The Central Idea
Spherical geometry is geometry carried out on the surface of a sphere.
The sphere is two-dimensional as a surface: a point on it can be located by two independent coordinates, even though the sphere sits naturally in three-dimensional space.
The key change is that the surface is curved. Because distance and straightness are interpreted along the surface, familiar rules from the flat plane must be reconsidered.
Sphere versus Ball
In ordinary mathematical language, a sphere usually means the surface consisting of all points a fixed distance from a center.
If the center is the origin and the radius is \(R>0\), the sphere in three-dimensional coordinates satisfies
\[ x^2+y^2+z^2=R^2. \]A ball includes the interior as well:
\[ x^2+y^2+z^2\le R^2. \]Spherical geometry concerns the surface, not the solid interior.
Intrinsic and Extrinsic Viewpoints
There are two useful ways to think about a sphere.
- The extrinsic viewpoint sees the sphere as an object embedded in ordinary three-dimensional space.
- The intrinsic viewpoint studies measurements made by an observer restricted to the surface itself.
Spherical geometry is primarily intrinsic. A traveler confined to Earth's surface measures distance along the surface rather than cutting through the planet in a straight three-dimensional chord.
What Plays the Role of a Straight Line?
On a flat plane, a straight line locally gives the shortest route between nearby points and continues without bending within the plane.
On a sphere, the corresponding paths are arcs of great circles.
A great circle is the intersection of the sphere with a plane that passes through the sphere's center.
Great Circles
The equator is a familiar example of a great circle. Every meridian of longitude, when continued through both poles to form a complete circle, is also a great circle.
If a plane through the center cuts a sphere of radius \(R\), the resulting circle also has radius \(R\).
Great circles are the largest circles that can be drawn on a given sphere.
Small Circles Are Different
A plane that cuts the sphere but does not pass through its center produces a small circle.
Most circles of latitude are small circles. The equator is the exceptional latitude circle that is a great circle.
A small circle is a perfectly valid circle on the sphere, but it does not play the role of a spherical straight line.
Spherical Lines
In the standard elementary model of spherical geometry, a spherical line is a complete great circle.
This definition immediately changes several familiar Euclidean facts. A Euclidean line extends forever without closing. A spherical line eventually returns to its starting point because a great circle is closed.
Its total length is the circumference of the sphere along that great circle:
\[ L=2\pi R. \]Great-Circle Arcs as Line Segments
Between two points on a sphere, a portion of a great circle plays the role of a line segment.
For two non-antipodal points, the shorter great-circle arc is the usual shortest surface path between them.
The longer arc lies on the same great circle but does not represent the shortest distance.
Antipodal Points
Two points are antipodal if they lie exactly opposite each other on the sphere.
If a point has position vector \(\mathbf{p}\) from the center, its antipode has position vector
\[ -\mathbf{p}. \]The north and south poles are antipodal. Every point on a sphere has exactly one antipodal point.
How Many Spherical Lines Pass through Two Points?
For two distinct points that are not antipodal, exactly one great circle passes through both points.
This is analogous to the Euclidean statement that exactly one line passes through two distinct points.
Antipodal points are the important exception: infinitely many great circles pass through a pair of antipodal points.
Why Antipodal Points Are Exceptional
A great circle comes from a plane through the center of the sphere.
If two surface points are antipodal, the line connecting them already passes through the center. Infinitely many planes can contain that same centerline, and each such plane cuts out a different great circle.
Therefore, antipodal points do not determine a unique spherical line.
Great Circles Always Meet
Any two distinct great circles intersect in two antipodal points.
The reason is three-dimensional: the two planes that generate the great circles both pass through the sphere's center. Distinct such planes intersect in a line through the center. That line meets the sphere at two opposite points.
There Are No Parallel Spherical Lines
In Euclidean geometry, distinct parallel lines never meet.
On the sphere, every pair of distinct spherical lines meets. Therefore, there are no parallel lines in the great-circle model of spherical geometry.
This is one of the clearest ways in which spherical geometry differs from Euclidean geometry.
The Parallel Postulate Changes
A common Euclidean formulation says that through a point not on a given line, exactly one parallel line can be drawn.
In spherical geometry, through a point not on a given spherical line, no spherical line can be drawn that avoids the original one.
