Difference between revisions of "Ellipsoid in three-dimensional Euclidean space"
From Diffgeom
(→Implicit and parametric descriptions) |
|||
Line 10: | Line 10: | ||
| Arbitrary || {{fillin}} || || || || This version need not be centered at the origin and need not be oriented parallel to the axes. | | Arbitrary || {{fillin}} || || || || This version need not be centered at the origin and need not be 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. | + | | 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> || || <math>x = x_0 + a\cos u \sin v, y = y_0 + b \sin u \sin v, z = z_0 + c \cos v</math>|| || This version need not be centered along the origin but is oriented parallel to the axes. |
|- | |- | ||
− | | 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 | + | | 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, y = b \sin u \sin v, z = c \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. |
|- | |- | ||
| 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>. | | 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>. | ||
|} | |} | ||
+ | |||
+ | ==Fundamental forms and curvatures== | ||
+ | |||
+ | For the table below, we consider the parametric description that is valid for | ||
+ | |||
+ | <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1</math> | ||
+ | |||
+ | namely the description: | ||
+ | |||
+ | <math>x = a\cos u \sin v, b = y \sin u \sin v, c = z \cos v</math> | ||
+ | |||
+ | {| class="sortable" border="1" | ||
+ | ! Quantity !! Meaning in general !! Value | ||
+ | |- | ||
+ | | <math>E,F,G</math> for [[first fundamental form]] using this parametrization || The Riemannian metric is given by <math>ds^2 = E \, du^2 + 2F \, du \, dv + G \, dv^2</math> || {{fillin}} | ||
+ | |- | ||
+ | | <math>e,f,g</math> for [[second fundamental form]] using this parametrization || {{fillin}} || {{fillin}} | ||
+ | |- | ||
+ | | [[principal curvature]]s using this parametrization || eigenvalues of the shape operator, which is the matrix <math>\begin{pmatrix} E & F \\ F & G \\\end{pmatrix}</math> || {{fillin}} | ||
+ | |- | ||
+ | | [[mean curvature]] || arithmetic mean of the principal curvatures || {{fillin}} | ||
+ | |- | ||
+ | | [[Gaussian curvature]] || product of the principal curvatures = determinant of shape operator = <math>EG - F^2</math> || {{fillin}} | ||
+ | |} | ||
+ | |||
+ | ==Verification of theorems== | ||
+ | |||
+ | ===Gauss-Bonnet theorem=== | ||
+ | |||
+ | {{fillin}} |
Revision as of 01:53, 6 August 2011
Contents
Definition
This surface type is not unique up to isometry or even up to similarity transformations, but rather, depends upon multiple nonzero parameters . 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 | This version need not be centered along the origin but is oriented parallel to the axes. | ||||
Up to rigid motions (rotations, translations, reflections) | , , 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 . | Here, and are ellipsoidal equivalents of the angles used in spherical polar coordinates. | This version is centered at the origin and oriented parallel to the axes. | ||
Up to similarity transformations | Here, . If we chose , we would get . | For this version, we can normalize one of the three values . In the form given here, we chose to normalize . |
Fundamental forms and curvatures
For the table below, we consider the parametric description that is valid for
namely the description:
Quantity | Meaning in general | Value |
---|---|---|
for first fundamental form using this parametrization | The Riemannian metric is given by | Fill this in later |
for second fundamental form using this parametrization | Fill this in later | Fill this in later |
principal curvatures using this parametrization | eigenvalues of the shape operator, which is the matrix | Fill this in later |
mean curvature | arithmetic mean of the principal curvatures | Fill this in later |
Gaussian curvature | product of the principal curvatures = determinant of shape operator = | Fill this in later |
Verification of theorems
Gauss-Bonnet theorem
Fill this in later