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In spherical geometry, a spherical circle (often shortened to circle) is the locus of points on a sphere at constant intrinsic distance (the intrinsic radius) from a given point on the sphere (the pole or intrinsic center). It is a curve of constant geodesic curvature relative to the sphere, analogous to a line or circle in the Euclidean plane; the curves analogous to straight lines are called great circles, and the curves analogous to planar circles are called small circles or lesser circles.

Fundamental concepts

Intrinsic characterization

A spherical circle with zero geodesic curvature is called a great circle, and is a geodesic analogous to a straight line in the plane. A great circle separates the sphere into two equal hemispheres, each with the great circle as its boundary. If a great circle passes through a point on the sphere, it also passes through the antipodal point. For any pair of distinct non-antipodal points, a unique great circle passes through both. Any two points on a great circle separate it into two arcs analogous to line segments in the plane; the shorter is called the minor arc and is the shortest path between the points, and the longer is called the major arc.

A circle with non-zero geodesic curvature is called a small circle, and is analogous to a circle in the plane. A small circle separates the sphere into two spherical disks or spherical caps, each with the circle as its boundary. For any triple of distinct non-antipodal points a unique small circle passes through all three. Any two points on the small circle separate it into two arcs, analogous to circular arcs in the plane.

Every circle has two antipodal poles (centers). A great circle is equidistant to its poles, while a small circle is closer to one pole than the other. Concentric circles are sometimes called parallels, because they each have constant distance to each-other, and in particular to their concentric great circle, and are in that sense analogous to parallel lines in the plane.

Extrinsic characterization

If the sphere is embedded in Euclidean space, the sphere's intersection with a plane is a circle, which can also be analyzed extrinsically to the sphere as a Euclidean circle: a locus of points in the plane at a constant Euclidean distance (the extrinsic radius) from a point in the plane (the extrinsic center). A great circle lies on a plane passing through the center of the sphere, so its extrinsic radius is equal to the radius of the sphere itself, and its extrinsic center is the sphere's center. A small circle lies on a plane not passing through the sphere's center, so its extrinsic radius is smaller than that of the sphere and its extrinsic center is an arbitrary point in the interior of the sphere. Parallel planes cut the sphere into parallel (concentric) small circles; the pair of parallel planes tangent to the sphere are tangent at the poles of these circles, and the diameter through these poles, passing through the sphere's center and perpendicular to the parallel planes, is called the axis of the parallel circles.

The sphere's intersection with a second sphere is also a circle, and the sphere's intersection with a concentric right circular cylinder or right circular cone is a pair of antipodal circles.

On the earth

In the geographic coordinate system on a globe, the parallels of latitude are small circles, with the Equator the only great circle. By contrast, all meridians of longitude, paired with their opposite meridian in the other hemisphere, form great circles.

Sphere-plane intersection

When the intersection of a sphere and a plane is not empty or a single point, it is a circle. This can be seen as follows:

Let S be a sphere with center O, P a plane which intersects S. Draw OE perpendicular to P and meeting P at E. Let A and B be any two different points in the intersection. Then AOE and BOE are right triangles with a common side, OE, and hypotenuses AO and BO equal. Therefore, the remaining sides AE and BE are equal. This proves that all points in the intersection are the same distance from the point E in the plane P, in other words all points in the intersection lie on a circle C with center E. This proves that the intersection of P and S is contained in C. Note that OE is the axis of the circle.

Now consider a point D of the circle C. Since C lies in P, so does D. On the other hand, the triangles AOE and DOE are right triangles with a common side, OE, and legs EA and ED equal. Therefore, the hypotenuses AO and DO are equal, and equal to the radius of S, so that D lies in S. This proves that C is contained in the intersection of P and S.

As a corollary, on a sphere there is exactly one circle that can be drawn through three given points.

The proof can be extended to show that the points on a circle are all a common angular distance from one of its poles.

Compare also conic sections, which can produce ovals.

Sphere-sphere intersection

To show that a non-trivial intersection of two spheres is a circle, assume (without loss of generality) that one sphere (with radius ${\displaystyle R}$) is centered at the origin. Points on this sphere satisfy

${\displaystyle x^{2}+y^{2}+z^{2}=R^{2}.}$

Also without loss of generality, assume that the second sphere, with radius ${\displaystyle r}$, is centered at a point on the positive x-axis, at distance ${\displaystyle a}$ from the origin. Its points satisfy

${\displaystyle (x-a)^{2}+y^{2}+z^{2}=r^{2}.}$

The intersection of the spheres is the set of points satisfying both equations. Subtracting the equations gives

${\displaystyle {\begin{aligned}(x-a)^{2}-x^{2}&=r^{2}-R^{2}\\a^{2}-2ax&=r^{2}-R^{2}\\x&={\frac {a^{2}+R^{2}-r^{2}}{2a}}.\end{aligned}}}$

In the singular case ${\displaystyle a=0}$, the spheres are concentric. There are two possibilities: if ${\displaystyle R=r}$, the spheres coincide, and the intersection is the entire sphere; if ${\displaystyle R\not =r}$, the spheres are disjoint and the intersection is empty. When a is nonzero, the intersection lies in a vertical plane with this x-coordinate, which may intersect both of the spheres, be tangent to both spheres, or external to both spheres. The result follows from the previous proof for sphere-plane intersections.

See also

• Line-plane intersection
• Line–sphere intersection

References

1. Proof follows Hobbs, Prop. 304
2. Hobbs, Prop. 308
3. Hobbs, Prop. 310

• Hobbs, C.A. (1921). Solid Geometry. G.H. Kent. pp. 397 ff.

Further reading

• Sykes, M.; Comstock, C.E. (1922). Solid Geometry. Rand McNally. pp. 81 ff.

• Rotational symmetry
• Spherical curves
• Circles