La primera forma fundamental I es un tensor 2-covariante, simétrico y definido sobre el espacio tangente a cada punto de la superficie S. Esta primera forma fundamental de hecho es el tensor métrico inducido por la métrica euclídea sobre la superficie. De hecho (S, I) constituye una variedad de Riemann con tensor métrico I.

Gracias a la primera forma fundamental podemos estimar longitudes de curvas definidas sobre la superficie, ángulos de intersección entre curvas y el resto de conceptos métricos habituales. Por razones históricas las componentes de la primera forma fundamental se designan por E, F y G:.

Además la forma cuadrática anterior es definida positiva, lo que implica que EG-F > 0. La primera forma anterior puede escribirse como una combinación lineal de productos tensoriales de las 1-formas coordenadas conforme a:.

Estas pueden calcularse explícitamente a partir de la parametrización:.

Length of a curve

Given a curve C completely contained in a surface S, its parametric equations can be expressed by:.

The length of this curve can be expressed by an integral of the derivatives of the functions u and v and the components of the first fundamental form:.

Angle between two curves

Similarly, given two curves C and C that intersect at a point P and whose parametric equations are:.

The angle α formed by the two curves at the point of intersection is defined by the equation:.

Where the derivatives are evaluated for the parameter values ​​and such that .

In particular, the angle formed by the coordinate lines associated with the coordinate system (u, v) is given by:.

In particular the coordinate system is called orthogonal if the coordinate lines are orthogonal (perpendicular) to each other at each point, that happens if and only if F = 0.

Area of ​​a region on the surface

Given a region Ω contained in a surface, its area is defined as:

If the surface is given by the explicit function z = f(x, y) then the above can be written simply as:.

The second fundamental form II of a surface is the projection onto the vector normal to the surface of the covariant derivative induced by the metric tensor or first fundamental form. It can be proven that this second fundamental form turns out to be a 2-covariant and symmetric tensor (that is, it gives rise to a bilinear form defined on the tangent space to the surface). For historical reasons the components of the second fundamental form are designated by L, M and N:.

Given a surface environment parameterized by the variables, the second fundamental form is also written, resulting in a rank two tensor, like the following linear combination:

of tensor products of the 1-coordinate forms. The components of the second fundamental form can be calculated explicitly from the parametric coordinates:

Cuando se tiene una curva sobre una superficie esta puede ser vista también como curva de a la que les son aplicables tanto las fórmulas de la geometría diferencial de curvas como las de la geometría diferencial de superficies. Eso permite relacionar la curvatura total de la curva con la curvatura de la curva vista o medida por un "habitante" de la superficie.

En concreto la curvatura total (χ) de una curva γ(t) puede ser descompuesta entre una componente tangencial a la superficie (y medible dentro de la misma), llamada curvatura geodésica (k), y una componente perpendicular a la superficie (que depende de cómo está curvada la superficie en el espacio y cuál es la dirección de la curva dentro de la superficie), llamada curvatura normal (k). De hecho se cumple que:.

Donde la curvatura geodésica y normal pueden calcularse a partir del ángulo que forman el vector normal a la superficie y el vector normal a la curva (n):.

La aceleración de cualquier punto material puede ser descompuesta en aceleración tangencial y aceleración normal. Si además el punto se mueve sobre la superficie, la aceleración normal puede descomponerse en aceleración propiamente normal y aceleración geodésica (debida al seguimiento que el punto hace de la superficie):.

Donde son respectivamente el vector tangente a la curva, el vector normal a la curva y el vector normal a la superficie. Esa ecuación muestra que las líneas geodésicas a la superficie son precisamente aquellas curvas para las cuales su curvatura total coincide con su curvatura normal.

Las curvaturas normal y geodésica de una curva sobre una superficie pueden calcularse fácilmente a partir de los vectores tangente a la curva y las normales a la curva y la superficie:.

Main curvatures

If we consider a point P on the surface and a whole collection of curves contained in the surface that pass through P, we observe that the normal curvature k of any of these curves in P varies between two extreme values ​​k < k < k. These two values ​​are in fact the k solutions of the following equation:.

A point is called umbilical if in it k = k. For a non-umbilical point P the directions tangent to the surface for which the maximum and minimum of the normal curvature are reached are always orthogonal. At each point these two orthogonal directions are called principal directions of curvature. A necessary and sufficient condition for the direction given by a vector to be principal is that if:.

