What is Ellipsoid: Definition and 97 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. 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...
  2. F

    Support Mapping of an Arbitrary Ellipsoid

    In this context, the support mapping of any convex geometry is any point on the geometry which results in the largest dot product to some direction vector. I would appreciate some help in computationally finding the support mapping of an arbitrary ellipsoid (some arbitrary orthonormal basis...
  3. G

    Find Shortest Distance to Point on Ellipsoid

    Right now I'm running this with a brute force program which takes points on an ellipsoid and checks the distance to the point, slightly readjusts, and keeps moving toward the minimum, but it takes far to long for the mass amount of points I want to run through the program. Is there an equation I...
  4. H

    Radiation from vibrating (rotating) ellipsoid

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  5. G

    Maximal volume of a Cuboid inscribed in an Ellipsoid

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  6. S

    Optimization of ellipsoid tube

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  7. C

    Volume of partial ellipsoid cut by plane

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  8. M

    Center of mass equation for an ellipsoid?

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  9. T

    Geodesic curves for an ellipsoid

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  10. R

    Surface Area of a part of a plane inside an ellipsoid.

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  11. M

    Moment of Inertia of an Ellipsoid

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  12. T

    Inscribe a polyhedron in an ellipsoid

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  13. U

    The VERY, VERY general equation of an ellipsoid - Who knows it?

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  14. B

    No problem, glad I could help! Good luck with your modeling!

    Ok so basically what I'm trying to model is an ellipsoid on a plane, the planes angle can be changed by the user and the ellisoid should move accordingly. But I have absolutely no idea where to start. I've tried finding equations etc but I could't find anything other than the equation of an...
  15. R

    Volume between cylinder and ellipsoid

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  16. W

    Solving Volume of Ellipsoid with Spherical Coordinates

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  17. M

    Surface area of a truncated ellipsoid

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  18. S

    Principal moments of inertia ellipsoid

    Problem: Find the principal moments of inertia of an ellipsoid. I began to label moments of intertia about minor axises. But I'm not sure where to go after that.
  19. W

    Tank Shape: Ellipsoid or Something Else?

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  20. mnb96

    Ellipsoid Equation: Finding the Classical Form with Rotated Radii

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  21. N

    Volumes of Revolution - Ellipsoid

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  22. P

    Describing and Sketching an Ellipsoid

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  23. H

    Finding Umbilic Points of an Ellipsoid & Lines of Curvature

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  24. H

    Finding Umbilic Points on an Ellipsoid & Their Connection to Lines of Curvature

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  25. P

    Coordinates of a point that touches the ellipsoid

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  26. P

    Find the coordinates of the point on the ellipsoid where the major axis meet

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  27. P

    Find the coordinates of the point on the ellipsoid where the major axis meet

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  28. K

    Outward flux through an ellipsoid

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  29. J

    Gravity anomalies : geoid v reference ellipsoid

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  30. W

    Radius of Ellipsoid with Known a,b,c and Phi, Theta

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  31. L

    Ellipsoid Creation and Rotation

    A little background first is that I'm currently a rising Sophomore at Winthrop University in South Carolina. I am a Computer Science and Mathematics Double Major. After finishing half-way through third semester calculus and dealing with 3-Dimensional space, vectors, planes, and surfaces, I...
  32. H

    Solve a Challenging Ellipsoid Problem Today!

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  33. O

    The widest point on an ellipsoid

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  34. B

    How Do You Calculate Ray and Ellipsoid Intersections?

    So I have an array p(t) = e + td, where e is the start position, t is some parameter, and d is the direction of the ray For a sphere with center c and radius R, the vector form equation is (p-c).(p-c)-R^2=0 This can be algebraically manipulated into: t = (-d.(e-c) +- sqrt((d.(e-c))^2 -...
  35. E

    Classical scattering off an ellipsoid

    Homework Statement Particles of scattered off the surface of an ellipsoid given by x^+y^2+z^2/f^2 = R^2, where f and R are constants. Find the differential cross-section.Homework Equations The Attempt at a Solution Let s be the impact parameter. I can find s as a function of the scattering...
  36. M

