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SphericalDistance[{θ1,ϕ1},{θ2,ϕ2}]

返回单位球面上的点 {θ1,ϕ1} 和 {θ2,ϕ2} 之间的大圆距离.

SphericalDistance[{θ1,1,θ1,2,…,ϕ1},{θ2,1,θ2,2,…,ϕ2}]

返回单位超球表面上任意维度的点之间的测地距离.

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Details and Options Details and Options
范例  
基本范例  
范围  
应用  
属性和关系  
可能存在的问题  
巧妙范例  
参见
历史
按以下格式引用：

SphericalDistance

SphericalDistance[{θ1,ϕ1},{θ2,ϕ2}]

返回单位球面上的点 {θ1,ϕ1} 和 {θ2,ϕ2} 之间的大圆距离.

SphericalDistance[{θ1,1,θ1,2,…,ϕ1},{θ2,1,θ2,2,…,ϕ2}]

返回单位超球表面上任意维度的点之间的测地距离.

更多信息

范例

打开所有单元 关闭所有单元

基本范例  (3)

单位球面上一对点之间的距离：

Wolfram Language code: SphericalDistance[{Pi / 4, 0}, {Pi / 3, Pi / 4}]

单位 3-sphere 上一对点之间的距离：

Wolfram Language code: SphericalDistance[{0.52, 0.63, 0.78}, {1.05, 1.57, 3.14}]

在球体的赤道上取几个点：

Wolfram Language code: pts = {{Pi / 2, 0}, {Pi / 2, Pi / 2}, {Pi / 2, Pi}, {Pi / 2, 3Pi / 2}}

证明它们到北极的距离相等：

Wolfram Language code: SphericalDistance[{0, 0}, pts]

范围  (5)

SphericalDistance 接受精确的数值输入：

Wolfram Language code: SphericalDistance[{Pi / 6, Pi / 5, Pi / 4}, {Pi / 3, Pi / 2, Pi}]

SphericalDistance 接受任意精度的数值输入：

Wolfram Language code: SphericalDistance[N[{Pi / 6, Pi / 5, Pi / 4}, 35], N[{Pi / 3, Pi / 2, Pi}, 35]]

SphericalDistance 也可处理符号输入：

Wolfram Language code: SphericalDistance[{a, b}, {c, d}]

计算矩形 DistanceMatrix，并用 ArrayPlot 显示结果：

Wolfram Language code: SeedRandom[1234]; ArrayPlot[SphericalDistance[RandomReal[Pi, {5, 4}], RandomReal[Pi, {10, 4}]]]

SphericalDistance 接受用 GeoPosition[{lat,lon}] 表示的点：

Wolfram Language code: SphericalDistance[GeoPosition[{80.0, 87.0}], GeoPosition[{50.0, 135.3}]]

使用一组用 GeoPosition 表示的点：

Wolfram Language code: SphericalDistance[GeoPosition[{{80.0, 87.0}, {50.0, 60.0}}], GeoPosition[{50.0, 135.3}]]

应用  (3)

在单位球的八分体内生成三个点：

Wolfram Language code: SeedRandom[1234]; pts = RandomReal[Pi / 2, {3, 2}]

求相应球面三角形的边长：

Wolfram Language code: distances = SphericalDistance@@@Subsets[pts, {2}]

验证球面三角不等式：

Wolfram Language code: AllTrue[{{{1, 2}, 3}, {{2, 3}, 1}, {{3, 1}, 2}}, Function[Greater[Total[distances[[First[#]]]], distances[[Last[#]]]]]]

在球坐标系中定义参数化轨迹：

Wolfram Language code: fun = Function[t, {Pi / 2 + Pi / 4 Cos[2t], t}]

用一系列点表示曲线并绘制：

Wolfram Language code: curve = N[fun /@ Range[0, 2Pi, 2Pi / 1000]];
Wolfram Language code: Graphics3D[{Darker[Blue], Thick, Line[Transpose[FromSphericalCoordinates[{1, θ, ϕ}] /. {θ -> First[Transpose[curve]], ϕ -> Last[Transpose[curve]]}]], Opacity[.75], LightGray, Sphere[]}, Lighting -> AmbientLight[White], ViewVertical -> {0, 0, 1}, ViewPoint -> {85, 315, 53}, Boxed -> False ]

估计曲线的弧长：

Wolfram Language code: Total[MapApply[SphericalDistance, Partition[curve, 2, 1]]]

取八面体顶点的角坐标：

Wolfram Language code: octahedron = Join[{{0, 0}, {Pi, 0}}, {Pi / 2, # Pi / 2}& /@ Range[0, 3]]

计算一个点到所有其他点的距离：

Wolfram Language code: SphericalDistance[First[octahedron], octahedron]

求任意一对点之间的对称距离矩阵：

Wolfram Language code: SphericalDistance[octahedron, octahedron]
Wolfram Language code: SymmetricMatrixQ[%]

属性和关系  (5)

计算沿大圆 的角距离：

Wolfram Language code: SphericalDistance[{Pi / 2, Pi / 5}, {Pi / 2, Pi / 2}]

