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The normal curvature, k n, is the curvature of the curve projected onto the plane containing the curve's tangent T and the surface normal u; the geodesic curvature, k g, is the curvature of the curve projected onto the surface's tangent plane; and the geodesic torsion (or relative torsion), τ r, measures the rate of change of the surface ...
Curvature form in a vector bundle ... of 1-forms and the above formula becomes the structure equation of E ... 2.5 Curvature form and structure equation, p ...
Note that this transformation formula is for the mean curvature vector, and the formula for the mean curvature in the hypersurface case is ~ = ( , ) where ...
A space curve; the vectors T, N, B; and the osculating plane spanned by T and N. In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space, or the geometric properties of the curve itself irrespective of any motion.
Radius of curvature and center of 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 ...
Furthermore, a surface which evolves under the mean curvature of the surface , is said to obey a heat-type equation called the mean curvature flow equation. The sphere is the only embedded surface of constant positive mean curvature without boundary or singularities. However, the result is not true when the condition "embedded surface" is ...
Through each tangent vector to M at p, there passes a normal plane P X which cuts out a curve in M. That curve has a certain curvature κ X when regarded as a curve inside P X. Provided not all κ X are equal, there is some unit vector X 1 for which k 1 = κ X 1 is as large as possible, and another unit vector X 2 for which k 2 = κ X 2 is as ...
Animation of the torsion and the corresponding rotation of the binormal vector. Let r be a space curve parametrized by arc length s and with the unit tangent vector T.If the curvature κ of r at a certain point is not zero then the principal normal vector and the binormal vector at that point are the unit vectors