# What is Ellipsoid: Definition and 96 Discussions

An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface;  that is, a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by either of the two following properties. Every planar cross section is either an ellipse, or is empty, or is reduced to a single point (this explains the name, meaning "ellipse-like"). It is bounded, which means that it may be enclosed in a sufficiently large sphere.
An ellipsoid has three pairwise perpendicular axes of symmetry which intersect at a center of symmetry, called the center of the ellipsoid. The line segments that are delimited on the axes of symmetry by the ellipsoid are called the principal axes, or simply axes of the ellipsoid. If the three axes have different lengths, the ellipsoid is said to be triaxial or rarely scalene, and the axes are uniquely defined.
If two of the axes have the same length, then the ellipsoid is an ellipsoid of revolution, also called a spheroid. In this case, the ellipsoid is invariant under a rotation around the third axis, and there are thus infinitely many ways of choosing the two perpendicular axes of the same length. If the third axis is shorter, the ellipsoid is an oblate spheroid; if it is longer, it is a prolate spheroid. If the three axes have the same length, the ellipsoid is a sphere.

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36. ### Establishing a smooth differential structure on the ellipsoid

Homework Statement Construct a C∞ natural differential structure on the ellipsoid \left\{(x_{1}, x_{2}, x_{3})\in E | \frac{x_{1}^{2}}{a^{2}}+\frac{x_{2}^{2}}{b^{2}}+ \frac{x_{3}^{2}}{c^{2}}=1\right\} Is this diffeomorphic to S2? Explain. Homework Equations Do I need to prove...
37. ### Finding straight line distance between an ellipsoid and a point

I have an ellipsoid representing the Earth (WGS84) and the current location of a spacecraft (somewhere above the surface). I am trying to find a method that allows me to calculate the straight line distance from the point to the surface of the ellipsoid. Any help would be appreciated...
38. ### Earth ellipsoid due to rotation

I recently saw a documentary, which claimed that if the Earth rotation slows down the water of the oceans will flood to the north and south because the centripetal force at the equator diminishes. In fact, earth’s radius is about 20 km longer at the equator than at the poles. However, I doubt...
39. ### Volume integral of an ellipsoid with spherical coordinates.

Homework Statement By making two successive simple changes of variables, evaluate: I =\int\int\int x^{2} dxdydz inside the volume of the ellipsoid: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=R^{2} Homework Equations dxdydz=r^2 Sin(phi) dphi dtheta dr The...
40. ### Finding the volume of an ellipsoid by using the volume of a sphere

Homework Statement So I keep coming across problems that suggest finding the Volume of an ellipsoid using the volume of a ball ie: Find the volume enclosed by the ellipsoid: (x/a)^2 + (y/b)^2 + (z/c)^2 = 1 by using the fact that the volume of the unit ball in R^3 is 4pi/3...
41. ### General formula for an angled ellipsoid

Homework Statement Hi, I'm writing a simple geophysics program in Fortran77. I'm trying to determine if a point (h,k,m) is within an angled ellipsoid. Theoretically I know the semi-axes of the ellipsoid (a,b,c), the value of the point (h,k,m), the azimuth (∅, +ve from the Y axis, 0≤∅<180°)...
42. ### Distance of a point to an Ellipsoid

I am working on a Matlab sim and I need to find the shorted distance of a point to an Elliposid surface. The point is defined as [X,Y,Z]. Elliposid center is defined as [Xc,Yc,Zc] Ellipsoid is defined as A B C E F G H I J (I don't if that's sufficient information for ellipsoid...
43. ### Maximum distance from point on ellipsoid

Homework Statement Find the point on an given ellipsoid that is the farthest to a given surface.(Distance between point on ellipsoid and surface should be max).Homework Equations ellipsoid: \left(x-3\right)^{2}\over{3}+y^{2}\over{4}+z^{2}\over{5} = 1 surface: 3x+4y^{2}+6z + 6=0 The Attempt...
44. ### Vector calculus question - surface of ellipsoid

Homework Statement Let E be the ellipsoid \frac{x^2}{a^2}+\frac{y^2}{b^2}+z^2=1 where a>\sqrt{2} and b>\sqrt{2}. Let S be the part of the surface of E defined by 0\le x\le1, 0\le y\le1, z>0 and let \mathbf{F} be the vector field defined by \mathbf{F}=(-y,x,0). Given that the surface area...
45. ### Find the maximum value of a rectangular box that can be inscribed in an ellipsoid

"Find the maximum value of a rectangular box that can be inscribed in an ellipsoid.." Homework Statement Find the maximum value of a rectangular box that can be inscribed in an ellipsoid x^2 /4 + y^2 /64 + z^2 /81 = 1 with sides parallel to the coordinate to the coordinate axes...
46. ### Convert ellipsoid from cartesian to spherical equation

Homework Statement In order to advance on a problem I'm working, I need to covert this ellipsoid from cartesian to spherical coordinates. \frac{x^2}{a^2} +\frac{y^2}{b^2} +\frac{z^2}{c^2} = 1 Homework Equations x^2 +y^2+z^2= \rho ^2 x=\rho sin \phi cos \theta y= \rho sin \phi sin...
47. ### Moment of Inertia for Ellipsoid

Homework Statement a)Evaluate ∫∫∫E dV, where E is the solid enclosed by the ellipsoid x^2/a^2+y^2/b^2+z^2/c^2 =1. Use the transformation x=au, y=bv, z=cw. b)If the solid in the above has density k find the moment of inertia about the z-axis. Homework Equations ∅=phi The Attempt...
48. ### Calc 3, area of an ellipsoid slice

This isn't homework or anything, I just want to understand the question better. Homework Statement The Attempt at a Solution I'm honestly not sure where to go with this. Is this an integral problem? As I understand it I'm finding the area of a slice, not a volume of the whole...
49. ### Differential geometry: smooth atlas of an ellipsoid

Homework Statement Consider the ellipsoid L \subsetE3 specified by (x/a)^2 + (y/b)^2 + (z/c)^2=1 (a, b, c \neq 0). Define f: L-S^{2} by f(x, y, z) = (x/a, y/b. z/c). (a) Verify that f is invertible (by finding its inverse). (b) Use the map f, together with a smooth atlas of S^{2}, to...
50. ### Ellipsoid algebra: converting forms

I have a matrix D (it happens to be in R^(nxm) where n>>m, but I don't think that is relevant at this point). I also have a vector t in R^n. I am interested in rewriting the set {x | (Dx-t)'(Dx-t) <= c} in standard ellipsoid form: --> {x | (x-z)'E(x-z)<=b} where E is an mxm positive...