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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  1. L

    I Quadrupole moment tensor calculation for ellipsoid

    Determine the element ##Q_{11}## of the quadrupole tensor for a homogeneously charged rotationally symmetric ellipsoid, $$\rho=\rho_{0}=\text { const. for } \frac{x_{1}^{2}}{a^{2}}+\frac{x_{2}^{2}}{a^{2}}+\frac{x_{3}^{2}}{c^{2}} \leq 1 $$ The formula is $$Q_{i j}=\int \rho(\mathbf{r})\left(3...
  2. Ben2

    I Oblate Ellipsoid: Can Earth Be Modified?

    I'm told Earth is an oblate spheroid. Is it possible for a planet to be an oblate ellipsoid (equation modified from (x/a)^2 + (y/b)^2 + (z/c)^2 = 1)? What would be the possible consequences, to include "tumbling"?
  3. ergospherical

    I Gravitational potential of an ellipsoid

    There is a formula for the potential ##\varphi## outside of a homogenous ellipsoid of density ##\mu## in Landau\begin{align*} \varphi = -\pi \mu abck \int_{\xi}^{\infty} \left(1- \dfrac{x^2}{a^2 + s} + \dfrac{y^2}{b^2 + s} + \dfrac{z^2}{c^2+s} \right) \frac{ds}{R_s} \ \ \ (1) \end{align*}where...
  4. S

    I General Method for Mapping an Ellipsoid to Unit Sphere

    I have been working on a problem for a while and my progress has slowed enough I figured I'd try reaching out for some more experience. I am trying to map a point on an ellipsoid to its corresponding point on a sphere of arbitrary size centered at the origin. I would like to be able to shift any...
  5. Adams2020

    I The surface area of an oblate ellipsoid

    In "An Introduction to Nuclear Physics by W. N. Cottingham, D. A. Greenwood" for the surface area of an oblate ellipsoid, the following equation is written for small values of ε : The book has said this without proof. I found the following formula for the desired shape: No matter how hard I...
  6. The Bill

    Geometry General Ellipsoid Area Formula: Detailed Explanation

    I'm looking for a source that fully derives the complete formula for the surface area of a general (triaxial) ellipsoid. I'd prefer a source that has more than just a full derivation, but also has a fair amount of prose discussion on this topic. Some historical context would be nice, as well...
  7. Buzz Bloom

    I Q: Volume of the largest ellipsoid in space which contains no stars?

    The answer to the primary question in the summary is the first step in seeking an answer to a more complicated question I plan to post in a separate thread later. This more complicated question is a consequence of the thread...
  8. LCSphysicist

    Pappus Theorem and Ellipsoid Fig One: Is My Integral Approach Correct?

    fig one: I just want to know if i am right in attack this problem by this integral: *pi Anyway, i saw this solution: In which it cut beta, don't know why. So i don't know.
  9. M

    I Area of a cylinder enclosing an ellipsoid

    I.m not absolutely sure if this comes under physics or maths, so apologies if I've put it in the wrong place. It is well known that if a sphere is exactly enclosed by a cylinder, the area of the curved surface of the cylinder is equal to hat of the sphere. Does this also apply if the cylinder...
  10. gary0000

    Rotating an ellipse to create a spheroid?

    I was able to find the equation of an ellipse where its major axis is shifted and rotated off of the x,y, or z axis. However, I could not find anywhere an equation for a spheroid that does not have its axis or revolution along the x,y, or z axis. How might I go about deriving such an...
  11. JD_PM

    What is the volume inside an ellipsoid between two intersecting planes?

    Homework Statement Find the volume between the planes ##y=0## and ##y=x## and inside the ellipsoid ##\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1##The Attempt at a Solution I understand we can approach this problem under the change of variables: $$x=au; y= bv; z=cw$$ Thus we get...
  12. D

    A How to Calculate Young's Modulus for Deformation of a Sphere into an Ellipsoid?