The Euclidean parallel postulate therefore does not hold on the sphere.
Distance along a Great Circle
Suppose two points subtend a central angle \(\theta\), measured in radians, where
\[ 0\le \theta\le \pi. \]The length of the shorter great-circle arc between them is
\[ \boxed{d=R\theta}. \]This is the spherical analogue of the arc-length formula.
Example: Quarter of a Great Circle
Suppose two points on a sphere of radius \(R\) are separated by a central angle of \(90^\circ\).
Convert to radians:
\[ 90^\circ=\frac{\pi}{2}. \]Therefore, their great-circle distance is
\[ d=R\left(\frac{\pi}{2}\right) =\boxed{\frac{\pi R}{2}}. \]Example: Antipodal Distance
Antipodal points are separated by a central angle of \(\pi\) radians.
Their shortest surface distance is therefore
\[ \boxed{d=\pi R}. \]This is half the circumference of a great circle.
Chord Distance Is Not Surface Distance
Two points on a sphere can also be joined by a straight chord through the surrounding three-dimensional space.
If their central angle is \(\theta\), the chord length is
\[ c=2R\sin\left(\frac{\theta}{2}\right). \]The spherical surface distance is instead
\[ d=R\theta. \]The two measurements answer different geometric questions.
Spherical Angles
When two great circles intersect, they determine an angle on the sphere.
One way to measure the angle is to compare the tangent directions of the two great circles at the point of intersection.
Equivalently, because each great circle lies in a plane through the center, the spherical angle can be related to the angle between those planes.
Spherical Triangles
A spherical triangle is formed by three suitable arcs of great circles joining three vertices on the sphere.
Its sides are curved when viewed from ordinary three-dimensional space, but intrinsically they are the spherical analogues of straight segments.
The interior angles are formed where pairs of great-circle arcs meet.
A Triangle with Three Right Angles
Spherical geometry allows a triangle that is impossible in the Euclidean plane.
Start at the north pole. Travel south along one meridian to the equator. Travel one quarter of the way around the equator. Then return to the north pole along the new meridian.
Each interior angle is \(90^\circ\), so the angle sum is
\[ 90^\circ+90^\circ+90^\circ =\boxed{270^\circ}. \]Triangle Angle Sums Are Greater than \(180^\circ\)
For an ordinary nondegenerate spherical triangle formed from the shorter great-circle arcs, the interior angle sum is greater than \(180^\circ\):
\[ \boxed{A+B+C>\pi}. \]In degrees,
\[ \boxed{A+B+C>180^\circ}. \]This contrasts with Euclidean triangles, whose angle sum is exactly \(180^\circ\).
Spherical Excess
The amount by which a spherical triangle's angle sum exceeds \(\pi\) radians is called its spherical excess.
\[ \boxed{E=A+B+C-\pi}. \]The excess is positive for a nondegenerate spherical triangle in the usual elementary setting.
Area and Spherical Excess
On a sphere of radius \(R\), the area of a spherical triangle is related directly to its spherical excess:
\[ \boxed{K=R^2E} \]when \(E\) is measured in radians.
This relationship is known as Girard's theorem. It shows that angle and area are linked in a way that has no direct Euclidean counterpart.
Example: Area from Three Right Angles
For the triangle with three right angles,
\[ A=B=C=\frac{\pi}{2}. \]The spherical excess is
\[ E=\frac{3\pi}{2}-\pi=\frac{\pi}{2}. \]Therefore, its area is
\[ K=R^2\left(\frac{\pi}{2}\right) =\boxed{\frac{\pi R^2}{2}}. \]Since the full sphere has area \(4\pi R^2\), this triangle occupies one-eighth of the sphere.
Larger Triangles Show More Curvature
Very small regions of a large sphere can look nearly flat. As a spherical triangle becomes small compared with the radius of the sphere, its angle sum approaches \(180^\circ\).
Larger triangles reveal the sphere's curvature more strongly and can have a noticeably larger angle sum.
This is an early example of the distinction between local and global geometry.
Locally Flat Does Not Mean Globally Flat
Standing on a small patch of Earth, the ground often appears flat. This is a local approximation.
Over sufficiently large distances, the global curvature becomes important. Routes converge, great circles intersect, and spherical triangle angle sums differ from their Euclidean counterparts.
