Spherical circle: Difference between revisions

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Revision as of 21:54, 22 July 2023 editJacobolus (talk \| contribs)Extended confirmed users35,790 editsm Jacobolus moved page Circle of a sphere to Spherical circle: in a search of academic literature, "spherical circle" is a bit more common than "circle of a/the sphere", and is also a bit more concise← Previous edit		Latest revision as of 16:41, 26 July 2024 edit undoCitation bot (talk \| contribs)Bots5,453,977 edits Altered pages. Added isbn. Formatted dashes. \| Use this bot. Report bugs. \| Suggested by Abductive \| Category:Spherical curves \| #UCB_Category 6/14
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	{{short description\|Mathematical expression of circle like slices of sphere}}		{{short description\|Mathematical expression of circle like slices of sphere}}
	{{Redirect\|Small circle\|the typographical symbol\|Degree symbol}}		{{Redirect\|Small circle\|the typographical symbol\|Degree symbol}}
			{{inline \|date=May 2024}}
	]		]
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	In ], a '''spherical circle''' (often shortened to '''circle''') is the ] of ]s on a ] at constant ] (the ''~~intrinsic~~ radius'') from a given point on the sphere (the ''pole'' or ''~~intrinsic~~ center''). It is a ] of constant ] relative to the sphere, analogous to a ] in the ]; the curves analogous to ] are called '']s'', and the curves analogous to planar ]s are called '''small circles''' or '''lesser circles'''.		In ], a '''spherical circle''' (often shortened to '''circle''') is the ] of ]s on a ] at constant ] (the ''spherical radius'') from a given point on the sphere (the ''pole'' or ''spherical center''). It is a ] of constant ] relative to the sphere, analogous to a ] in the ]; the curves analogous to ] are called '']s'', and the curves analogous to planar ]s are called '''small circles''' or '''lesser circles'''. If the sphere is embedded in three-dimensional ], its circles are the ] of the sphere with ], and the great circles are intersections with planes passing through the ] of the sphere.

	== Fundamental concepts ==		== Fundamental concepts ==
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	=== Intrinsic characterization ===		=== Intrinsic characterization ===

	A spherical circle with zero geodesic curvature is called a ''great circle'', and is a ] analogous to a straight line in the plane. A great circle separates the sphere into two equal '']s'', each with the great circle as its boundary. If a great circle passes through a point on the sphere, it also passes through the ] (the unique furthest other point on the sphere). 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 ]s 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 spherical circle with zero geodesic curvature is called a ''great circle'', and is a ] analogous to a straight line in the plane. A great circle separates the sphere into two equal '']'', each with the great circle as its boundary. If a great circle passes through a point on the sphere, it also passes through the ] (the unique furthest other point on the sphere). 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 ]s 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 '']s'', 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 ]s in the plane.		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 '']s'', 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 ]s 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. ] 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 ]s in the plane.		Every circle has two antipodal poles (or centers) intrinsic to the sphere. A great circle is equidistant to its poles, while a small circle is closer to one pole than the other. ] 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 ]s in the plane.

	=== Extrinsic characterization ===		=== Extrinsic characterization ===
		⚫	]

	If the sphere is ] in ], the sphere's ] with a ] is a circle, which can ~~also~~ be interpreted extrinsically to the sphere as a Euclidean circle: a locus of points in the plane at a constant ] (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 ] through these poles, passing through the sphere's center and perpendicular to the parallel planes, is called the ''axis'' of the parallel circles.		If the sphere is ] ] in ], the sphere's ] with a ] is a circle, which can be interpreted extrinsically to the sphere as a Euclidean circle: a locus of points in the plane at a constant ] (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 ] 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 ] or ] is a pair of antipodal circles.		The sphere's intersection with a second sphere is also a circle, and the sphere's intersection with a concentric ] or ] is a pair of antipodal circles.
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	In the ] on a globe, the ] of ] are small circles, with the ] the only great circle. By contrast, all ] of ], paired with their opposite meridian in the other ], form great circles.		In the ] on a globe, the ] of ] are small circles, with the ] the only great circle. By contrast, all ] of ], paired with their opposite meridian in the other ], form great circles.

