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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 (the intrinsic center). It is a curve of constant curvature relative to the sphere.

A circle with zero curvature relative to the sphere is called a great circle, and is a geodesics analogous to a straight line in the Euclidean plane. 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.

A circle with non-zero curvature relative to the sphere is called a small circle or lesser circle, and is analogous to a circle in the plane. For any triple of distinct non-antipodal points a unique small circle passes through all three.

Every circle has two antipodal centers, sometimes called its poles. 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 lines in the plane.

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 in Euclidean space 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.

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.

Related terminology

The diameter of the sphere which passes through the center of the circle is called its axis and the endpoints of this diameter are called its poles. A circle of a sphere can also be defined as the set of points at a given angular distance from a given pole.

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