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单词 Oblate spheroidal coordinates
释义

Oblate spheroidal coordinates

中文百科

扁球面坐标系

扁球面坐标系是一种三维正交坐标系。设置二维椭圆坐标系包含于xz-平面；两个焦点F_{1}与F_{2}的直角坐标分别为( - a,\ 0,\ 0)与(a,\ 0,\ 0)。将椭圆坐标系绕着z-轴旋转，则可以得到扁球面坐标系。（假若，绕着y-轴旋转，则可以得到长球面坐标系。）椭圆坐标系的两个焦点，变为一个半径为a的圆圈，包含于三维空间的xy-平面。称这圆圈为焦圆，又称为参考圆。扁球面坐标系可以被视为椭球坐标系的极限案例，其两个最大的半轴的长度相同。

当边界条件涉及扁球面或旋转双曲面时，扁球面坐标时常可以用来解析偏微分方程序。例如，关于佩兰摩擦因子（Perrin friction factors）的计算，扁球面坐标扮演了极重要的角色。让·佩兰因此而荣获1926年诺贝尔物理奖。佩兰摩擦因子决定了分子的旋转扩散（rotational diffusion）。这进程又影响了许多科技，像蛋白质核磁共振光谱学（protein NMR），的可行性。应用这进程，我们可以推论分子的流体动力体积与形状。扁球面坐标也时常用来解析电磁学（例如，扁球形带电的分子的电容率），声学（例如，声音通过圆孔时产生的散射），流体动力学（水通过消防水带的喷口），扩散理论（红热的钱币在水里的冷却），等等方面的问题。

英语百科

Oblate spheroidal coordinates 扁球面坐标系

Figure 1: Coordinate isosurfaces for a point P (shown as a black sphere) in oblate spheroidal coordinates (μ, ν, φ). The z-axis is vertical, and the foci are at ±2. The red oblate spheroid (flattened sphere) corresponds to μ = 1, whereas the blue half-hyperboloid corresponds to ν = 45°. The azimuth φ = −60° measures the dihedral angle between the green x-z half-plane and the yellow half-plane that includes the point P. The Cartesian coordinates of P are roughly (1.09, −1.89, 1.66).
Figure 2: Plot of the oblate spheroidal coordinates μ and ν in the x-z plane, where φ is zero and a equals one. The curves of constant μ form red ellipses, whereas those of constant ν form cyan half-hyperbolae in this plane. The z-axis runs vertically and separates the foci; the coordinates z and ν always have the same sign. The surfaces of constant μ and ν in three dimensions are obtained by rotation about the z-axis, and are the red and blue surfaces, respectively, in Figure 1.
Figure 3: Coordinate isosurfaces for a point P (shown as a black sphere) in the alternative oblate spheroidal coordinates (σ, τ, φ). As before, the oblate spheroid corresponding to σ is shown in red, and φ measures the azimuthal angle between the green and yellow half-planes. However, the surface of constant τ is a full one-sheet hyperboloid, shown in blue. This produces a two-fold degeneracy, shown by the two black spheres located at (x, y, ±z).

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 a 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.
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更新时间：2025/6/23 8:16:55