Editing Sphere (section)

==Curves on a sphere {{anchor|Curves}}==
[[File:Ellipso-eb-ku.svg|upright=1.2|thumb|Plane section of a sphere: one circle]]
[[File:Kugel-zylinder-kk.svg|thumb|Coaxial intersection of a sphere and a cylinder: two circles]]

===Circles===
{{Main|Circle of a sphere}}
Circles on the sphere are, like circles in the plane, made up of all points a certain distance from a fixed point on the sphere. The intersection of a sphere and a plane is a circle, a point, or empty.<ref>{{MathWorld |id=SphericSection |title=Spheric section}}</ref> Great circles are the intersection of the sphere with a plane passing through the center of a sphere: others are called small circles.

More complicated surfaces may intersect a sphere in circles, too: the intersection of a sphere with a [[surface of revolution]] whose axis contains the center of the sphere (are ''coaxial'') consists of circles and/or points if not empty. For example, the diagram to the right shows the intersection of a sphere and a cylinder, which consists of two circles. If the cylinder radius were that of the sphere, the intersection would be a single circle. If the cylinder radius were larger than that of the sphere, the intersection would be empty.

===Loxodrome===
{{Main|Rhumb line}}
[[File:Loxodrome.png|thumb|upright=0.5|Loxodrome]]

In [[navigation]], a ''loxodrome'' or ''rhumb line'' is a path whose [[bearing (navigation)|bearing]], the angle between its tangent and due North, is constant. Loxodromes project to straight lines under the [[Mercator projection]]. Two special cases are the [[meridian (geography)|meridians]] which are aligned directly North–South and [[circle of latitude|parallels]] which are aligned directly East–West. For any other bearing, a loxodrome spirals infinitely around each pole. For the Earth modeled as a sphere, or for a general sphere given a [[spherical coordinate system]], such a loxodrome is a kind of [[spherical spiral]].<ref>{{cite web | url=https://mathworld.wolfram.com/Loxodrome.html | title=Loxodrome }}</ref>

===Clelia curves===
{{Main|Clélie}}
[[File:Kugel-spirale-1-2.svg|thumb|upright=0.8|Clelia spiral with {{math|1=''c'' = 8}}]]

Another kind of spherical spiral is the Clelia curve, for which the [[longitude]] (or azimuth) <math>\varphi</math> and the [[colatitude]] (or polar angle) <math>\theta</math> are in a linear relationship, {{tmath|1=\varphi = c\theta}}. Clelia curves project to straight lines under the [[equirectangular projection]]. [[Viviani's curve]] ({{tmath|1=c=1}}) is a special case. Clelia curves approximate the [[ground track]] of satellites in [[polar orbit]].

===Spherical conics===
{{Main|Spherical conic}}
The analog of a [[conic section]] on the sphere is a [[spherical conic]], a [[quartic function|quartic]] curve which can be defined in several equivalent ways.
*The intersection of a sphere with a quadratic cone whose vertex is the sphere center
*The intersection of a sphere with an [[cylinder#cylindrical surfaces|elliptic or hyperbolic cylinder]] whose axis passes through the sphere center
*The locus of points whose sum or difference of [[great-circle distance]]s from a pair of [[focus (geometry)|foci]] is a constant

Many theorems relating to planar conic sections also extend to spherical conics.

===Intersection of a sphere with a more general surface===
[[File:Is-spherecyl5-s.svg|thumb|upright=0.8|General intersection sphere-cylinder]]

If a sphere is intersected by another surface, there may be more complicated spherical curves.
;Example: sphere–cylinder
{{Main|Sphere–cylinder intersection}}
The intersection of the sphere with equation <math>\; x^2+y^2+z^2=r^2\;</math> and the cylinder with equation <math> \;(y-y_0)^2+z^2=a^2, \; y_0\ne 0\; </math> is not just one or two circles. It is the solution of the non-linear system of equations
:<math>x^2+y^2+z^2-r^2=0</math>
:<math>(y-y_0)^2+z^2-a^2=0\ .</math>
(see [[implicit curve]] and the diagram)