计算沿大圆 的角距离：

Wolfram Language code: SphericalDistance[{Pi / 5, Pi}, {Pi / 2, Pi}]

两个结果皆为分量之差：

Wolfram Language code: Pi / 2 - Pi / 5

在球面上选择两个点：

Wolfram Language code: SeedRandom[1234]; {a, b} = RandomReal[Pi, {2, 2}]

转换为单位球面上的笛卡尔坐标：

Wolfram Language code: {p, q} = FromSphericalCoordinates[Prepend[#, 1]& /@ {a, b}]

比较 SphericalDistance 和 VectorAngle 的结果：

Wolfram Language code: SphericalDistance[a, b]
Wolfram Language code: VectorAngle[p, q]
Wolfram Language code: % - %%

EuclideanDistance 的结果总是小于 SphericalDistance 的结果：

Wolfram Language code: EuclideanDistance[p, q]
Wolfram Language code: SphericalDistance[a, b]

找到猎户座中的两颗星：

Wolfram Language code: stars = {Entity["Star", "Rigel"], Entity["Star", "Betelgeuse"]};
Wolfram Language code: AstroGraphics[stars, AstroReferenceFrame -> "Equatorial", AstroRange -> Quantity[15, "AngularDegrees"]]

计算它们的角距，单位为弧度：

Wolfram Language code: Apply[SphericalDistance, Times[{90, 0} - AstroPosition[#, "Equatorial"]["Data"][[{2, 1}]], Degree]& /@ stars]

与 AstroAngularSeparation 的结果相比较：

Wolfram Language code: Normal[AstroAngularSeparation[#1, #2, Entity["Star", "Sun"]]&@@stars ]

用 GeoPosition 指定坐标：

Wolfram Language code: p = GeoPosition[{80.0, 87.0}]; q = GeoPosition[{50.0, 135.3}];
Wolfram Language code: SphericalDistance[p, q]

计算椭圆形地球上的 GeoDistance：

Wolfram Language code: GeoDistance[p, q]

除以地球的平均半径，得到非常相近的结果：

Wolfram Language code: Divide[%, Entity["Planet", "Earth"]["Radius"]]

计算球体上任意点之间的符号距离：

Wolfram Language code: SphericalDistance[{a, b}, {c, d}]

与 VectorAngle 的结果相比较，在实数上进行化简：

Wolfram Language code: Simplify[VectorAngle[FromPolarCoordinates[{1, a, b}], FromPolarCoordinates[{1, c, d}]], {a, b, c, d}∈Reals]

可能存在的问题  (1)

SphericalDistance 的不同用法可能会返回不同的结果：

Wolfram Language code: res1 = SphericalDistance[{a1, a2, a3}, {{b1, b2, b3}}]
Wolfram Language code: res2 = SphericalDistance[{a1, a2, a3}, {b1, b2, b3}]

检查两个结果是否相同：

Wolfram Language code: First[res1] == res2//Simplify

巧妙范例  (1)

列出四面体顶点的球坐标：

Wolfram Language code: tetrahedronPts = Function[{ {#1, #2}, {Pi - #1, -#2}, {Pi - #1, 3#2}, {#1, -3#2} }][ArcTan[Sqrt[2]], Pi / 4]

绘制四面体，同时在顶点之间绘制弧线：

Wolfram Language code: With[{cartesian = FromSphericalCoordinates[ Prepend[#, 1]]& /@ tetrahedronPts}, Graphics3D[{Line /@ Subsets[cartesian, {2}], Red, Line[Nest[Riffle[#, Normalize @* Mean /@ Partition[#, 2, 1] ]&, #, 5]]& /@ Subsets[cartesian, {2}], LightGray, Opacity[.5], Sphere[]}, Lighting -> AmbientLight[White], Boxed -> False, ViewVertical -> {0, 0, 1}, ViewPoint -> {2, -3, 4} ] ]

验证所有点之间的角距都相等：

Wolfram Language code: SphericalDistance[tetrahedronPts, tetrahedronPts]
Wolfram Research (2023)，SphericalDistance，Wolfram 语言函数，https://reference.wolfram.com/language/ref/SphericalDistance.html.

文本

Wolfram Research (2023)，SphericalDistance，Wolfram 语言函数，https://reference.wolfram.com/language/ref/SphericalDistance.html.

CMS

Wolfram 语言. 2023. "SphericalDistance." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/SphericalDistance.html.

APA

Wolfram 语言. (2023). SphericalDistance. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/SphericalDistance.html 年

BibTeX

@misc{reference.wolfram_2026_sphericaldistance, author="Wolfram Research", title="{SphericalDistance}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/SphericalDistance.html}", note=[Accessed: 03-November-2026]}

BibLaTeX

@online{reference.wolfram_2026_sphericaldistance, organization={Wolfram Research}, title={SphericalDistance}, year={2023}, url={https://reference.wolfram.com/language/ref/SphericalDistance.html}, note=[Accessed: 03-November-2026]}