How do you calculate the mean curvature?

How do you calculate the mean curvature?

Half of the sum of the principal curvatures (cf. Principal curvature) k1 and k2, calculated at a point A of this surface: H(A)=k1+k22.

What does zero mean curvature mean?

A surface is a minimal surface if and only if the mean curvature is zero.

How do you calculate normal curvature?

That is, if we slice the cylinder along the vector v( q) through a normal to the surface at a point P, then the curvature at P of the curve formed by the intersection is kn (q) = -cos2(q) .

What is the curvature of a sphere?

For the unit sphere, both principal curvatures are 1 and hence the Gauss curvature is 1. For a unit cylinder, the principal curvatures are 1 and 0 and hence the Gauss curvature is 0.

How do you calculate the curvature of a surface?

One way to examine how much a surface bends is to look at the curvature of curves on the surface. Let γ(t) = σ(u(t),v(t)) be a unit-speed curve in a surface patch σ. Thus, ˙γ is a unit tangent vector to σ, and it is perpendicular to the surface normal n at the same point.

What is the curvature of a function?

The curvature measures how fast a curve is changing direction at a given point. There are several formulas for determining the curvature for a curve. The formal definition of curvature is, κ=∥∥∥d→Tds∥∥∥ where →T is the unit tangent and s is the arc length.

What is the mean curvature of a minimal surface?

curvature zero
On a minimal surface, the curvature along the principal curvature planes are equal and opposite at every point. This makes the mean curvature zero.

What is the normal curvature?

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 curvature of a circle?

At every point on a circle, the curvature is the reciprocal of the radius; for other curves (and straight lines, which can be regarded as circles of infinite radius), the curvature is the reciprocal of the radius of the circle that most closely conforms to the curve at the given point (see figure).

What is the radius of a curvature?

In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or combinations thereof.

What does curvature mean in physics?

Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.

What do you mean by minimal surface?

In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).