A geometry can therefore behave approximately Euclidean at small scales while remaining non-Euclidean globally.
Longitude and Latitude
Latitude and longitude provide a convenient coordinate system for points on a sphere or approximately spherical planet.
- Latitude measures angular position north or south of the equatorial plane.
- Longitude measures angular position around the rotational axis relative to a chosen reference meridian.
These coordinates are useful, but the coordinate grid itself does not make every coordinate curve a spherical line.
Meridians and Parallels of Latitude
Complete meridians are great circles because their planes pass through the sphere's center.
The equator is also a great circle.
Other latitude circles are small circles, so despite being called parallels of latitude in geography, they are not parallel lines in the spherical-geometric sense.
Navigation and Great-Circle Routes
For long-distance travel on an idealized spherical Earth, the shortest surface route between non-antipodal locations follows the shorter arc of a great circle.
On a flat map projection, that route may appear curved. The apparent curve comes from flattening a spherical surface onto a plane.
A straight line on a map and a shortest path on the globe need not be the same route.
Map Projections Cannot Preserve Everything
Flattening a sphere onto a plane necessarily introduces distortion.
Different map projections may preserve or prioritize different properties, such as local angles, areas, selected distances, or directions, but no single flat map can preserve every geometric property of the sphere everywhere.
This is one practical reason spherical geometry matters in geography and navigation.
Spherical Geometry and Euclidean Geometry
Some Euclidean ideas have close spherical analogues, while others change substantially.
- Points remain fundamental geometric objects.
- Great circles replace Euclidean straight lines.
- Great-circle arcs replace ordinary line segments.
- Distinct spherical lines always intersect.
- Spherical triangle angle sums exceed \(180^\circ\).
- Distance is measured along the curved surface.
Spherical Geometry Is Not a Distorted Euclidean Plane
It is tempting to think of spherical geometry as ordinary plane geometry drawn badly on a curved object. That misses the central point.
The sphere has its own natural notions of distance, straightest path, triangle, angle, and area. Once those notions are defined intrinsically, spherical geometry becomes a coherent mathematical system.
Common Error: Treating Every Circle on a Sphere as a Line
Not every circle drawn on the sphere is a spherical line.
In the standard elementary model, only great circles play that role because they arise from planes through the center and form the relevant geodesic paths.
Most latitude circles are therefore not spherical lines.
Common Error: Measuring through the Sphere
Spherical distance is not normally the straight three-dimensional chord between two surface points.
The intrinsic distance stays on the surface and follows the shortest great-circle arc.
Always identify which notion of distance the problem requires.
Common Error: Assuming Triangle Angles Must Sum to \(180^\circ\)
The Euclidean triangle-sum theorem depends on Euclidean geometry.
A spherical triangle can have an angle sum greater than \(180^\circ\), and that excess carries geometric information about the triangle's area.
Common Error: Calling Geographic Latitude Lines Parallel Spherical Lines
Geographic terminology can be misleading in a geometry lesson. Circles of latitude are often called parallels, but most are small circles and are not lines in spherical geometry.
Great circles, the spherical lines, always intersect other distinct great circles.
Common Error: Forgetting the Antipodal Exception
Two ordinary distinct, non-antipodal points determine a unique great circle.
Antipodal points do not. Infinitely many great circles pass through the same antipodal pair.
This exception matters whenever uniqueness is part of a statement.
A Reliable Procedure for Basic Spherical Problems
- Identify the sphere's radius \(R\).
- Determine whether the relevant path is a great-circle arc.
- Express angular separation in radians when using arc-length formulas.
- Use \(d=R\theta\) for shortest great-circle distance when \(0\le\theta\le\pi\).
- Check whether the points are antipodal.
- Distinguish surface distance from chord distance.
- For spherical triangles, remember that the angle sum is not fixed at \(\pi\).
- Use the geometry of the sphere rather than importing Euclidean assumptions automatically.
Example: Comparing Arc and Chord
Let two points on a sphere of radius \(10\) be separated by a central angle of \(60^\circ\).
First convert the angle:
\[ 60^\circ=\frac{\pi}{3}. \]The great-circle distance is
\[ d=10\left(\frac{\pi}{3}\right) =\boxed{\frac{10\pi}{3}}. \]The chord distance is
\[ c=2(10)\sin\left(\frac{\pi}{6}\right)=10. \]The surface path is longer because it must remain on the sphere.
