What are the points in spherical coordinates

Spherical coordinates

In this article we look at the Spherical coordinates and their transformation With Cartesian coordinates more accurate. This also includes the transformations of the Differentials, of Surfaces-, volume- and Line element as well as the transformation of the Basis vectors, of Nabla- and des Laplace operator.

The most important thing on the subject Spherical coordinates we also have a short one Video prepared for you. This promotes the imagination and can bring you closer to the topic more easily.

  • Spherical coordinates definition
    in the text
  • Convert spherical coordinates
    in the text
  • Transformation of differentials
    in the text
  • Transformation of basis vectors and vector operators
    in the text
  • in the text

Spherical coordinates definition

In Spherical coordinates becomes a Point in space through his Distance from the origin of coordinates, and two angles specified. These two angles are defined differently depending on the convention.

If the distance from the origin is kept constant, one speaks of spherical coordinates. The points under consideration then lie on a spherical surface.

Spherical coordinate system

To be able to describe a point in space is a Coordinate system of necessity. In Spherical coordinates is this Coordinate system determined by the following points:

  • the center or the origin of the coordinate system.
  • a directed straight line through the origin. This is called Polar axis and their direction is called Polar direction designated. The polar axis also gives the so-called Equatorial plane in front. This level lies orthogonal to the pole direction and runs through the origin.
  • a Reference direction, i.e. one Half straight in the equatorial plane.

Around Conversions with the Cartesian coordinates To simplify, the points mentioned can be defined in such a way that the origin of the spherical coordinate system corresponds to that of the Cartesian system. Furthermore, the -Axis as polar axis used so that the -Level the Equatorial plane corresponds to. The positive -Axis can also be used as Reference direction to get voted.

Spherical coordinate representation

As mentioned earlier, a period in Spherical coordinates among other things by two angles specified. How these are chosen, however, differs depending on the Convention. In the following, the convention should be followed which is specified in the mathematics and the physics is common. A Point in the room is then through the following three coordinates given:

A point in space is thus determined by specifying these three coordinates in Spherical coordinates described.

Convert spherical coordinates

To simplify invoices are common Conversions required between the Cartesian and the spherical coordinate system. These are to be shown in both directions below.

Convert spherical coordinates to Cartesian coordinates

Do you want that Spherical coordinatesin Cartesian coordinates convert, the following formulas are obtained through geometrical considerations:

Convert Cartesian coordinates to spherical coordinates

Will the Azimuth angle between and ( and ) specified, the Spherical coordinates can be calculated as follows:

Other conventions

In the described convention, corresponds to Polar angle not the latitude, which describes the angle between the position vector and the equatorial plane. In contrast to the polar angle, according to the above convention, it takes values ​​between and