    I need to calculate Young's modulus based on deformation of sphere into ellipsoid. I assume the deforming force acting along one axes. Initial dimensions of the object before (sphere) and after (ellipsoid) deformation are known.Does anyone know or familiar with good reference?
  13. D

    Integration for the Volume of an Ellipsoid

    Homework Statement Let E be the ellipsoid x^2 + 2xy +5y^ +4z^2 = 1 Find the Volume of E Homework Equations None, just various integration methods. The Attempt at a Solution I know we're not supposed to say 'I don't know where to start' but with this one I really don't. If the best approach...
  14. S

    Integral over a rotating ellipsoid

    Homework Statement Calculate ##\int x^2 dV## over an ellipsoid with semi-axes a, b and c along x, y and z. rotating around the z axis with an angular speed ##\omega##. Homework EquationsThe Attempt at a Solution I managed to calculate this in the case when it is not rotating and I got...
  15. harpazo

    MHB Ellipsoid: Change of Variables

    Evaluate ∫∫∫ over E, 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. I decided to replace x with au, y with bv and z with cw in the ellipsoid. After simplifying, I got u^2 + v^2 + w^2 = 1. What is the next step...
  16. T

    B About the surface area of a prolate ellipsoid

    Is there any limit for which we can approximately write the surface area of a prolate ellipsoid to be 4piA*B comparing with the spherical 4piR*R??
  17. toforfiltum

    Equation of ellipsoid and graph

    Homework Statement Equation of ellipsoid is: ##\frac{x^2}{4} + \frac{y^2}{9} + z^2 = 1## First part of the question, they asked to graph the equation. I have a question about this, I know that ##-1\leq z \leq 1##. So what happens when the constant 1 gets smaller after minusing some value of...
  18. H

    Rolling of body cone depends on whether ellipsoid is prolate or oblate?

    From the last few sentences of the below attached paragraph, when the inertia ellipsoid is prolate, the body cone rolls outside the space cone; when it is oblate, the body cone rolls inside the space cone. Whether the body cone rolls outside or inside the space cone should depend on whether the...
  19. Conservation

    Finding the intersection of an ellipsoid and a plane

    Homework Statement Find the curve that is the intersection of x-y-z>-10 and x2+y2/4+z2/9=36. Homework EquationsThe Attempt at a Solution The best idea I have is to define x as x=y+z-10 and substitute it into the ellipsoid equation to get a function defined by y and z; the trouble is that...
  20. P

    Matrix Representation of a Uniform Sphere Centered at the Origin

    What is the basic matrix form for a uniform (unit) sphere centered at the origin? Given a vector that specifies the radii (1,1,1) == (r1,r2,r3), I would like the matrix that implies no rotation (is it [[1,0,0],[0,1,0],[0,0,1]]?) and covers the rest of the necessary parameters. I am testing...
  21. TheDemx27

    Linear Charge Distribution on a Needle?

    http://www.colorado.edu/physics/phys3320/phys3320_sp12/AJPPapers/AJP_E&MPapers_030612/Griffiths_ConductingNeedle.pdf I was reading this paper, and was confused by a result in section 2-A. (Heck they even mention they weren't expecting it themselves). The purpose of the paper is to find the...
  22. Dethrone

    MHB Lagrange Multiplier Ellipsoid

    Use Lagrange multipliers to find $a,b,c$ so that the volume $V=\frac{4\pi}{3}abc$ of an ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$, passing through the point $(1,2,1)$ is as small as possible. I just need to make sure my setup is correct. $\triangledown...
  23. bananabandana

    Calculating Flux through Ellipsoid

    Homework Statement Let ## E ## be the ellipsoid: $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+z^{2}=1 $$ Let ## S ## be the part of the surface of ## E ## defined by: $$ 0 \leq x \leq 1, \ 0 \leq y \leq 1, \ z > 0 $$ Let F be the vector field defined by $$ F=(-y,x,0)$$ A) Explain why ##...
  24. Pull and Twist

    MHB Work required to empty water out of a vertical ellipsoid tank.