Example: Spherical Excess
Suppose a spherical triangle has angles
\[ A=80^\circ,\qquad B=70^\circ,\qquad C=60^\circ. \]The angle sum is
\[ 80^\circ+70^\circ+60^\circ=210^\circ. \]The excess is
\[ 210^\circ-180^\circ=\boxed{30^\circ}. \]In radians, this is
\[ E=\frac{\pi}{6}. \]On a sphere of radius \(R\), the triangle's area is therefore
\[ \boxed{K=\frac{\pi R^2}{6}}. \]Practice 1: Identify the Spherical Line
Which of the following is a spherical line in the standard great-circle model: the equator, the latitude circle at \(30^\circ\) north, or both?
Practice 2: Great-Circle Distance
Two points on a sphere of radius \(12\) are separated by a central angle of \(\frac{\pi}{4}\). Find their shortest great-circle distance.
Practice 3: Antipodal Points
How many great circles pass through a pair of antipodal points?
Practice 4: Parallel Lines
In the standard spherical model, how many spherical lines through a point can remain parallel to a given spherical line?
Practice 5: Triangle Angle Sum
A spherical triangle has angles \(100^\circ\), \(70^\circ\), and \(50^\circ\). Find its spherical excess in degrees.
Practice 6: Area from Spherical Excess
A spherical triangle on a sphere of radius \(6\) has spherical excess \(\frac{\pi}{9}\). Find its area.
Practice 7: Quarter-Sphere Triangle
A spherical triangle has three right angles. What fraction of the sphere's total surface area does it occupy?
Practice 8: Surface versus Chord
Two points are separated by \(\theta=\frac{\pi}{2}\) on a sphere of radius \(R\). Write formulas for both their shorter great-circle distance and their straight chord distance.
Practice Answers
Practice 1
The equator is a great circle and therefore a spherical line. The latitude circle at \(30^\circ\) north is a small circle.
Answer: the equator only.
Practice 2
\[ d=R\theta =12\left(\frac{\pi}{4}\right) =\boxed{3\pi}. \]Practice 3
Infinitely many great circles pass through a pair of antipodal points.
Practice 4
None. Any two distinct great circles intersect, so there are no parallel spherical lines.
Practice 5
The angle sum is
\[ 100^\circ+70^\circ+50^\circ=220^\circ. \]Therefore, the spherical excess is
\[ \boxed{220^\circ-180^\circ=40^\circ}. \]Practice 6
\[ K=R^2E =6^2\left(\frac{\pi}{9}\right) =36\left(\frac{\pi}{9}\right) =\boxed{4\pi}. \]Practice 7
Three right angles give spherical excess \(\frac{\pi}{2}\), so the area is
\[ \frac{\pi R^2}{2}. \]Divide by the sphere's total area \(4\pi R^2\):
\[ \frac{\pi R^2/2}{4\pi R^2}=\boxed{\frac18}. \]Practice 8
Great-circle distance:
\[ d=R\left(\frac{\pi}{2}\right) =\boxed{\frac{\pi R}{2}}. \]Chord distance:
\[ c=2R\sin\left(\frac{\pi}{4}\right) =\boxed{R\sqrt{2}}. \]What to Remember
- Spherical geometry studies geometry on the surface of a sphere.
- Great circles play the role of spherical straight lines.
- The shortest route between non-antipodal points follows a shorter great-circle arc.
- Great-circle distance is \(d=R\theta\) when \(\theta\) is in radians.
- Two non-antipodal points determine one great circle.
- Antipodal points lie on infinitely many great circles.
- Distinct great circles always intersect, so spherical lines have no parallels.
- Spherical triangles can have angle sums greater than \(180^\circ\).
- Spherical excess is \(E=A+B+C-\pi\).
- For a sphere of radius \(R\), spherical triangle area satisfies \(K=R^2E\).
Where We Go Next
Spherical geometry shows one way that geometry can differ from the Euclidean plane: there are no parallel spherical lines, and triangle angle sums are larger than \(180^\circ\).
The next lesson, Hyperbolic Geometry, explores another non-Euclidean system in which the behavior of parallel lines and triangle angle sums changes in the opposite direction.