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- Oblate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the non-focal axis of the ellipse, i.e., the symmetry axis that separates the foci. Thus, the two foci are transformed into a ring of radius in the x-y plane. (Rotation about the other axis produces prolate spheroidal coordinates.) Oblate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two largest semi-axes are equal in length. Oblate spheroidal coordinates are often useful in solving partial differential equations when the boundary conditions are defined on an oblate spheroid or a hyperboloid of revolution. For example, they played an important role in the calculation of the Perrin friction factors, which contributed to the awarding of the 1926 Nobel Prize in Physics to Jean Baptiste Perrin. These friction factors determine the rotational diffusion of molecules, which affects the feasibility of many techniques such as protein NMR and from which the hydrodynamic volume and shape of molecules can be inferred. Oblate spheroidal coordinates are also useful in problems of electromagnetism (e.g., dielectric constant of charged oblate molecules), acoustics (e.g., scattering of sound through a circular hole), fluid dynamics (e.g., the flow of water through a firehose nozzle) and the diffusion of materials and heat (e.g., cooling of a red-hot coin in a water bath) (en)
- 扁球面坐標系（英語：Oblate spheroidal coordinates）是一種三維正交坐標系。設定二維橢圓坐標系包含於xz-平面；兩個焦點與的直角坐標分別為與。將橢圓坐標系繞著z-軸旋轉，則可以得到扁球面坐標系。（假若，繞著y-軸旋轉，則可以得到長球面坐標系。）橢圓坐標系的兩個焦點，變為一個半徑為的圓圈，包含於三維空間的xy-平面。稱這圓圈為焦圓，又稱為參考圓。扁球面坐標系可以被視為橢球坐標系的極限案例，其兩個最大的半軸的長度相同。 當邊界條件涉及扁球面或旋轉雙曲面時，扁球面坐標時常可以用來解析偏微分方程式。例如，關於（Perrin friction factors）的計算，扁球面坐標扮演了極重要的角色。讓·佩蘭因此而榮獲1926年諾貝爾物理獎。佩蘭摩擦因子決定了分子的（rotational diffusion）。這程序又影響了許多科技，像蛋白質核磁共振光譜學（protein NMR），的可行性。應用這程序，我們可以推論分子的流體動力體積與形狀。扁球面坐標也時常用來解析電磁學（例如，扁球形帶電的分子的電容率），聲學（例如，聲音通過圓孔時產生的散射），流體動力學（水通過消防水帶的噴口），擴散理論（紅熱的錢幣在水裏的冷卻），等等方面的問題。 (zh)
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- 扁球面坐標系（英語：Oblate spheroidal coordinates）是一種三維正交坐標系。設定二維橢圓坐標系包含於xz-平面；兩個焦點與的直角坐標分別為與。將橢圓坐標系繞著z-軸旋轉，則可以得到扁球面坐標系。（假若，繞著y-軸旋轉，則可以得到長球面坐標系。）橢圓坐標系的兩個焦點，變為一個半徑為的圓圈，包含於三維空間的xy-平面。稱這圓圈為焦圓，又稱為參考圓。扁球面坐標系可以被視為橢球坐標系的極限案例，其兩個最大的半軸的長度相同。 當邊界條件涉及扁球面或旋轉雙曲面時，扁球面坐標時常可以用來解析偏微分方程式。例如，關於（Perrin friction factors）的計算，扁球面坐標扮演了極重要的角色。讓·佩蘭因此而榮獲1926年諾貝爾物理獎。佩蘭摩擦因子決定了分子的（rotational diffusion）。這程序又影響了許多科技，像蛋白質核磁共振光譜學（protein NMR），的可行性。應用這程序，我們可以推論分子的流體動力體積與形狀。扁球面坐標也時常用來解析電磁學（例如，扁球形帶電的分子的電容率），聲學（例如，聲音通過圓孔時產生的散射），流體動力學（水通過消防水帶的噴口），擴散理論（紅熱的錢幣在水裏的冷卻），等等方面的問題。 (zh)
- Oblate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the non-focal axis of the ellipse, i.e., the symmetry axis that separates the foci. Thus, the two foci are transformed into a ring of radius in the x-y plane. (Rotation about the other axis produces prolate spheroidal coordinates.) Oblate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two largest semi-axes are equal in length. (en)
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- Oblate spheroidal coordinates (en)
- 扁球面坐標系 (zh)
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