface, however, the great circle is locally straight.
Parallel Transport Intuition
Another way to think about geodesics is to carry a direction vector along the path while keeping it as parallel as the curved geometry allows.
Along a geodesic, the tangent direction transports itself without intrinsic turning.
This idea becomes precise through the mathematical machinery of parallel transport and covariant derivatives.
The Geodesic Equation
In advanced coordinate geometry, a geodesic can be described by the geodesic equation
\[ \boxed{ \frac{d^2x^k}{ds^2} +\Gamma^k_{ij} \frac{dx^i}{ds} \frac{dx^j}{ds} =0 }. \]Here \(s\) is an affine parameter, often arc length, and the quantities \(\Gamma^k_{ij}\) are called Christoffel symbols.
This formula is introduced here for recognition, not for full computation.
What the Geodesic Equation Means
In ordinary Cartesian coordinates on flat Euclidean space, the relevant Christoffel symbols vanish, so the geodesic equation reduces to
\[ \frac{d^2x^k}{ds^2}=0. \]Its solutions are straight lines with constant coordinate velocity.
In curved coordinates or curved spaces, the extra Christoffel-symbol terms account for how the coordinate system and metric change from place to place.
Coordinates Can Make Straight Paths Look Complicated
Even flat Euclidean space can have nonzero Christoffel symbols when expressed in curvilinear coordinates such as polar coordinates.
Therefore, a complicated coordinate equation does not by itself prove that the underlying space is curved.
Curvature is a geometric property of the metric, not merely an artifact of choosing complicated coordinates.
Geodesics and Motion
Geodesics also describe natural motion in many physical theories.
A free object that experiences no applied force can often be modeled as following the straightest available path in the relevant geometry.
In general relativity, freely falling objects follow spacetime geodesics. The geometry is four-dimensional and Lorentzian rather than the ordinary two-dimensional surface geometry developed here, but the conceptual bridge is the same: geometry determines natural unforced paths.
Geodesics and Navigation
Geodesic ideas appear whenever travel is constrained to a curved or otherwise structured space.
Applications include:
- air and sea navigation;
- satellite positioning;
- robot motion on surfaces;
- computer graphics and mesh processing;
- geographic information systems;
- optimization on curved manifolds;
- models of spacetime in physics.
Geodesic Distance
The geodesic distance between two points is the infimum of the lengths of admissible paths joining them.
In many familiar settings, this infimum is achieved by a minimizing geodesic.
Symbolically, if \(\gamma\) ranges over paths from \(P\) to \(Q\), then
\[ d(P,Q)=\inf_{\gamma:P\to Q} L(\gamma). \]This definition generalizes ordinary straight-line distance.
Intrinsic Distance versus Chord Distance
On a sphere, two surface points can also be joined by a straight chord through three-dimensional space.
The chord length is not the same as the spherical geodesic distance.
If the central angle is \(\theta\), then the chord length is
\[ c=2R\sin\left(\frac{\theta}{2}\right), \]while the shorter surface geodesic has length
\[ s=R\theta. \]For \(0<\theta\leq\pi\), the surface route is longer than the chord because it is constrained to remain on the sphere.
Example: Chord versus Geodesic Distance
Let \(R=10\) and \(\theta=\pi/2\).
The spherical geodesic distance is
\[ s=10\left(\frac{\pi}{2}\right)=5\pi. \]The chord length is
\[ c=20\sin\left(\frac{\pi}{4}\right) =20\left(\frac{\sqrt2}{2}\right) =10\sqrt2. \]Thus,
\[ \boxed{s=5\pi}, \qquad \boxed{c=10\sqrt2}. \]Geodesic Completeness
A geometric space is called geodesically complete when its geodesics can be extended indefinitely in their natural parameter in both directions.
The Euclidean plane, the complete sphere, and the standard hyperbolic plane are geodesically complete.
A surface with an edge or missing point may fail to be geodesically complete because a geodesic can reach the missing boundary in finite length.
Closed Geodesics
A closed geodesic returns to its starting point and closes into a loop.
Every great circle on a sphere is a closed geodesic.
By contrast, ordinary straight lines in the Euclidean plane do not close. Whether closed geodesics exist therefore depends on the global geometry and topology of the space.
Geodesics and Topology Are Different Ideas
Geometry and topology interact, but they answer different questions.
A metric determines lengths, angles, and geodesics. Topology describes continuity, neighborhoods, connectedness, and global structure without requiring specific distance measurements.
Two spaces can have the same topology while supporting different metrics and therefore different geodesics.
Common Error: Geodesic Means Globally Shortest Everywhere
This is too strong.
Geodesics are locally straight and locally distance-minimizing under suitable conditions. A long geodesic segment may fail to be the globally shortest route between its endpoints.
The longer great-circle arc between two non-antipodal points is the standard example.
Common Error: Geodesics Must Look Straight
A geodesic only needs to be straight according to the intrinsic geometry.
Great circles on a sphere, circular arcs in the Poincaré disk, and helices on a cylinder can all represent geodesics even though they appear curved in an external picture.
Common Error: Every Circle on a Sphere Is a Geodesic
Only great circles are spherical geodesics.
A latitude circle other than the equator is a small circle. It turns within the surface and therefore has nonzero geodesic curvature.
Common Error: Chord Distance Equals Surface Distance
A chord passes through the surrounding Euclidean space, while a surface geodesic remains on the surface.
These are different admissible path problems and generally produce different distances.
Common Error: A Model's Euclidean Lines Are Its Geodesics
In the Poincaré disk, most hyperbolic geodesics are Euclidean circular arcs, not Euclidean straight lines.
The hyperbolic metric, not an ordinary ruler placed on the drawing, determines straightness and distance.
