The Central Idea
Distance on a surface is normally measured by the length of the shortest allowable path that stays on that surface.
If two points are joined by many surface paths, the intrinsic distance is the smallest path length among them:
\[ \boxed{d(p,q)=\inf_{\gamma} L(\gamma)}. \]Here, \(\gamma\) ranges over surface paths joining \(p\) to \(q\), and \(L(\gamma)\) is the length of a path.
On familiar smooth surfaces, a shortest path is usually a segment of a geodesic.
Intrinsic Distance versus Ambient Distance
A curved surface may sit inside ordinary three-dimensional space. That creates two different notions of distance.
- Ambient distance measures the ordinary straight-line distance through the surrounding space.
- Intrinsic distance measures the shortest route that remains on the surface.
The distinction is essential. If two cities lie on Earth, a tunnel through the planet and an airplane following the surface correspond to different distance concepts.
A Simple Sphere Example
Let two points lie on a sphere of radius \(R\). Suppose the angle between their radius vectors, measured at the center of the sphere, is \(\theta\) radians.
The straight-line chord through the surrounding three-dimensional space has length
\[ \boxed{c=2R\sin\left(\frac{\theta}{2}\right)}. \]The shorter great-circle arc along the sphere has length
\[ \boxed{s=R\theta}. \]Except when the points coincide, the surface arc is longer than the chord.
Why Radians Matter
The arc-length formula
\[ s=R\theta \]assumes that \(\theta\) is measured in radians.
If an angle is given in degrees, first convert it using
\[ \theta_{\text{rad}}=\theta_{\text{deg}}\frac{\pi}{180}. \]For example, \(60^\circ\) equals
\[ 60\frac{\pi}{180}=\frac{\pi}{3}. \]Great Circles
A great circle is the intersection of a sphere with a plane passing through the sphere's center.
The equator is a great circle. Most lines of latitude are not.
Great circles play the role of straight lines in spherical geometry. Their arcs are geodesic paths, and the shorter great-circle arc normally gives the intrinsic distance between two non-antipodal points on a sphere.
Great-Circle Distance
If the central angle between two points is \(\theta\), then their great-circle distance is
\[ \boxed{d=R\theta}, \]where \(0\leq\theta\leq\pi\) for the shorter route.
This formula reduces the distance problem to finding the central angle.
Example: Distance from a Central Angle
Two points on a sphere of radius \(12\) are separated by a central angle of \(75^\circ\).
Convert the angle to radians:
\[ 75^\circ\frac{\pi}{180^\circ}=\frac{5\pi}{12}. \]Then
\[ d=12\left(\frac{5\pi}{12}\right)=5\pi. \]Therefore,
\[ \boxed{d=5\pi}. \]Central Angles from Vectors
Suppose points \(p\) and \(q\) lie on a sphere of radius \(R\) centered at the origin. Their position vectors satisfy
\[ \|p\|=\|q\|=R. \]The dot-product formula gives
\[ p\cdot q=R^2\cos\theta. \]Therefore,
\[ \boxed{\theta=\arccos\left(\frac{p\cdot q}{R^2}\right)}. \]The spherical distance is then
\[ \boxed{d=R\arccos\left(\frac{p\cdot q}{R^2}\right)}. \]Example: Distance from Position Vectors
On the unit sphere, let
\[ p=(1,0,0),\qquad q=(0,1,0). \]Their dot product is
\[ p\cdot q=0. \]Thus,
\[ \theta=\arccos(0)=\frac{\pi}{2}. \]Since \(R=1\),
\[ \boxed{d=\frac{\pi}{2}}. \]Latitude and Longitude
Points on a sphere are often described by latitude and longitude.
Let the two locations have latitudes \(\phi_1,\phi_2\) and longitudes \(\lambda_1,\lambda_2\), all measured in radians.