    This is another problem I am having difficulty with... I set it up like I've been working the book problems, especially the sphere problems, but can't seem to get the right answer. I feel that I am calculating the radius incorrectly. I know I am supposed to us {x}^{2}+{y}^{2}={r}^{2} and r=1...
  25. B

    Finding flux through ellipsoid in Cylindrical Coordinates

    Homework Statement Using Cylindrical coordinates, find the total flux through the surface of the ellipsoid defined by x2 + y2 + ¼z2 = 1 due to an electric field E = xx + yy + zz (bold denoting vectors | x,y,z being the unit vectors) Calculate ∇⋅E and then confirm the Gauss's Law Homework...
  26. 1

    Find the outward flux of a vector field across an ellipsoid

    Homework Statement [/B] Find the outward flux of the vector field ## \vec F = y^2e^{z^2+y^2} i + x^2 e^{z^2+x^2} j + z^2 e^{x^2+y^2} k## across that part of the ellipsoid $$ x^2 + y^2 + 4z^2 = 8$$ which lies in the region ##0 ≤ z ≤ 1## (Note: The two “horizontal discs” at the top and bottom are...
  27. AwooOOoo

    Radius of Triaxial / Scalene Ellipsoid

    Hi, I have been referencing this (https://www.physicsforums.com/threads/radius-of-ellipsoid.251321/) previous post to calculate the radius of a Triaxial Ellipsoid (a>b>c), but I'm running into some issues. Let 0 ≤ ϕ ≤ π 0 ≤ θ ≤ 2π and x=r * cos(θ) * sin(ϕ) (1) y=r * sin(θ) * sin(ϕ)...
  28. P

    Scattering from a hard ellipsoid

    Homework Statement a,b is constant and s is impact parameter, θ is scattering angle. i know that ψ in the picture is <ψ=(π-θ)/2> Homework Equations differential scattering crossection dσ/dΩ = (s/sinΘ) I ds/dθ I and σ(θ)=∫(dσ/dΩ)dΩ The Attempt at a Solution i guessed, first step is that...
  29. C

    Regarding volume of an ellipsoid bounded by 2 planar cutting planes

    Homework Statement Hi I require to compute the volume of a ellipsoid that is bounded by two planes. The first horizontal (xy) plane is cutting directly along the mid-section of the ellipsoid. The second horizontal plane is at a z = h below the first horizontal plane. The volume of the...
  30. J

    Ellipsoid intersected by cylinder

    Homework Statement Find volume of the ellipsoid ##x^2 +2(y^2+z^2) \le 10## intersected by the cylinder ##y^2 + z^2 \le 1 ## The Attempt at a Solutionseems like changing to cylindrical coordinates would be best so I have \left\{ \begin{array}{cc} r^2cos\theta + 2r^2sin^2\theta +2z^2 \le 10...
  31. C

    The curve formed by the intersection of paraboloid and ellipsoid

    I will state the specifics to this problem if necessary. I need to find the parametric equations for the the tan line at point, P(x1,y1,z1) on the curve formed from paraboloid intersection with ellipsoid. The parametric equations for the level surfaces that make up paraboloid and ellipsoid...
  32. N

    Calculating Earth's Shape as an Ellipsoid

    Homework Statement I'm an engineering student, and my professor of the mechanics course gave a homework to my class last week, we were intended to calculate the real shape of the Earth (as an ellipsoid) by taking the centrifugal force in account, using the equation a' = a -wX(wXr). For that...
  33. L

    Volume of an ellipsoid using double integrals

    Homework Statement Using double integrals, calculate the volume of the solid bound by the ellipsoid: x²/a² + y²/b² + z²/c² = 1 2. Relevant data must be done using double integrals The Attempt at a Solution i simply can't find a way to solve this by double integrals, i did with triple...
  34. J

    Ellipsoid Equation for Object Motion?

    Homework Statement Hey, I'm doing some physics programming for a game, and could use some general help getting a formula. I'm not great with mathematics/physics, but I know enough to comprehend any feedback. Any help is greatly appreciated! So I have an object free-floating in 3D space at...
  35. skate_nerd

    MHB Area of a given interval and volume of an ellipsoid

    I am given a pretty basic ellipsoid: $$\frac{x^2}{16}+\frac{y^2}{9}+\frac{z^2}{1}=1$$ First, for each number t in the interval \(-1\leq{t}\leq{1}\) I need to find the area A(t) of the plane cross-section made by \(z=t\). This I know should be a function of \(t\). After that I have to find the...
  36. S

    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. S

    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. U

    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. E

    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. F

    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. A

    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. J

    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. S

    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. F

    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. S

    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. ArcanaNoir

    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. L

    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. A

    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. G

    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. D

    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...