So that this direction is the main direction of curvature implies that:

Curvature lines

A line of curvature is a connected regular curve contained in a regular surface in which all its tangent vectors generate a principal direction on the surface.

The Gaussian curvature of a surface is a real number (P) that measures the intrinsic curvature at each regular point P of a surface. This curvature can be calculated from the determinants of the first and second fundamental shapes of the surface:.

This Gaussian curvature generally varies from one point to another on the surface and is related to the principal curvatures of each point (k and k), through the relationship K = k**k.

An interesting case of a surface is the sphere, which has the same curvature at all its points.

Calculating the Gaussian curvature of a sphere (2-sphere). From the previous formula it is easily arrived at that for a sphere of radius r, the Gaussian curvature is the same at all points and equal to .

While we note that there are surfaces that have constant curvature, Gaussian curvature must be viewed as a relationship

where (a differentiable function over S) that assigns each surface its Gaussian curvature function.

The actual way to define Gaussian curvature is using the shape operator") of the surface S:.

Where are the coordinate tangent vectors and they are being evaluated at position p.

With the derivative (Jacobian) of the form operator.

one obtains a self-adjoint linear transformation - called Weingarten transformation - and thus, the Gaussian curvature is a determinant of L, i.e.

It is relatively easy to verify that it matches the definition given above.

In terms of the components of the Riemann curvature tensor for differentiable 2-manifolds, one finds the relation.

Example, the Gaussian curvature of a torus "Torus (geometry)") is where the parameterization has been used:.

• - Differential geometry of curves.

Literature

• - Girbau, J.: "Geometria differential i relativitat", Ed. Universitat Autònoma de Catalunya, 1993. ISBN 84-7929-776-X.

• - Spiegel, M. & Abellanas, L.: "Formulas and tables of applied mathematics", Ed. McGraw-Hill, 1988. ISBN 84-7615-197-7.

• - M. do Carmo: "Differential geometry of curves and surfaces".

External links

• - Portal:Mathematics. Content related to Mathematics.

• - Springer-Verlag online encyclopedia [1].

La primera forma fundamental I es un tensor 2-covariante, simétrico y definido sobre el espacio tangente a cada punto de la superficie S. Esta primera forma fundamental de hecho es el tensor métrico inducido por la métrica euclídea sobre la superficie. De hecho (S, I) constituye una variedad de Riemann con tensor métrico I.

Gracias a la primera forma fundamental podemos estimar longitudes de curvas definidas sobre la superficie, ángulos de intersección entre curvas y el resto de conceptos métricos habituales. Por razones históricas las componentes de la primera forma fundamental se designan por E, F y G:.

Además la forma cuadrática anterior es definida positiva, lo que implica que EG-F > 0. La primera forma anterior puede escribirse como una combinación lineal de productos tensoriales de las 1-formas coordenadas conforme a:.

Estas pueden calcularse explícitamente a partir de la parametrización:.

Length of a curve

Given a curve C completely contained in a surface S, its parametric equations can be expressed by:.

The length of this curve can be expressed by an integral of the derivatives of the functions u and v and the components of the first fundamental form:.

Angle between two curves

Similarly, given two curves C and C that intersect at a point P and whose parametric equations are:.

The angle α formed by the two curves at the point of intersection is defined by the equation:.

Where the derivatives are evaluated for the parameter values ​​and such that .

In particular, the angle formed by the coordinate lines associated with the coordinate system (u, v) is given by:.

In particular the coordinate system is called orthogonal if the coordinate lines are orthogonal (perpendicular) to each other at each point, that happens if and only if F = 0.

Area of ​​a region on the surface

Given a region Ω contained in a surface, its area is defined as:

If the surface is given by the explicit function z = f(x, y) then the above can be written simply as:.

The second fundamental form II of a surface is the projection onto the vector normal to the surface of the covariant derivative induced by the metric tensor or first fundamental form. It can be proven that this second fundamental form turns out to be a 2-covariant and symmetric tensor (that is, it gives rise to a bilinear form defined on the tangent space to the surface). For historical reasons the components of the second fundamental form are designated by L, M and N:.

