What is the Gaussian curvature of a sphere?

What is the Gaussian curvature of a sphere?

The Gaussian radius of curvature is the reciprocal of Κ. For example, a sphere of radius r has Gaussian curvature 1r2 everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus.

How do you find Gaussian curvature?

The Gaussian curvature of σ is K = κ1κ2, and its mean curvature is H = 1 2 (κ1 + κ2). To compute K and H, we use the first and second fundamental forms of the surface: Edu2 + 2F dudv + Gdv2 and Ldu2 + 2Mdudv + Ndv2.

What is the Gaussian curvature of a cylinder?

Normal curvatures for a plane surface are all zero, and thus the Gaussian curvature of a plane is zero. For a cylinder of radius r, the minimum normal curvature is zero (along the vertical straight lines), and the maximum is 1/r (along the horizontal circles). Thus, the Gaussian curvature of a cylinder is also zero.

What is the difference between Gaussian curvature and mean curvature?

The mean curvature is “linear” in the curvatures, while the Gaussian curvature is “quadratic”. So for weak curvatures (for example :protein-membrane, membrane ), the mean curvature is bigger. Because Gaussian curvature is suppressed by one extra 1/length.

What is the Gaussian curvature of the surface given by z f/x y )?

The Gauss curvature is given by K=κ1κ2 and the mean curvature is given by H=12(κ1+κ2).

What does Gauss great theorem say about the curvature of a surface?

Gauss’s remarkable theorem, the one which I like to imagine made him giggle with joy, is that an ant living on a surface can work out its curvature without ever having to step outside the surface, just by measuring distances and doing some math.

What is the curvature of a circle of radius R?

and thus the curvature of a circle of radius r is 1r provided that the positive direction on the circle is anticlockwise; otherwise it is −1r .

What are the conditions for Gaussian surface?

A gaussian surface must exist where the electric field is either parallel or perpendicular to the surface vector. This makes the cosines in all the dot products equal to simply zero or one. The electric field that passes through the parts of the gaussian surface where the flux is non-zero has a constant magnitude.

What are the three Gaussian surfaces?

Common Gaussian surfaces

  • Spherical surface.
  • Cylindrical surface.
  • Gaussian pillbox.

How do you find the radius of a sphere?

The radius is half the diameter, so use the formula r = D/2. This is identical to the method used for calculating the radius of a circle from its diameter. If you have a sphere with a diameter of 16 cm, find the radius by dividing 16/2 to get 8 cm. If the diameter is 42, then the radius is 21.

Is Gaussian surface a real surface?

A gaussian surface is around a symmetric charge distribution is any imaginary closed surface such that the intensity of electric field at all points on the surface is same.

When a surface has a constant zero Gaussian curvature, then it is a developable surface and the geometry of the surface is Euclidean geometry . When a surface has a constant positive Gaussian curvature, then it is a sphere and the geometry of the surface is spherical geometry .

What is the Gaussian curvature of a cylindrical tube?

For example, the Gaussian curvature of a cylindrical tube is zero, the same as for the “unrolled” tube (which is flat). On the other hand, since a sphere of radius R has constant positive curvature R−2 and a flat plane has constant curvature 0, these two surfaces are not isometric, even locally.

What is the Gaussian curvature of a hyperboloid?

The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus . Gaussian curvature is an intrinsic measure of curvature, depending only on distances that are measured on the surface, not on the way it is isometrically embedded in Euclidean space.

What is the Gaussian curvature of embedded smooth surface in R3?

The Gaussian curvature of an embedded smooth surface in R3 is invariant under the local isometries. For example, the Gaussian curvature of a cylindrical tube is zero, the same as for the “unrolled” tube (which is flat).