Polar coordinate system

In mathematics and geodesy is meant by a polar coordinate system (also: circular coordinate system) a two-dimensional coordinate system in which each point is defined in a plane defined by the distance from a given fixed point and the angle from a fixed direction.

The fixed point is called the pole; it corresponds to the origin in a Cartesian coordinate system. The hazards arising from the pole in the fixed direction beam is called the polar axis. The distance from the pole is usually referred to with or radius or radial coordinate and is called, the angle is denoted by or and is called angular coordinate, polar angle, or azimuth.

Polar coordinates form a special case of orthogonal curvilinear coordinates. They are helpful when it is easier to describe by angles and distances of the ratio between two points than would be the case with x - and y- coordinates. In geodesy, polar coordinates are the most common method of calibration points (polar method). In the radio navigation, the principle is often called " Rho Theta " ( for distance and angle measurement) respectively.

In mathematics, the angular coordinate is measured in the mathematically positive sense ( counterclockwise). A Cartesian coordinate system is used at the same time, as is usually the point of origin as a pole, and the X -axis and the polar axis. The angular coordinate is thus measured from the x axis towards the y-axis. In geodesy and navigation, the azimuth is measured from the north direction from clockwise.

• 3.1 cylindrical coordinates 3.1.1 Conversion between cylindrical coordinates and Cartesian coordinates
• 3.1.2 basis vectors
• 3.1.3 Jacobian
• 3.1.4 Vector Analysis 3.1.4.1 gradient
• 3.1.4.2 divergence
• 3.1.4.3 rotation

• 4.1 Conversion to Cartesian coordinates
• 4.2 Jacobian

History

The terms angle and radius were already used by the ancients in the first millennium BC. The Greek astronomer Hipparchus ( 190-120 BCE) created a table of trigonometric functions tendons to find the length of the chord for each angle. Using this basis, it was possible for him to use the polar coordinates in order to use it to set the position of certain stars can. However, his work consisted of only a part of the coordinate system.

In his treatise On Spirals, Archimedes describes a spiral line with a function whose radius changes depending on its angle. However, the work of the Greeks consisted of more than a full coordinate system.

There are several features in order to define the polar coordinate system as part of a formal coordinate system. The entire history on this topic is summarized in the book Origin of Polar Coordinates ( origin of the polar coordinates) of the Harvard professor Julian Coolidge and explained .. Accordingly led Grégoire de Saint- Vincent and Bonaventura Cavalieri independently in this conception of the mid-17th century. Saint -Vincent wrote in 1625 on a private basis on this subject and published his work in 1647, while Cavalieri published his drafting in 1635, with a corrected version was published in 1653. Cavalieri used polar coordinates initially to solve a problem in terms of the area of ​​the Archimedean spiral. A little later, Blaise Pascal used polar coordinates to calculate the length of parabolic angles.

In the work Method of Fluxions ( Fluxionsmethode ) (written 1671, published 1736) considered Sir Isaac Newton, the transformation between polar coordinates, which he cites as the "Seventh Manner; For Spirals ", (seventh method; For spirals ) moved, and nine other coordinate systems.

This was followed by Jacob Bernoulli, who in the journal Acta Eruditorum ( 1691) used a system that consisted of a straight line and a point on this line, which he called the polar axis or pole. The coordinates were laid down therein by the distance from the pole and the angle to the polar axis. Bernoulli's work ranged up to the formulation of the circle of curvature of curves, which he expressed by the aforementioned coordinates.

The term polar coordinates in use today was finally introduced by Gregorio Fontana and used in Italian writings of the 18th century. In the following, George Peacock took over in 1816 this name in the English language when he translated the work of Sylvestre Lacroix differential and integral calculus ( differential and integral calculus ) in his language.

However, Alexis -Claude Clairaut was the first who thought about polar coordinates in three dimensions, their development, however, only the Swiss mathematician Leonhard Euler succeeded.

Plane polar coordinates: circular coordinates

Definition

The polar coordinates of a point in the Euclidean plane are in relation to a specified coordinate origin ( a point in the plane ) and a direction ( a ray starting at the origin ).

The polar coordinate system is thus clearly established that an excellent point, also called the pole, is the origin / origin of the coordinate system. Further, risks resulting from the beam is excellent as so-called polar axis. Ultimately, nor must one direction ( of two possible ), which is perpendicular to the polar axis of this, are defined to be positive to set the direction of rotation / rotation / orientation. Now can an angle is the polar angle, defined between any beam which passes through the pole, and this excellent polar axis. It is clear to the pole, regardless of which is selected as the square of it only to on integral rotations. On the polar axis itself still carried an arbitrary but fixed scaling to define the radial unit length. Now each pair can be assigned to a point in the plane unambiguously and giving you look at the first component as radial length and the second as the polar angle. Such a pair is referred to as number (not necessarily unique) Polar coordinates of a point in this plane.

The coordinate r is a length, a radius ( occasionally ) refers (in practice as a distance), and the coordinate of a (polar ) angle or, in practice also known as azimuth.

In mathematics, the angle is usually defined in a counterclockwise direction as positive if one looks vertically down onto the plane ( clock ). So goes the direction of rotation from east to north ( and further west ), with the north facing upwards. As angular doing the Radiant is the preferred unit of angle, because it is then analytically to handle most elegantly. The polar axis points in the graphs of the coordinate system typically to the east.