Define the longitude difference
\[ \Delta\lambda=\lambda_2-\lambda_1. \]The spherical law of cosines gives the central angle \(\theta\) by
\[ \boxed{ \cos\theta =\sin\phi_1\sin\phi_2 +\cos\phi_1\cos\phi_2\cos(\Delta\lambda) }. \]The Spherical Law of Cosines Distance Formula
Once the central angle is known, multiply by the sphere radius:
\[ \boxed{ d=R\arccos\left( \sin\phi_1\sin\phi_2 +\cos\phi_1\cos\phi_2\cos(\Delta\lambda) \right) }. \]This formula models Earth as a sphere. It is useful for conceptual work and many approximate distance calculations.
Example: Two Points on the Equator
Suppose two points lie on the equator, so
\[ \phi_1=\phi_2=0. \]If their longitudes differ by \(40^\circ\), then the central angle is also \(40^\circ\).
On a sphere of radius \(R\), the distance is
\[ d=R\left(40\frac{\pi}{180}\right) =\frac{2\pi R}{9}. \]Therefore,
\[ \boxed{d=\frac{2\pi R}{9}}. \]The Haversine Formula
Another standard way to compute spherical distance uses the haversine function,
\[ \operatorname{hav}(x)=\sin^2\left(\frac{x}{2}\right). \]For two latitude-longitude points,
\[ a= \sin^2\left(\frac{\Delta\phi}{2}\right) +\cos\phi_1\cos\phi_2 \sin^2\left(\frac{\Delta\lambda}{2}\right). \]Then
\[ \theta=2\arctan2\left(\sqrt a,\sqrt{1-a}\right) \]and
\[ \boxed{d=R\theta}. \]Why the Haversine Formula Is Useful
The spherical law of cosines is mathematically direct, but numerical computations with very small separations can lose precision when the cosine of the central angle is extremely close to \(1\).
The haversine formulation is often more stable for nearby points and has a long history in navigation and geodesy.
Modern software may use still more specialized formulas, especially when Earth is modeled as an ellipsoid rather than a sphere.
Chord Distance and Arc Distance
For a central angle \(\theta\), the chord distance is
\[ c=2R\sin\left(\frac{\theta}{2}\right), \]while the shorter spherical distance is
\[ s=R\theta. \]For \(0<\theta\leq\pi\),
\[ \boxed{c\leq s}. \]Equality occurs only in the limiting case when the two points coincide.
Example: Comparing Chord and Surface Distance
Let \(R=10\) and \(\theta=\pi/3\).
The chord length is
\[ c=2(10)\sin\left(\frac{\pi}{6}\right)=10. \]The surface distance is
\[ s=10\left(\frac{\pi}{3}\right)=\frac{10\pi}{3}\approx10.47. \]Thus the route constrained to the sphere is slightly longer:
\[ \boxed{10<\frac{10\pi}{3}}. \]Why Nearby Surface Distance Looks Euclidean
For small \(\theta\),
\[ \sin\left(\frac{\theta}{2}\right) \approx\frac{\theta}{2}. \]Therefore,
\[ c=2R\sin\left(\frac{\theta}{2}\right) \approx R\theta=s. \]This is another expression of local flatness: over sufficiently short distances, curved geometry can closely resemble Euclidean geometry.
Antipodal Points
Two points on a sphere are antipodal if they lie directly opposite one another through the center.
Their central angle is
\[ \theta=\pi. \]Therefore their spherical distance is
\[ \boxed{d=\pi R}. \]Unlike most pairs of points, antipodal points have infinitely many shortest great-circle routes between them.
Shortest Geodesics Are Not Always Unique
A geodesic is locally straight, but global shortest-path behavior can be more complicated.
On a sphere, non-antipodal points have a unique shorter great-circle arc. Antipodal points are joined by infinitely many minimizing semicircles.
This distinction is a first example of how global geometry can differ from local geometry.
A Geodesic Can Stop Being Globally Shortest
Continue traveling along a great circle beyond the antipodal point. The path is still geodesic, but it is no longer the shortest route back to the starting point's neighborhood.
Thus,
- being geodesic is a local property;
- being distance minimizing is a global comparison.
The two ideas often agree over short segments but should not be treated as identical.
Distance on a General Curved Surface
On an arbitrary smooth surface, there may be no simple great-circle formula.
