Search results
Results From The WOW.Com Content Network
Its center lies on the inner normal line, and its curvature defines the curvature of the given curve at that point. This circle, which is the one among all tangent circles at the given point that approaches the curve most tightly, was named circulus osculans (Latin for "kissing circle") by Leibniz .
Assume that this surface is regular, meaning that the vectors r u and r v are linearly independent. Complete this to a basis { r u , r v , n }, by selecting a unit vector n normal to the surface. It is possible to express the second partial derivatives of r (vectors of R 3 {\displaystyle \mathbb {R^{3}} } ) with the Christoffel symbols and the ...
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.
By analyzing the class of curves which lie on such a surface, and the degree to which the surfaces force them to curve in ℝ 3, one can associate to each point of the surface two numbers, called the principal curvatures. Their average is called the mean curvature of the surface, and their product is called the Gaussian curvature.
T is the unit tangent, P the unit normal, and B the unit binormal. A Frenet frame is a moving reference frame of n orthonormal vectors e i (t) which are used to describe a curve locally at each point γ(t). It is the main tool in the differential geometric treatment of curves because it is far easier and more natural to describe local ...
The normal section of a surface at a particular point is the curve produced by the intersection of that surface with a normal plane. [1] [2] [3] The curvature of the normal section is called the normal curvature. If the surface is bow or cylinder shaped, the maximum and the minimum of these curvatures are the principal curvatures.
From just the curvature and torsion, the vector fields for the tangent, normal, and binormal vectors can be derived using the Frenet–Serret formulas. Then, integration of the tangent field (done numerically, if not analytically) yields the curve.
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 ...