Conversion between polar coordinates and Cartesian coordinates

Conversion of polar coordinates into Cartesian coordinates

If you choose a Cartesian coordinate system with the same origin as the polar coordinate system, while the x - axis in the direction of the polar axis, and finally the positive y - axis in the direction of positive rotation sense - as in the above picture shown on the right - so results for the Cartesian coordinate of a point:

Conversion from Cartesian coordinates to polar coordinates

The conversion from Cartesian coordinates to polar coordinates is a little harder because you're mathematically it always relies on trigonometric one (not the full range of comprehensive round angle ) inverse function. First, however, the radius of the Pythagorean theorem as follows simply be calculated as:

In determining the angle of two special features of the polar coordinates need to be considered:

• For the angle is not uniquely determined, but could take on any real value. For a unique transformation rule he is often defined to be 0. The following formulas are given to simplify their derivation and representation provided.
• For the angle is only up to integer multiples of determined because the angle and ( for ) describe the same point. For the purpose of simple and unique transformation rule, the angle is limited to a half-open interval of length. Usually, this depending on the application or the intervals selected.

For the calculation of each of the equations

Be used. However, the angle is thus not uniquely determined, even in the interval, or because none of the three functions, and in these intervals is injective. The last equation is also not defined for. Therefore, a case distinction is needed, which depends on the quadrant in which the point is located, that is, from the signs of and.

Calculation of the angle in the interval ( - π, π ]

Using the arctangent can be determined in the interval, as follows:

Some programming languages ​​provide an arctangent function atan2 with two arguments on which considers the case distinctions represented internally and calculates the correct value for for any values ​​of and.

Also with the help of complex numbers can be explicitly represented by:

With the right side is calculated by.

Using the inverse cosine to get out with only two different cases:

By using the signum function, you can avoid an explicit case distinction in the formula:

By exploiting the fact that, in a circle, a center angle is always twice as large as the corresponding angle at circumference, the argument may also be calculated using the arc tangent function with less different cases:

Calculation of the angle in the interval [0, 2π )

The calculation of the angle in the interval may be carried out in principle in such a way that the angle is calculated first as described above in the interval and, only if it is negative, or is increased to:

By modification of the first formula above can be directly determined in the interval as follows:

The formula with the inverse cosine is also in this case with only two different cases of:

Jacobian

From the conversion formulas from polar coordinates into Cartesian coordinates obtained for the Jacobian as a determinant of the Jacobian matrix:

Surface element

The Jacobian is obtained for the surface element in polar coordinates:

Line element

From the above transformation equations

Follow

For the Cartesian line element

What follows in polar coordinates

Velocity and acceleration in polar coordinates

This is broken down into a radial movement and a perpendicular " transverse " component. For the velocity vector

With the unit vectors and.

For the acceleration

Spatial polar coordinates: cylindrical coordinates and spherical coordinates

Cylindrical coordinates

Cylindrical coordinates or cylindrical coordinates are plane polar coordinates, that are complemented by a third coordinate substantially. This third co-ordinate describing the height of a point vertically above (or below) the plane of the polar coordinate system and generally referred to by z. The coordinate ( referred to as R in the following figure ) no longer is the distance of a point from the origin, but the z-axis.

Conversion between cylindrical coordinates and Cartesian coordinates

If so aligns a Cartesian coordinate system, that the z- axes coincide, showing the x - axis in the direction and the angle from the x - axis to the y- axis increases ( right-wing ) then results in the following conversion formulas:

For the conversion from Cartesian coordinates to cylindrical coordinates are obtained for and the same formulas as in the polar coordinates.

Basis vectors

The basis vectors and are mutually orthonormal and form in this order, a legal system.

Jacobian

The addition of the rectilinear coordinate z has no effect on the Jacobian:

Consequently, the result for the volume element dV:

This also corresponds to the square root of the sum of the determinant of the metric tensor, with the aid of the coordinate transformation can be calculated (see Laplace -Beltrami operator).

Vector Analysis

The following representations of the nabla operator can be applied in the given form directly to scalar or vector-valued fields in cylindrical coordinates. The procedure is in this case analogous to vector analysis in Cartesian coordinates.

Gradient

The representation of the gradient transfers as follows from Cartesian to cylindrical coordinates:

Divergence

The divergence additional terms are added, arising from the derivatives of the of, and dependent unit vectors:

Rotation

Spherical coordinates

See main article: spherical coordinates

Spherical coordinates are plane polar coordinates that are complemented by a third coordinate substantially. This is done by specifying an angle of the third axis. The third coordinate is an angle between the vector to the point P and the Z- axis. is exactly zero when P is in the z- axis.

N-dimensional polar coordinates

It can also be a generalization of polar coordinates with an n- dimensional space with Cartesian coordinates specify. To do leads for each new dimension ( inductive construction via the same ) one another angle specifies the angle between the vector and the new, positive coordinate axis for. The same procedure can be constructed inductively from the number line in a consistent manner the angular coordinate of the 2- dimensional space, where the radial coordinate and negative values ​​, so therefore would be entirely approved.

Conversion to Cartesian coordinates

A conversion rule from these coordinates to Cartesian coordinates would then be:

As can be demonstrated, these polar coordinates go for the case n = 2 in the usual polar coordinates and for n = 3 in the spherical coordinates over.

Jacobian

The Jacobian of the transformation from polar coordinates into Cartesian coordinates is:

So is the n-dimensional volume element:

Note: n-dimensional cylindrical coordinates are simply a product / composition of k -dimensional spherical coordinates and (nk) -dimensional Cartesian coordinates with k ≥ 2 and nk ≥ 1