    Finding the Equation for the Plane of Equidistant Points: Solving for b and c

    Homework Statement Find an equation for the plane consisting of all points that are equidistant from the points (1,0,-2) and (3,4,0) Homework Equations The Attempt at a Solution I found the midpoint ant (4, 4, -2), which I believe is the center. However, I have no idea on how...
  37. E

    Probability Ellipsoid: Explaining Magnetic Field Axes

    Is anyone familiar with the concept of a probability tensor or a probability ellipsoid? I am learning about them in the context of NMR techniques. Here is a page describing them: http://www3.interscience.wiley.com/cgi-bin/abstract/107633228/ABSTRACT?CRETRY=1&SRETRY=0 My question is, why...
  38. T

    Precession of ellipsoid Question

    Homework Statement A uniform symmetric ellipsoid (Mass M) has a large semi axis c and small semi axis a. A particle of mass m<<M is moving along a straight line parallel to the x-axis with speed v(i). Its y-coordinate is a/2 and its z-coordinate it c/2. After an inelastic collision, it sticks...
  39. H

    What is the Maximum Inscribed Sphere for Ellipsoid?

    I'm trying to find the largest sphere that be inscribed inside the ellipsoid with equation 3x^2 + 2y^2 + z^2 = 6. Homework Equations I know I will need at least 2 equations. One of them is the constraining equation (f(x) = a, where 'a' is a constant) and the other is the equation you...
  40. H

    Solving Raindrop Paths on Ellipsoid with 4x^2+y^2+4z^2=16

    We have an ellipsoid with the equation 4x^2 + y^2+ 4z^2 = 16, and it is raining. Gravity will make the raindrops slide down the dome as rapidly as possible. I have to describe the curves whose paths the raindrops follow. This is probably more vector calculus than physics, but i wasn't sure...
  41. M

    Finding the Normal to an Ellipsoid - A Step-by-Step Guide

    Hello! I am trying to find the normal to an ellipsoid of the form x^2/a^2 + y^2/b^2 + z^2 = 1 What I did is the following: Let psi = x^2/a^2 + y^2/b^2 + z^2 then grad psi (and thus the normal) is: grad (psi) = (2x/a^2, 2y/b^2, 2z) = n Could anyone tell me whether that...
  42. N

    How can we find the volume of an ellipsoid using an integral?

    Well, I have a small problem. I know the general formula for the volume of an ellipsoid. But I have a task to find it with the help of an integral. Can you explain me how to do this?
  43. tandoorichicken

    Deriving Expression for Volume of Ellipsoid

    I've been asked to derive an expression for the volume of an ellipsoid. I know what the expression is, I just don't know how to get there from the information given. All that is given is that it is defined by \frac{x^2}{a^2} +\frac{y^2}{b^2} +\frac{z^2}{c^2} \leq 1, a standard...
  44. W

    Find the volume of the ellipsoid

    Find the volume of the ellipsoid x^2 + y^2 + 10z^2 = 16 solve for z... z=sqrt((16-x^2-y^2)/(10)) z = sqrt((16-r^2)/10) so to find the volume, my integral looks like this: latex doesn't seem to be working, so this could look messy... 2*int (from 0-2pi)*int(from 0-1)*...
  45. C

    Integrating an Improper Divergent Integral & Ellipsoid Volume

    I need help with two questions. Find a divergent improper integral whose value is neither infinity nor -infinity. 2. Find the volume of an ellipsoid (a^2*x^2) + (b^2*8y^2) + (c^2*z^2) = a^2*b^2*c^2 using integration.
  46. tandoorichicken

    General Equation for an Ellipsoid

    What is the general equation for an ellipsoid (i.e., the general equation of a sphere is (x-h)^2 + (y-j)^2 + (z-k)^2 = r^2 Where (h, j, k) is the center of the sphere) ?
  47. P

    Max Vol Q: Find the Volume of Rectangular Box Inscribed in Ellipsoid

    "Hi, I have a question on max vol. q. Its invloved with multivariable calculus. Q) Find the volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid 9x^2+36y^2 + 4z^2 = 36. What i did was i found the three x,y and z-intersection points...
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