		⚫	== References ==

			* {{cite journal \|mode=cs2 \|last=
⚫	==References==
			Allardice \|first=Robert Edgar \|author-link=Robert Edgar Allardice \|year=
	{{reflist}}
			1883 \|title=Spherical Geometry \|journal=Proceedings of the Edinburgh Mathematical Society \|volume=2 \|pages=8–16 \|doi=10.1017/S0013091500037020 \|doi-access=free }} <!-- hathi trust: https://babel.hathitrust.org/cgi/pt?id=inu.30000021035997&view=1up&seq=16 -->
			* {{cite book \|mode=cs2 \|last=
			Casey \|first=John \|author-link=John Casey (mathematician) \|year=
			1889 \|title=A treatise on spherical trigonometry \|publisher=Hodges, Figgis, & co. \|isbn=
			978-1-4181-8047-8 \|url=https://archive.org/details/treatiseonspheri00seri/ }}
			* {{cite journal \|mode=cs2 \|last=
			Papadopoulos \|first=Athanase \|author-link=Athanase Papadopoulos \|year=
			2014 \|title=On the works of Euler and his followers on spherical geometry \|journal=Gaṇita Bhārati \|volume=36 \|pages=53–108 \|arxiv=1409.4736 }}
			* {{cite book \|mode=cs2 \|last1=
			Todhunter \|first1=Isaac \|authorlink1=Isaac Todhunter \|last2=
			Leathem \|first2=John Gaston \|year=
			1901 \|title=Spherical Trigonometry \|edition=Revised \|publisher=MacMillan \|url=https://archive.org/details/sphericaltrigono00todh/ }}

	]		]
	]		]


	{{geometry stub}}		{{geometry stub}}

Latest revision as of 16:41, 26 July 2024

Mathematical expression of circle like slices of sphere "Small circle" redirects here. For the typographical symbol, see Degree symbol.

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In spherical geometry, a spherical circle (often shortened to circle) is the locus of points on a sphere at constant spherical distance (the spherical radius) from a given point on the sphere (the pole or spherical 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. If the sphere is embedded in three-dimensional Euclidean space, its circles are the intersections of the sphere with planes, and the great circles are intersections with planes passing through the center of the sphere.

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 (the unique furthest other point on the sphere). 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 (or centers) intrinsic to the sphere. 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

$BC^{2}=AB^{2}+AC^{2}$ , where C is the center of the sphere, A is the center of the small circle, and B is a point in the boundary of the small circle. Therefore, knowing the radius of the sphere, and the distance from the plane of the small circle to C, the radius of the small circle can be determined using the Pythagorean theorem.

{\displaystyle BC^{2}=AB^{2}+AC^{2}} — $BC^{2}=AB^{2}+AC^{2}$ , where C is the center of the sphere, A is the center of the small circle, and B is a point in the boundary of the small circle. Therefore, knowing the radius of the sphere, and the distance from the plane of the small circle to C, the radius of the small circle can be determined using the Pythagorean theorem.

If the sphere is isometrically embedded in Euclidean space, the sphere's intersection with a plane is a circle, which can be interpreted 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.

Applications

Geodesy

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.

References

Allardice, Robert Edgar (1883), "Spherical Geometry", Proceedings of the Edinburgh Mathematical Society, 2: 8–16, doi:10.1017/S0013091500037020
Casey, John (1889), A treatise on spherical trigonometry, Hodges, Figgis, & co., ISBN 978-1-4181-8047-8
Papadopoulos, Athanase (2014), "On the works of Euler and his followers on spherical geometry", Gaṇita Bhārati, 36: 53–108, arXiv:1409.4736
Todhunter, Isaac; Leathem, John Gaston (1901), Spherical Trigonometry (Revised ed.), MacMillan

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