Instead, we measure the length of curves that lie on the surface and then search for the shortest one.
If a surface path is written as
\[ \gamma(t),\qquad a\leq t\leq b, \]its length is
\[ \boxed{L(\gamma)=\int_a^b\|\gamma'(t)\|\,dt}. \]Surface Coordinates and the Metric
A surface can often be parametrized by two coordinates \((u,v)\):
\[ X=X(u,v). \]Small changes in \(u\) and \(v\) produce a small displacement on the surface. The squared infinitesimal distance takes the form
\[ \boxed{ds^2=E\,du^2+2F\,du\,dv+G\,dv^2}. \]This expression is called the first fundamental form or, more broadly, the surface metric.
What the Metric Coefficients Mean
If
\[ X_u=\frac{\partial X}{\partial u}, \qquad X_v=\frac{\partial X}{\partial v}, \]then
\[ E=X_u\cdot X_u, \qquad F=X_u\cdot X_v, \qquad G=X_v\cdot X_v. \]These coefficients tell us how coordinate changes translate into actual surface lengths and angles.
Path Length from a Surface Metric
If a curve on the surface is described by
\[ u=u(t),\qquad v=v(t), \]then
\[ du=u'(t)dt, \qquad dv=v'(t)dt. \]Its length becomes
\[ \boxed{ L=\int_a^b \sqrt{ E(u')^2+2F u'v'+G(v')^2 }\,dt }. \]This is the general intrinsic path-length formula in local coordinates.
The Sphere Metric
A sphere of radius \(R\) can be described using latitude-like angle \(\phi\) and longitude \(\lambda\).
Its metric can be written
\[ \boxed{ ds^2=R^2\,d\phi^2+R^2\cos^2\phi\,d\lambda^2 }. \]This formula explains why one degree of longitude represents less physical distance near the poles than near the equator.
Distance Along a Meridian
Along a meridian, longitude is constant, so
\[ d\lambda=0. \]The sphere metric reduces to
\[ ds=R|d\phi|. \]Therefore the distance between latitudes \(\phi_1\) and \(\phi_2\) along the same meridian is
\[ \boxed{d=R|\phi_2-\phi_1|}. \]Distance Along a Parallel of Latitude
Along a fixed latitude \(\phi\), we have
\[ d\phi=0. \]The metric becomes
\[ ds=R\cos\phi\,|d\lambda|. \]Therefore an arc along that parallel spanning longitude difference \(|\Delta\lambda|\) has length
\[ \boxed{L=R\cos\phi\,|\Delta\lambda|}. \]Except for the equator, however, a parallel of latitude is not a great circle and generally does not provide the shortest route between its endpoints.
Path Length Is Not Automatically Distance
A curve can have a perfectly valid length without being the shortest path.
Suppose two points lie on the same non-equatorial latitude. Traveling along that parallel gives one surface path and therefore one path length.
The intrinsic distance between the points is the minimum over all allowable surface paths, and a great-circle route may be shorter.
Geodesic Distance
When a shortest surface path is a geodesic segment, its length is called geodesic distance.
In many smooth geometric settings, distance between nearby points can be found by solving for the appropriate geodesic and then evaluating its length.
This unifies the flat-plane line segment, the spherical great-circle arc, and shortest paths on more general surfaces.
The Cylinder as an Intrinsically Flat Surface
A circular cylinder looks curved in three-dimensional space, but it can be unrolled onto a plane without stretching.
This means its intrinsic distance problems can often be converted into ordinary planar distance problems.
The cylinder is therefore a useful reminder that visible bending and intrinsic curvature are not the same thing.
Unrolling a Cylinder
Consider a cylinder of radius \(R\). Use coordinates
\[ (\theta,z), \]where \(\theta\) is the angular position and \(z\) is height.
When the cylinder is unrolled, the horizontal coordinate becomes
\[ x=R\theta. \]Thus the surface metric is
\[ \boxed{ds^2=R^2d\theta^2+dz^2}. \]Example: Distance on a Cylinder
Suppose two points on a cylinder of radius \(3\) differ in angle by \(\Delta\theta=\pi/2\) and in height by \(4\).
