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An IP user has set a question in section "Plane sections" of the article, about the proof that planes sections of an ellipsoid are circles, single points of empty point. I have put it in the field "reason=" of a template {{ clarification needed}}. As I am not sure wether a detailed proof deserves to appear in the article. I'll answer here.
Let E be an ellipsoid, P be a plane, f be an affine transformation that maps the unit sphere onto E, and g be the inverse transformation. The image by g of the intersection of E and P is the intersection C of the unit sphere g(E) and the plane g(P). It is thus either a circle, a point, or the empty set. Thus the intersection of E and P is f(C), that is an ellipse, a point or the empty circle.
I do not know if such a proof deserve to be put in the article. Therefore, I leave to the community to decide what should be done. D.Lazard ( talk) 14:57, 18 September 2017 (UTC)
@ D.Lazard: the edit 858602570 is correct, Geoid, an important object modeled by an elipsoid. It is not an elipsoid, it is modeled as an elipsoid (instead a sphere). The most used elipsoid-Goid model in nowadays, is the WGS84 elipsoid. -- Krauss ( talk)
As a general rule, the "See also" section should not repeat links that appear in the article's body or its navigation boxes. D.Lazard ( talk) 11:19, 8 September 2018 (UTC)
An editor as inserted in the list item about Poinsot's ellipsoid a comment about a so called "Mac Cullagh ellipsoid". I have reverted this edit, and I'll revert it again for the following reasons:
Thus I'll revert again the mention of "Mac Cullagh ellipsoid" in this article. If you disagree, please, read carefully WP:BRD and do not start WP:Edit warring.
I'll also revert the insertion of Geoid in section "See also", as the linked article does not contain the word "ellipsoid", and is linked in other articles appearing in this see also section. It thus not useful for any reader to link this article. D.Lazard ( talk) 19:11, 16 November 2018 (UTC)
MacCullagh ellipsoid is now at WP:AfD. D.Lazard ( talk) 12:38, 19 November 2018 (UTC)
I think the formula in this section for e_1 is incorrect by setting its z-coordinate to 0. If so, after inverse affine transformation, the corresponding z-coodinate is still 0, which is not necessary.— Preceding unsigned comment added by 24.188.214.97 ( talk) 07:27, 3 December 2018 (UTC)
The equation for the surface normal in the Parametric Representation section appears to be incorrect for non-spherical ellipsoids. When I try to replicate, the normals point roughly outward but do not match the surface curvature except at the poles and the equator. It is as if the map from the isotropic sphere to the anisotropic ellipsoid is not taken into account. 98.69.156.214 ( talk) 03:20, 10 April 2020 (UTC)
The volume of an ellipsoid is 2/3 the volume of a circumscribed elliptic cylinder, and π/6 the volume of the circumscribed box.
These do not circumscribe uniquely; our preferential embedding in at least the box case can likely be specified as the one having minimum volume. — MaxEnt 00:45, 26 May 2020 (UTC)
The top of article describes ellipsoid as an affine image of a sphere. Since the position of the sphere is not fixed, linear transformations (if understood as homogeneous degree 1 maps) suffice. This is better for introduction as not everyone shares the mathematics terminology that linear transformations must be homogeneous or involve a choice of origin, and "linear" is more intuitive. "Affine" is a more math-specific term and the Wiki link to a page about transformations preserving "an affine structure" is ridiculous as an explanation for people who may not know what is an ellipsoid. It is more of a definition for purists that can be elaborated inside the article. 73.89.25.252 ( talk) 04:39, 14 June 2020 (UTC)
In the "as a quadric" section, the following is stated:
I am not a mathematician, but am a scientist and I tried using this fact to find the size of an ellipsoid. I suspect that the statement of the eigenvalues being the reciprocal squares of the semi-axes is only true when the matrix A is real (or maybe when the eigenvectors of the matrix are real). Hopefully someone here can check this requirement. I can show a simple example proving that some condition is missing in the statement. If you start with a complex positive definite matrix A, the expression
is identical to the alternative expression
because the imaginary components cancel out on the left hand side, when doing the matrix multiplication, if the matrix A is Hermitian. Therefore, both matrices A and B = Re[A] are positive-definite matrices, and they both describe the SAME ellipsoid, because the equation is identical when matrix-vector multiplication is carried out. However, in general, the two matrices have different eigenvalues! So the statement that its eigenvalues are equal to the inverse squares of the semi-axes of the ellipsoid cannot be simultaneously a correct statement for both matrices. By calculating an explicit example, I found that when I take Re[A], then its eigenvalues DO coincide with the inverse squares of the semi-axes, but not when I keep A complex. Long story sort: some condition is needed before stating that the eigenvalues are the inverse squared semi-axes. Probably requirement of real eigenvectors or similar. El pak ( talk) 16:55, 1 July 2021 (UTC)
In Kreyszig, Advanced Engineering Mathematics, 4th ed, on p. 431 there is a parametric representation of a sphere and one is given as part of a problem for an ellipsoid but there is no interpretation of it in terms of the spheroid. In the parameterization θ is not an angle between the equator and a point on the ellipsoid but rather it is a parameter similar to Kepler's eccentric anomaly. ~~~~ Jbergquist ( talk) 23:26, 19 June 2023 (UTC)
In recent edits, there seems to be some confusion about how the semi-axes relate to the eigenvalues of the quadric. Suppose that the ellipsoid is represented by a symmetric 3x3 matrix E, in that the ellipsoid is the set of points (column vectors) x such that xT E x = 1. Then the eigenvalues of E are a-2, b-2, c-2, where a, b, c are the semi-axis lengths. Mgnbar ( talk) 23:09, 24 July 2024 (UTC)
This article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
This page has archives. Sections older than 365 days may be automatically archived by Lowercase sigmabot III when more than 10 sections are present. |
An IP user has set a question in section "Plane sections" of the article, about the proof that planes sections of an ellipsoid are circles, single points of empty point. I have put it in the field "reason=" of a template {{ clarification needed}}. As I am not sure wether a detailed proof deserves to appear in the article. I'll answer here.