A Reliable Procedure for Surface Shortest-Path Problems
- Identify the surface or geometric space.
- Determine the metric or distance rule being used.
- Identify the geodesics available in that geometry.
- List candidate geodesic paths between the endpoints.
- Compute or compare their intrinsic lengths.
- Choose the candidate with minimum length when a global shortest path is required.
- Check for symmetry, wrapping, antipodal points, or other causes of nonuniqueness.
Example: Choosing the Shorter Great-Circle Arc
Two points on a sphere of radius \(R=12\) are separated along a great circle by a central angle
\[ \theta=\frac{2\pi}{3}. \]The shorter arc has length
\[ L_1=12\left(\frac{2\pi}{3}\right)=8\pi. \]The complementary longer arc has angle
\[ 2\pi-\frac{2\pi}{3}=\frac{4\pi}{3}, \]so its length is
\[ L_2=12\left(\frac{4\pi}{3}\right)=16\pi. \]Both lie on the same geodesic, but the geodesic distance is
\[ \boxed{8\pi}. \]Example: Several Cylinder Candidates
A cylinder has circumference \(C=10\). In an unrolled representation, two points differ vertically by \(h=4\) and horizontally by \(w=3\).
Candidate horizontal displacements are
\[ 3+10k, \qquad k\in\mathbb{Z}. \]For \(k=0\),
\[ d_0=\sqrt{3^2+4^2}=5. \]For \(k=-1\),
\[ d_{-1}=\sqrt{(-7)^2+4^2}=\sqrt{65}. \]For \(k=1\),
\[ d_1=\sqrt{13^2+4^2}=\sqrt{185}. \]Therefore the shortest of these candidates is
\[ \boxed{5}. \]Why Geodesics Matter Conceptually
Geodesics provide a single language for straightness across many geometries.
They allow us to say:
- a Euclidean line is intrinsically straight;
- a great circle is intrinsically straight on a sphere;
- a Poincaré arc can be intrinsically straight in hyperbolic space;
- a helix can be intrinsically straight on a cylinder.
The visual form changes, but the structural idea remains the same.
Practice 1: Spherical Geodesic Distance
A sphere has radius \(R=9\). Two points are separated by a central angle \(\theta=\pi/4\). Find their spherical geodesic distance.
Practice 2: Antipodal Distance
Find the geodesic distance between antipodal points on a sphere of radius \(7\).
Practice 3: Chord and Surface Distance
A sphere has radius \(R=8\), and two points subtend an angle \(\theta=\pi/3\). Find both the chord length and the shorter spherical geodesic distance.
Practice 4: Cylinder Geodesic
After a cylinder is unrolled, two surface points differ horizontally by \(5\) units and vertically by \(12\) units. Find the length of the straight segment joining them in the unrolled surface.
Practice 5: Identify the Geodesic
Which of the following are geodesics on a sphere?
- the equator;
- a latitude circle at \(30^\circ\) north;
- a complete meridian circle;
- an arbitrary small circle.
Practice 6: Local versus Global
Explain why the longer great-circle arc joining two non-antipodal points can lie on a geodesic without being the geodesic distance between the points.
Practice 7: Poincaré Disk
Describe the complete geodesics in the Poincaré disk model.
Practice 8: Geodesic Curvature
What is the geodesic curvature \(k_g\) of a geodesic?
Practice Answers
Practice 1
Use
\[ d=R\theta. \]Therefore,
\[ d=9\left(\frac{\pi}{4}\right) =\boxed{\frac{9\pi}{4}}. \]Practice 2
Antipodal points have central angle \(\pi\), so
\[ d=7\pi. \]Thus,
\[ \boxed{7\pi}. \]Practice 3
The chord length is
\[ c=2R\sin\left(\frac{\theta}{2}\right) =16\sin\left(\frac{\pi}{6}\right) =16\left(\frac12\right) =\boxed{8}. \]The surface geodesic distance is
\[ s=R\theta =8\left(\frac{\pi}{3}\right) =\boxed{\frac{8\pi}{3}}. \]Practice 4
In the unrolled surface,
\[ d=\sqrt{5^2+12^2} =\sqrt{169} =\boxed{13}. \]Practice 5
The geodesics are 1 and 3: the equator and a complete meridian circle are great circles.
The other listed circles are small circles and are not spherical geodesics.
Practice 6
The longer arc follows the same locally straight great circle, so it is a geodesic segment in the local sense. However, the shorter arc has smaller total length and therefore realizes the geodesic distance between the two points.
Practice 7
Complete geodesics in the Poincaré disk are Euclidean diameters and Euclidean circular arcs that meet the boundary circle orthogonally.
Practice 8
A geodesic has
\[ \boxed{k_g=0}. \]What to Remember
- A geodesic is an intrinsically straight path.
- In the Euclidean plane, geodesics are ordinary straight lines.
- On a sphere, geodesics are great circles.
- In the Poincaré disk, geodesics are diameters or circular arcs orthogonal to the boundary.
- On an unrolled cylinder, geodesics become Euclidean straight lines.
- A geodesic is fundamentally a local notion and need not always be globally shortest.
- Geodesic distance is obtained by minimizing intrinsic path length.
- Chord distance and surface geodesic distance are different measurements.
- A geodesic has zero geodesic curvature.
- The metric determines which paths are geodesics.
- Coordinates may make a geodesic look complicated without changing its intrinsic straightness.
- The geodesic equation generalizes zero-acceleration motion to curved geometry.
Where We Go Next
Geodesics tell us how straight paths behave once the geometry of a space is fixed. Their behavior also gives clues about how the space itself bends.
The next lesson, Curvature and Local Geometry, develops the idea of curvature more directly and examines how a space can look nearly Euclidean in a small neighborhood while still having substantial global curvature.