After unrolling, the horizontal separation is
\[ 3\left(\frac{\pi}{2}\right)=\frac{3\pi}{2}. \]The straight-line distance in the unrolled plane is
\[ d=\sqrt{\left(\frac{3\pi}{2}\right)^2+4^2}. \]Therefore,
\[ \boxed{d=\sqrt{\frac{9\pi^2}{4}+16}}. \]Multiple Wraps on a Cylinder
Because the angular coordinate repeats every \(2\pi\), there can be many straight-line images of the same endpoint in the unrolled plane.
Candidate horizontal separations have the form
\[ R(\Delta\theta+2\pi k), \qquad k\in\mathbb Z. \]The shortest surface route is obtained by choosing the integer \(k\) that gives the smallest planar distance.
Distance on a Cone
A cone is another surface that can be cut and flattened into a planar sector without stretching away from its tip.
Many geodesic-distance problems on a cone can therefore be solved by unfolding the cone, drawing a straight line in the planar sector, and then wrapping the result back onto the surface.
The cone's tip requires special care because the geometry is singular there.
Surfaces That Cannot Be Flattened Without Distortion
A sphere cannot be flattened into a plane without changing distances.
This is why every flat map of Earth introduces some form of distortion. Different map projections preserve different properties, but no ordinary planar projection preserves all distances, areas, and angles globally.
Distance calculations should therefore use the geometry of the surface or a projection whose distortion is understood.
Map Distance versus Surface Distance
A straight line drawn on a map is not automatically the shortest route on Earth.
Its meaning depends on the map projection. A projection can transform a spherical geodesic into a curve, while a visually straight map segment may correspond to a different kind of route on the sphere.
The underlying surface geometry, not the drawing alone, determines the intrinsic distance.
The Earth Is Not Exactly a Sphere
Modeling Earth as a sphere is often sufficient for introductory geometry, but Earth is better approximated by an oblate ellipsoid.
High-precision geodesy therefore uses ellipsoidal models and algorithms designed for those models.
The conceptual framework remains the same: define the geometry, identify allowable surface paths, and determine the shortest path length according to that geometry.
Spherical Models versus Ellipsoidal Models
A spherical model has one radius \(R\) and permits especially simple formulas such as
\[ d=R\theta. \]An ellipsoid has direction-dependent geometry. Its curvature varies with location, and distance calculations are more complicated.
Choosing a model is therefore part of choosing the mathematical assumptions appropriate to the desired accuracy.
Approximate Distance on a Small Patch
On a sufficiently small region of a smooth surface, a tangent plane can provide a useful approximation.
If the region is small compared with the local radius of curvature, then planar distance and surface distance may be nearly equal.
As the region grows, accumulated curvature makes the flat approximation less reliable.
Distance and Curvature
Curvature influences how geodesics spread apart or come together and how distances compare with familiar Euclidean expectations.
- Positive curvature tends to make geodesics reconverge.
- Zero curvature gives locally Euclidean behavior.
- Negative curvature tends to make geodesics separate more rapidly.
These effects connect distance geometry to the triangle-angle phenomena studied in the previous lesson.
Distance Functions Must Satisfy Basic Rules
A mathematical distance function \(d(p,q)\) normally satisfies four basic properties.
- \(d(p,q)\geq0\).
- \(d(p,q)=0\) exactly when \(p=q\).
- \(d(p,q)=d(q,p)\).
- \(d(p,r)\leq d(p,q)+d(q,r)\).
The fourth property is the triangle inequality.
The Triangle Inequality on a Surface
Suppose \(p\), \(q\), and \(r\) lie on a curved surface. Traveling from \(p\) to \(r\) by way of \(q\) cannot be shorter than the direct intrinsic distance between \(p\) and \(r\):
\[ \boxed{d(p,r)\leq d(p,q)+d(q,r)}. \]The shapes of the minimizing paths may differ from Euclidean line segments, but the metric principle remains.
Distance as an Optimization Problem
Distance on a curved surface is fundamentally an optimization problem.