Given a surface environment parameterized by the variables, the second fundamental form is also written, resulting in a rank two tensor, like the following linear combination:

of tensor products of the 1-coordinate forms. The components of the second fundamental form can be calculated explicitly from the parametric coordinates:

Cuando se tiene una curva sobre una superficie esta puede ser vista también como curva de a la que les son aplicables tanto las fórmulas de la geometría diferencial de curvas como las de la geometría diferencial de superficies. Eso permite relacionar la curvatura total de la curva con la curvatura de la curva vista o medida por un "habitante" de la superficie.

En concreto la curvatura total (χ) de una curva γ(t) puede ser descompuesta entre una componente tangencial a la superficie (y medible dentro de la misma), llamada curvatura geodésica (k), y una componente perpendicular a la superficie (que depende de cómo está curvada la superficie en el espacio y cuál es la dirección de la curva dentro de la superficie), llamada curvatura normal (k). De hecho se cumple que:.

Donde la curvatura geodésica y normal pueden calcularse a partir del ángulo que forman el vector normal a la superficie y el vector normal a la curva (n):.

La aceleración de cualquier punto material puede ser descompuesta en aceleración tangencial y aceleración normal. Si además el punto se mueve sobre la superficie, la aceleración normal puede descomponerse en aceleración propiamente normal y aceleración geodésica (debida al seguimiento que el punto hace de la superficie):.

Donde son respectivamente el vector tangente a la curva, el vector normal a la curva y el vector normal a la superficie. Esa ecuación muestra que las líneas geodésicas a la superficie son precisamente aquellas curvas para las cuales su curvatura total coincide con su curvatura normal.

Las curvaturas normal y geodésica de una curva sobre una superficie pueden calcularse fácilmente a partir de los vectores tangente a la curva y las normales a la curva y la superficie:.

Main curvatures

If we consider a point P on the surface and a whole collection of curves contained in the surface that pass through P, we observe that the normal curvature k of any of these curves in P varies between two extreme values ​​k < k < k. These two values ​​are in fact the k solutions of the following equation:.

A point is called umbilical if in it k = k. For a non-umbilical point P the directions tangent to the surface for which the maximum and minimum of the normal curvature are reached are always orthogonal. At each point these two orthogonal directions are called principal directions of curvature. A necessary and sufficient condition for the direction given by a vector to be principal is that if:.

So that this direction is the main direction of curvature implies that:

Curvature lines

A line of curvature is a connected regular curve contained in a regular surface in which all its tangent vectors generate a principal direction on the surface.

The Gaussian curvature of a surface is a real number (P) that measures the intrinsic curvature at each regular point P of a surface. This curvature can be calculated from the determinants of the first and second fundamental shapes of the surface:.

This Gaussian curvature generally varies from one point to another on the surface and is related to the principal curvatures of each point (k and k), through the relationship K = k**k.

An interesting case of a surface is the sphere, which has the same curvature at all its points.

Calculating the Gaussian curvature of a sphere (2-sphere). From the previous formula it is easily arrived at that for a sphere of radius r, the Gaussian curvature is the same at all points and equal to .

While we note that there are surfaces that have constant curvature, Gaussian curvature must be viewed as a relationship

where (a differentiable function over S) that assigns each surface its Gaussian curvature function.

The actual way to define Gaussian curvature is using the shape operator") of the surface S:.

Where are the coordinate tangent vectors and they are being evaluated at position p.

With the derivative (Jacobian) of the form operator.

one obtains a self-adjoint linear transformation - called Weingarten transformation - and thus, the Gaussian curvature is a determinant of L, i.e.

It is relatively easy to verify that it matches the definition given above.

In terms of the components of the Riemann curvature tensor for differentiable 2-manifolds, one finds the relation.

Example, the Gaussian curvature of a torus "Torus (geometry)") is where the parameterization has been used:.

• - Differential geometry of curves.

Literature

• - Girbau, J.: "Geometria differential i relativitat", Ed. Universitat Autònoma de Catalunya, 1993. ISBN 84-7929-776-X.

• - Spiegel, M. & Abellanas, L.: "Formulas and tables of applied mathematics", Ed. McGraw-Hill, 1988. ISBN 84-7615-197-7.

• - M. do Carmo: "Differential geometry of curves and surfaces".

External links

• - Portal:Mathematics. Content related to Mathematics.

• - Springer-Verlag online encyclopedia [1].