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.
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. 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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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. 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.
Foundations of Arithmetic Differential Geometry
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. 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.
Namely, two directions on a surface at a given point are said to be conjugate if the straight lines generated by these directions are conjugate with respect to the osculating paraboloid at this point.
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Orthogonal conjugate directions are called principal. At a given point on the surface there are, as a rule, two principal directions. Flat points and special elliptic points cf. Umbilical point , at which all directions are principal, form an exception to this rule. A curve on a surface for which the tangent direction is principal at all points is said to be a curvature line. At non-elliptic points of the surface there exist self-conjugate directions. They are known as asymptotic directions cf. Asymptotic direction. A curve on a surface for which the direction is asymptotic at all points is known as an asymptotic line.
The concept of the envelope of a family of surfaces is introduced similarly to that of the envelope of a family of curves in a plane. However, the family of surfaces may be a one-parameter or a two-parameter family. In the theory of surfaces, the envelope of a one-parameter family of planes is of special significance. An important role in the theory of surfaces is played by two differential quadratic forms, the fundamental forms of a surface. Let denote the vector of a point on a surface, and let denote the unit vector of the normal to the surface; the fundamental forms are then written as.
The coefficients of the first and the second fundamental form are usually denoted by and , respectively. The first fundamental form yields the distance on the surface between a point and an infinitesimally close point :. The length of a curve defined on the surface by the equations is computed with the aid of the first fundamental form:. The first fundamental form of the surface defines the angles between curves on the surface. In particular: the formula. It is seen, accordingly, that the coordinate net on the surface is orthogonal if.
Foundations of Differential Geometry - Shoshichi Kobayashi, Katsumi Nomizu - Google книги
The area of a piece of the surface is also given by the first fundamental form and, for a domain on the surface, is computed by the formula. Figure 9. The definition of regular surfaces. The function uniquely defines the geometric surface but not vice versa - a surface can have several representations, or even parametric representations which do not fulfill the above mentioned criteria.
Those representations, which fulfills the criteria mentioned in the definition, are called regular representations. A technical note: partial derivatives of a vector valued function is computed analogously to the one-parameter case, by separately differentiating the coordinate-functions with respect to the actual parameter. The topological mapping over the parameter domain can be performed in the simplest way by an orthogonal projection.
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This way we obtain a domain in the parameter plane. Consider a topological mapping of this domain to another domain in this plane. The relation between the surface and cannot be simply described by a projection any more of course, however this can also yield a regular representation of the surface. Regular surfaces form such a small subset of surfaces that well-known surfaces such as sphere or torus are not in this subset, because cannot be mapped onto a simply connected open subset of the plane.
These surfaces however, can be constructed by a union of a finite number of regular surfaces, e. Analogous solution can be found for other, non-regular surfaces as well. Thus we can define the notion of surface as union of finite number of regular surfaces and for any point of the surface has a sufficiently small neighborhood, which is a regular surface piece on the surface. A surface is connected if any two points of the surface can be connected by a regular curve on the surface.
Explicit representation. Consider a Cartesian coordinate-system in and an explicit function with two variables:. Those points the coordinates of which are form a surface. This representation is also called Euler-Monge-type form. Implicit representation. The role of connections respectively curvature attached to metrics is played by certain adelic respectively global objects attached to the corresponding matrices. One of the main conclusions of the theory is that the spectrum of the integers is "intrinsically curved"; the study of this curvature is then the main task of the theory.
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