We begin with many candidate paths, assign each a length, and seek the least possible value.
\[ d(p,q)=\inf_{\gamma:p\to q}L(\gamma). \]This viewpoint connects geometry with calculus of variations, differential equations, numerical optimization, robotics, navigation, and physics.
How Geodesic Equations Enter
On a smooth surface, the paths that make the length functional stationary satisfy geodesic equations.
In local coordinates \(x^1,x^2\), these can be written schematically as
\[ \frac{d^2x^k}{dt^2} +\Gamma^k_{ij} \frac{dx^i}{dt} \frac{dx^j}{dt}=0. \]The coefficients \(\Gamma^k_{ij}\) encode how the coordinate system and metric vary across the surface. A later course in differential geometry can develop these equations systematically.
Exact Formulas versus Numerical Methods
Some surfaces have enough symmetry to produce exact distance formulas.
Examples include the plane, sphere, and many developable surfaces after suitable unfolding.
On irregular surfaces, distance may be computed numerically by approximating the surface with a mesh or grid and then solving a shortest-path problem.
Distance on a Triangulated Mesh
Computer graphics and engineering often represent a curved surface by many small triangles.
A simple approximation can treat mesh vertices and edges as a weighted graph, with edge lengths serving as weights.
Graph shortest-path algorithms then estimate surface distance. More advanced methods approximate geodesics across triangle interiors rather than forcing paths to remain on mesh edges.
Applications in Navigation
Long-distance navigation must account for Earth's geometry.
Great-circle and ellipsoidal geodesic calculations appear in aviation, marine navigation, mapping, surveying, and satellite systems.
A route that appears curved on a flat map may be the shortest path on the curved Earth model.
Applications in Robotics and Motion Planning
A robot constrained to move on a curved object, terrain surface, or manifold cannot simply use straight-line distance through empty space.
The feasible path must remain in the allowed configuration or physical surface. Geodesic and shortest-path methods therefore become part of motion planning and control.
Applications in Computer Graphics
Surface distance is useful for texture placement, mesh processing, shape analysis, deformation, segmentation, and locating neighborhoods on curved models.
Two points that are close through three-dimensional space may be far apart along the surface if the surface folds between them.
Applications in Biology and Medicine
Biological structures frequently have curved surfaces. Distances along the cortex, organs, membranes, or anatomical meshes can differ substantially from direct three-dimensional chord distances.
Intrinsic distance provides a way to measure separation while respecting the geometry of the structure itself.
Common Error: Using Degrees Directly in \(s=R\theta\)
Suppose \(R=8\) and the central angle is \(45^\circ\).
Writing
\[ s=8(45) \]is incorrect because the formula requires radians.
Convert first:
\[ 45^\circ=\frac{\pi}{4}. \]Then
\[ \boxed{s=8\left(\frac{\pi}{4}\right)=2\pi}. \]Common Error: Using the Chord as Surface Distance
The chord joins two spherical points through the surrounding three-dimensional space. It does not remain on the sphere.
If travel is constrained to the surface, use the shorter great-circle arc, not the chord.
Common Error: Assuming Every Latitude Circle Is a Geodesic
The equator is a great circle and therefore a spherical geodesic.
Other parallels of latitude lie in planes that do not pass through the sphere's center. They are not great circles and are not spherical geodesics.
Common Error: Assuming Every Geodesic Segment Is Globally Shortest
A geodesic is locally straight. A sufficiently long geodesic segment may fail to be globally minimizing.
The long arc of a great circle provides the basic spherical example: it is geodesic, but the complementary shorter arc connects the same endpoints with less length.
Common Error: Measuring from a Map without Considering Projection
Flat maps distort the sphere. The amount and type of distortion depend on the projection.
A ruler measurement on a map is therefore not automatically proportional to true surface distance unless the projection and scale justify that use.
Common Error: Treating Earth as Exactly Spherical
The spherical model is an approximation.
It is often appropriate for classroom problems and rough calculations, but high-precision surveying and navigation require a more accurate Earth model.
Mathematical modeling always includes a decision about which geometric assumptions are adequate for the task.
A Reliable Procedure for Spherical Distance from a Central Angle
- Identify the sphere radius \(R\).
