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Items related to Lectures on the Differential Geometry of Curves and Lectures on the Differential Geometry of Curves and Surfaces. Cartan in relation to differential geometry. Cartan established the theory of spaces with projective connections and affine connections cf. Affine connection ; Projective connection.

In Russia a school of differential geometry was established by F. Minding and K. Their studies largely concern the problems of isometric deformation cf. Deformation, isometric of surfaces, i.

These studies were continued by many Russian and Soviet geometers. The major subject of the theory of curves are the so-called differentiable curves. These are curves which can locally be specified by equations of the type. The degree of differentiability of the curve is given by the degree of differentiability of and. For one and the same curve there are uncountably many ways of parametric description of the type 1. Of these, the so-called natural parametrization, in which the length of an arc of the curve, counted from some given point, serves as the parameter, is especially important.

A point on a curve is said to be regular if, by a suitable choice of a Cartesian coordinate system , the curve can be defined, in a neighbourhood of that point, by equations of the form. Otherwise it is called singular cf. Singular point.

Thank you for visiting nature. The definition of Gauss and Mean curvature. Understand curvature and torsion of space curves. The following is analogous to the result which says that the derivative of a turning angle of a. Chapter 3.

In differential geometry the study of a curve mainly concerns a neighbourhood of a regular point. For a point on a curve defined by the general equation 1 to be regular, it is necessary and sufficient that the inequality. A number of fundamental concepts in the theory of curves is based on the concept of contact of sets, which may be explained as follows. Let and be two sets with a common point. One says that the sets and have a contact of order at if. If is a curve while is a straight line passing through a point of the curve, then if , the contact condition defines to be the tangent to the curve at Fig.

A smooth differentiable curve has a definite tangent at each of its points. The direction of the tangent at a point of a curve specified by 1 coincides with the direction of the vector. In differential geometry the equations of the tangent are derived for the various ways in which the curve is analytically specified.

In particular, if the curve is defined by equations 1 , the equations of the tangent at the point corresponding to the value of the parameter are. If is a plane passing through a point of the curve , the contact condition for defines the osculating plane to the curve Fig. A twice differentiable curve has an osculating plane at each of its points. It is either unique or else any plane passing through the tangent to the curve is osculating. When moving along a curve, the tangent to the curve undergoes a rotation. The velocity of this rotation, as the motion along the curve takes place at velocity one, is called the curvature of the curve.

If the curve has been parametrized as in 1 , its curvature is given by the formula. Straight lines, and only straight lines, have a curvature which is everywhere equal to zero.

A twice differentiable curve has a unique osculating plane at all points at which the curvature is non-zero. As one moves along the curve in a neighbourhood of such a point, the osculating plane rotates, and the tangent to the curve is the instantaneous axis of this rotation. The velocity of rotation of the osculating plane, for a uniform motion along the curve at velocity one, is called the torsion of the curve. The sign of the torsion depends on the direction of the rotation.

A thrice differentiable curve has a definite torsion at each of its points where the curvature is non-zero. If the curve has been parametrized as in 1 , its torsion is given by the formula. The torsion of a planar curve is zero at all points. Conversely, a curve with torsion identically equal to zero is planar. A straight line perpendicular to the tangent and passing through the point of contact is known as a normal to the curve. The normal lying in the osculating plane is called the principal normal , while the normal which is perpendicular to the osculating plane is known as the binormal.

If the edges of this frame at the given point are considered as the axes of a Cartesian coordinate system, then the equation of the curve in a natural parametrization in a neighbourhood of this point will have the form. The unit vectors of the tangent, of the principal normal and of the binormal to a curve vary during a motion along the curve. A suitable choice of directions of these vectors, in accordance with the definitions of the curvature and the torsion, yields the formulas.

A curve with non-zero curvature is defined, up to its location in space, by specifying its curvature and its torsion as functions of the arc length of the curve. This is why the system of equations. Natural equation. Planar curves, i.

In the case of planar curves it is possible to distinguish between the directions of rotation of the tangent moving along the curve, so that the curvature can be given a sign depending on the direction of this rotation. The curvature of a planar curve defined by the equations is given by the formula. The plus or minus sign is chosen in accordance with a convention, but must be the same all along the curve.

The important concept of the osculating circle is introduced for planar curves.

https://togkapemesu.cf This is a circle with a contact of order with the curve Fig. It exists at each point of a twice differentiable curve with a non-zero curvature.

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The centre of the osculating circle is known as the centre of curvature, while its radius is known as the radius of curvature. The radius of curvature is the quantity inverse to the curvature. The locus of the centres of curvatures of a curve is known as the evolute. The curve which orthogonally intersects the tangents to the curve is called the evolvent cf. Evolvent of a plane curve Fig. The evolute of the evolvent of a curve is the curve itself.

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