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Jul 12, The Architect marked it as to-read Shelves: m-diffgeo. Oscar marked it as to-read Oct 31, Eric Johnson added it Nov 25, Boradicus added it Dec 05, Ethylene Glycol marked it as to-read Jul 17, 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 Treatise on the Differential Geometry of Curves and Surfaces

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.

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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.

This is a circle with a contact of order with the curve Fig.

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It exists at each point of a twice differentiable curve with a non-zero curvature. 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. A curve is called the envelope of a family of curves , depending on a parameter , if at each of its points it is tangent to at least one curve of the family, and if each of its segments is tangent to an infinite set of these curves.

## Differential geometry - Encyclopedia of Mathematics

The theory of surfaces usually deals with differentiable surfaces. These are surfaces which can be locally defined by equations of the type. The degree of differentiability of the surface is given by the corresponding degree of differentiability of these functions. A surface has uncountably many ways in which it can be defined by parametric equations such as 3. A point of a surface is said to be regular ordinary if, in a neighbourhood of it, a suitable choice of the coordinates enables one to give the surface in the form.

In differential geometry the study of the surface is mainly conducted in a neighbourhood of the regular ordinary points. For a point of a surface given by the equations 3 to be regular, it is necessary and sufficient that the rank of the matrix. If a surface is given by equations of the type 3 , it is usually assumed, without stating this explicitly, that this condition is satisfied. If either or is fixed, the equations 3 define curves on the surface. Such curves are said to be coordinate lines on the surface.

The parameters and are called surface coordinates or curvilinear Gaussian coordinates. The concept of a tangent plane to a surface is defined in terms of the concept of contact. It is a plane passing through a point on the surface having contact of order with the surface at that point. A smooth differentiable surface has a unique tangent plane at each regular point. The tangent plane of a surface defined by equations 3 , under condition 4 at the point , is defined by the equation. The straight line which passes through a point of the surface and is perpendicular to the tangent plane at this point is called the normal to the surface.

If is a vector with coordinates , then.

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The important concept of an osculating paraboloid is introduced for surfaces. This is a paraboloid the axis of which is the normal to the surface at the given point and with contact of order to the surface at this point. A twice differentiable surface has a unique osculating paraboloid at each of its points, which may degenerate to a parabolic cylinder or a plane. If the surface is described in Cartesian coordinates, with the given point as the coordinate origin, while the tangent plane at this point is taken as the -plane, the equation of the surface in a neighbourhood of this point will be.

Depending on the type of the osculating paraboloid, the points of the surface are divided into elliptic points, hyperbolic points, parabolic points, and flat points cf. Elliptic point ; Flat point ; Hyperbolic point ; Parabolic point. The importance of the osculating paraboloid is due to the fact that it reproduces the form of the surface up to infinitesimals of the second order the tangent plane reproduces this form up to infinitesimals of the first order.

The osculating paraboloid is used to introduce the concept of conjugate directions on a surface.