- Determine the smaller central angle \(\theta\).
- Convert \(\theta\) to radians if necessary.
- Use \(d=R\theta\).
- Keep units consistent.
- Check that \(0\leq\theta\leq\pi\) for the shorter route.
A Reliable Procedure for Latitude-Longitude Distance
- Choose the spherical radius appropriate to the model.
- Convert all latitudes and longitudes to radians.
- Compute \(\Delta\phi\) and \(\Delta\lambda\).
- Use the spherical law of cosines or haversine formula.
- Determine the central angle \(\theta\).
- Multiply by \(R\) to obtain distance.
- State that the result is based on a spherical approximation when relevant.
A Reliable Procedure for a General Curved Surface
- Identify the surface and the geometry being used.
- Choose convenient coordinates or a parametrization.
- Determine the metric or infinitesimal line element.
- Describe candidate paths between the endpoints.
- Compute or approximate their lengths.
- Find the minimizing path, usually a geodesic segment.
- Compare multiple candidates when global nonuniqueness is possible.
- Use numerical methods if no practical exact formula exists.
Example: Full Spherical Workflow
Two points lie on a sphere of radius \(20\). Their unit direction vectors have dot product \(1/2\).
Since the vectors are unit directions,
\[ \cos\theta=\frac12. \]Therefore,
\[ \theta=\frac{\pi}{3}. \]The surface distance is
\[ d=20\left(\frac{\pi}{3}\right). \]Hence,
\[ \boxed{d=\frac{20\pi}{3}}. \]Example: Meridian Distance
Two points lie on the same meridian of a sphere of radius \(6400\) km. Their latitudes are \(15^\circ\) and \(52^\circ\).
The latitude difference is
\[ 37^\circ=37\frac{\pi}{180}. \]Therefore,
\[ d=6400\left(37\frac{\pi}{180}\right). \]So
\[ \boxed{d\approx4133\text{ km}}. \]This value uses a spherical model of Earth.
Example: A Non-Geodesic Latitude Route
Two points lie at latitude \(60^\circ\) on a sphere of radius \(R\), with a longitude difference of \(90^\circ\).
Traveling directly along the parallel of latitude gives length
\[ L=R\cos60^\circ\left(\frac{\pi}{2}\right). \]Since \(\cos60^\circ=1/2\),
\[ \boxed{L=\frac{\pi R}{4}}. \]This is the length of that chosen route. It should not automatically be called the intrinsic distance, because the parallel is not a great circle.
Practice 1: Central Angle
A sphere has radius \(9\). Find the shorter surface distance between two points separated by a central angle of \(120^\circ\).
Practice 2: Quarter Circle
Two points on a sphere of radius \(14\) are separated by a central angle of \(\pi/2\). Find their spherical distance.
Practice 3: Chord versus Arc
On a sphere of radius \(6\), two points are separated by central angle \(\pi/3\). Find both the chord distance and the shorter surface distance.
Practice 4: Antipodal Points
Find the intrinsic distance between antipodal points on a sphere of radius \(11\). How many shortest great-circle paths join them?
Practice 5: Vector Method
On the unit sphere, let
\[ p=(0,0,1),\qquad q=(1,0,0). \]Find the spherical distance between the points.
Practice 6: Same Meridian
Two points lie on the same meridian of a sphere of radius \(10\). Their latitudes differ by \(36^\circ\). Find the distance along the meridian.
Practice 7: Same Parallel
On a sphere of radius \(8\), find the length of a route that follows the latitude \(60^\circ\) parallel through a longitude change of \(90^\circ\). Do not assume this route is geodesic.
Practice 8: Cylinder
On a cylinder of radius \(2\), two points differ by angle \(\pi\) and height \(3\). After unrolling the cylinder using the shorter angular separation, find the corresponding surface distance.
Practice 9: Identify the Error
A student says that two points on a sphere are \(5\) units apart because the straight-line chord between them has length \(5\). Explain why this does not necessarily give their intrinsic spherical distance.