Let E be an ellipsoid, P be a plane, f be an affine transformation that maps the unit sphere onto E, and g be the inverse transformation. The image by g of the intersection of E and P is the intersection C of the unit sphere g(E) and the plane g(P). It is thus either a circle, a point, or the empty set. Thus the intersection of E and P is f(C), that is an ellipse, a point or the empty circle.
I do not know if such a proof deserve to be put in the article. Therefore, I leave to the community to decide what should be done. D.Lazard ( talk) 14:57, 18 September 2017 (UTC)
@ D.Lazard: the edit 858602570 is correct, Geoid, an important object modeled by an elipsoid. It is not an elipsoid, it is modeled as an elipsoid (instead a sphere). The most used elipsoid-Goid model in nowadays, is the WGS84 elipsoid. -- Krauss ( talk)
As a general rule, the "See also" section should not repeat links that appear in the article's body or its navigation boxes. D.Lazard ( talk) 11:19, 8 September 2018 (UTC)
An editor as inserted in the list item about Poinsot's ellipsoid a comment about a so called "Mac Cullagh ellipsoid". I have reverted this edit, and I'll revert it again for the following reasons:
Thus I'll revert again the mention of "Mac Cullagh ellipsoid" in this article. If you disagree, please, read carefully WP:BRD and do not start WP:Edit warring.
I'll also revert the insertion of Geoid in section "See also", as the linked article does not contain the word "ellipsoid", and is linked in other articles appearing in this see also section. It thus not useful for any reader to link this article. D.Lazard ( talk) 19:11, 16 November 2018 (UTC)
MacCullagh ellipsoid is now at WP:AfD. D.Lazard ( talk) 12:38, 19 November 2018 (UTC)
I think the formula in this section for e_1 is incorrect by setting its z-coordinate to 0. If so, after inverse affine transformation, the corresponding z-coodinate is still 0, which is not necessary.— Preceding unsigned comment added by 24.188.214.97 ( talk) 07:27, 3 December 2018 (UTC)
The equation for the surface normal in the Parametric Representation section appears to be incorrect for non-spherical ellipsoids. When I try to replicate, the normals point roughly outward but do not match the surface curvature except at the poles and the equator. It is as if the map from the isotropic sphere to the anisotropic ellipsoid is not taken into account. 98.69.156.214 ( talk) 03:20, 10 April 2020 (UTC)
The volume of an ellipsoid is 2/3 the volume of a circumscribed elliptic cylinder, and π/6 the volume of the circumscribed box.
These do not circumscribe uniquely; our preferential embedding in at least the box case can likely be specified as the one having minimum volume. — MaxEnt 00:45, 26 May 2020 (UTC)
The top of article describes ellipsoid as an affine image of a sphere. Since the position of the sphere is not fixed, linear transformations (if understood as homogeneous degree 1 maps) suffice. This is better for introduction as not everyone shares the mathematics terminology that linear transformations must be homogeneous or involve a choice of origin, and "linear" is more intuitive. "Affine" is a more math-specific term and the Wiki link to a page about transformations preserving "an affine structure" is ridiculous as an explanation for people who may not know what is an ellipsoid. It is more of a definition for purists that can be elaborated inside the article. 73.89.25.252 ( talk) 04:39, 14 June 2020 (UTC)
In the "as a quadric" section, the following is stated:
I am not a mathematician, but am a scientist and I tried using this fact to find the size of an ellipsoid. I suspect that the statement of the eigenvalues being the reciprocal squares of the semi-axes is only true when the matrix A is real (or maybe when the eigenvectors of the matrix are real). Hopefully someone here can check this requirement. I can show a simple example proving that some condition is missing in the statement. If you start with a complex positive definite matrix A, the expression
is identical to the alternative expression
because the imaginary components cancel out on the left hand side, when doing the matrix multiplication, if the matrix A is Hermitian. Therefore, both matrices A and B = Re[A] are positive-definite matrices, and they both describe the SAME ellipsoid, because the equation is identical when matrix-vector multiplication is carried out. However, in general, the two matrices have different eigenvalues! So the statement that its eigenvalues are equal to the inverse squares of the semi-axes of the ellipsoid cannot be simultaneously a correct statement for both matrices. By calculating an explicit example, I found that when I take Re[A], then its eigenvalues DO coincide with the inverse squares of the semi-axes, but not when I keep A complex. Long story sort: some condition is needed before stating that the eigenvalues are the inverse squared semi-axes. Probably requirement of real eigenvectors or similar. El pak ( talk) 16:55, 1 July 2021 (UTC)
In Kreyszig, Advanced Engineering Mathematics, 4th ed, on p. 431 there is a parametric representation of a sphere and one is given as part of a problem for an ellipsoid but there is no interpretation of it in terms of the spheroid. In the parameterization θ is not an angle between the equator and a point on the ellipsoid but rather it is a parameter similar to Kepler's eccentric anomaly. ~~~~ Jbergquist ( talk) 23:26, 19 June 2023 (UTC)
In recent edits, there seems to be some confusion about how the semi-axes relate to the eigenvalues of the quadric. Suppose that the ellipsoid is represented by a symmetric 3x3 matrix E, in that the ellipsoid is the set of points (column vectors) x such that xT E x = 1. Then the eigenvalues of E are a-2, b-2, c-2, where a, b, c are the semi-axis lengths. Mgnbar ( talk) 23:09, 24 July 2024 (UTC)