Difference between revisions of "Ellipsoid in three-dimensional Euclidean space"

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(Implicit and parametric descriptions)
(Implicit and parametric descriptions)
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| Up to rotations || <math>\frac{(x - x_0)^2}{a^2} + \frac{(y - y_0)^2}{b^2} + \frac{(z - z_0)^2}{c^2} = 1</math> || || || || This version need not be centered along the origin but is oriented parallel to the axes.
 
| Up to rotations || <math>\frac{(x - x_0)^2}{a^2} + \frac{(y - y_0)^2}{b^2} + \frac{(z - z_0)^2}{c^2} = 1</math> || || || || This version need not be centered along the origin but is oriented parallel to the axes.
 
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| Up to rigid motions (rotations, translations, reflections) || <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1</math> || || || || This version is centered at the origin and oriented parallel to the axes.
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| Up to rigid motions (rotations, translations, reflections) || <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1</math> || <math>a</math>, <math>b</math>, <math>c</math> are positive numbers that represent the lengths of the three ellipsoidal semi-axes. Note that because we allow rigid motions, we can permute and rearrange to force <math>a \ge b \ge c</math>. || <math>x = a\cos u \sin v, b = y \sin u \sin v, c = z \cos v</math> || Here, <math>u</math> and <math>v</math> are ellipsoidal equivalents of the angles <math>\theta, \phi</math> used in [[spherical polar coordinates]]. || This version is centered at the origin and oriented parallel to the axes.
 
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| Up to similarity transformations || <math>x^2 + \frac{y^2}{q^2} + \frac{z^2}{r^2} = 1</math> || Here, <math>q = b/a, r = c/a</math> || || || For this version, we can normalize one of the three values <math>a,b,c</math>. In the form given here, we chose to normalize <math>a</math>.
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| Up to similarity transformations || <math>x^2 + \frac{y^2}{q^2} + \frac{z^2}{r^2} = 1</math> || Here, <math>q = b/a, r = c/a</math>. If we chose <math>a \ge b \ge c</math>, we would get <math>0 < q,r \le 1</math>. || || || For this version, we can normalize one of the three values <math>a,b,c</math>. In the form given here, we chose to normalize <math>a</math>.
 
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Revision as of 01:42, 6 August 2011

Definition

This surface type is not unique up to isometry or even up to similarity transformations, but rather, depends upon multiple nonzero parameters a,b,c. If we're considering the surface up to rigid isometries, the parameters are unique up to permutations. If we're considering the surface up to similarity transformations, the parameters are unique up to the action of permutations and projective equivalence.

Implicit and parametric descriptions

Degree of generality Implicit description What the parameters mean Parametric description What the additional parameters mean Comment
Arbitrary Fill this in later This version need not be centered at the origin and need not be oriented parallel to the axes.
Up to rotations \frac{(x - x_0)^2}{a^2} + \frac{(y - y_0)^2}{b^2} + \frac{(z - z_0)^2}{c^2} = 1 This version need not be centered along the origin but is oriented parallel to the axes.
Up to rigid motions (rotations, translations, reflections) \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 a, b, c are positive numbers that represent the lengths of the three ellipsoidal semi-axes. Note that because we allow rigid motions, we can permute and rearrange to force a \ge b \ge c. x = a\cos u \sin v, b = y \sin u \sin v, c = z \cos v Here, u and v are ellipsoidal equivalents of the angles \theta, \phi used in spherical polar coordinates. This version is centered at the origin and oriented parallel to the axes.
Up to similarity transformations x^2 + \frac{y^2}{q^2} + \frac{z^2}{r^2} = 1 Here, q = b/a, r = c/a. If we chose a \ge b \ge c, we would get 0 < q,r \le 1. For this version, we can normalize one of the three values a,b,c. In the form given here, we chose to normalize a.