Practice 10: Geodesic versus Minimizing
A traveler follows a great circle from one point on a sphere to another, but chooses the longer of the two great-circle arcs. Is the route geodesic? Is it globally distance minimizing? Explain.
Practice Answers
Practice 1
Convert \(120^\circ\) to radians:
\[ 120^\circ=\frac{2\pi}{3}. \]Then
\[ d=9\left(\frac{2\pi}{3}\right)=6\pi. \]Therefore,
\[ \boxed{d=6\pi}. \]Practice 2
\[ d=14\left(\frac{\pi}{2}\right). \]Hence,
\[ \boxed{d=7\pi}. \]Practice 3
The chord is
\[ c=2(6)\sin\left(\frac{\pi}{6}\right)=6. \]The surface distance is
\[ s=6\left(\frac{\pi}{3}\right)=2\pi. \]Therefore,
\[ \boxed{c=6,\qquad s=2\pi}. \]Practice 4
Antipodal points have central angle \(\pi\), so
\[ \boxed{d=11\pi}. \]Infinitely many great circles pass through an antipodal pair, so there are infinitely many shortest semicircular routes.
Practice 5
The dot product is
\[ p\cdot q=0. \]Hence
\[ \theta=\arccos(0)=\frac{\pi}{2}. \]On the unit sphere,
\[ \boxed{d=\frac{\pi}{2}}. \]Practice 6
Convert \(36^\circ\) to radians:
\[ 36^\circ=\frac{\pi}{5}. \]Therefore,
\[ d=10\left(\frac{\pi}{5}\right)=2\pi. \]Thus,
\[ \boxed{d=2\pi}. \]Practice 7
Use
\[ L=R\cos\phi\,|\Delta\lambda|. \]Then
\[ L=8\left(\frac12\right)\left(\frac{\pi}{2}\right)=2\pi. \]Therefore,
\[ \boxed{L=2\pi}. \]This is the length along the chosen parallel, not necessarily the intrinsic distance between the endpoints.
Practice 8
The horizontal separation after unrolling is
\[ R\Delta\theta=2\pi. \]The vertical separation is \(3\), so
\[ d=\sqrt{(2\pi)^2+3^2}. \]Thus,
\[ \boxed{d=\sqrt{4\pi^2+9}}. \]Practice 9
A chord travels through the surrounding space rather than remaining on the sphere. Intrinsic spherical distance is measured along the shortest surface path, normally the shorter great-circle arc. The chord length can therefore be smaller than the spherical distance.
Practice 10
The longer arc lies on a great circle, so it is still a geodesic path. However, it is not globally minimizing because the complementary shorter great-circle arc joins the same endpoints with less length.
What to Remember
- Intrinsic distance is measured within the surface.
- Ambient straight-line distance can differ from surface distance.
- On a sphere, shortest routes normally follow shorter great-circle arcs.
- For central angle \(\theta\) in radians, spherical distance is \(d=R\theta\).
- The chord length is \(2R\sin(\theta/2)\).
- Latitude-longitude distance can be found from the spherical law of cosines or haversine formula.
- Antipodal points are separated by distance \(\pi R\).
- Antipodal points have infinitely many shortest great-circle paths.
- A geodesic is locally straight but need not remain globally minimizing forever.
- Path length and distance are not the same concept unless the path is minimizing.
- A surface metric determines infinitesimal lengths intrinsically.
- General surface distance is the infimum of path lengths joining the endpoints.
- Cylinders and cones can often be unfolded to solve intrinsic distance problems.
- A sphere cannot be flattened globally without metric distortion.
- Earth is often modeled as a sphere for introductory work and as an ellipsoid for higher precision.
- Small surface patches can often be approximated by Euclidean geometry.
Where We Go Next
Spherical and curved-surface distance shows that geometry depends on the space in which measurement is performed. Length, straightness, and shortest paths all depend on the underlying geometric structure.
The next lesson, Projective Geometry, changes the emphasis. Instead of making distance and angle primary, projective geometry studies incidence, alignment, perspective, and properties that remain meaningful under projection. This provides another major example of how changing the foundational assumptions produces a different but internally coherent geometry.