staymathematicsMedium,Polar coordinate systemIs atwo-dimensionalCoordinate system。Any position in the coordinate system can be determined byincluded angleOpposite to a sectionorigin- polardistanceTo represent.Polar coordinate system has a wide range of applications, includingmathematics、Physics、engineering、navigation、aviationas well asrobotDomain.Polar coordinate system is particularly useful when the relationship between two points is easily expressed by the included angle and distance;And inRectangular coordinate system This relationship can only be usedtrigonometric functionTo represent.For many types of curves, the polar equation is the simplest form of expression. Even for some curves, only the polar equation can be expressed.
Chinese name
Polar equation
Foreign name
Polar equation of a curve
Applicable fields
mathematics
Applied discipline
mathematics
Definition
The reference system introduced by the position of a point in space
as everyone knows,GreekThe concept of angle and radian was first used.astronomerHipachas(190-120 BC) made a table to calculate the chord length function of the chord corresponding to each corner.Moreover, someone once quoted his polar coordinate system to determine the position of stars.In terms of spiral,ArchimedesDescribed his famous spiral, an equation of radius changing with angle.The Greeks made a contribution, although the entire coordinate system was not established in the end.
There are many opinions about who first applied the polar coordinate system as a formal coordinate system.A more detailed history of this issue,Harvard UniversityprofessorJulian Coolidge The origin of polar coordinate system.Greiger de Saint VincentandBonaventura Cavalieri, is considered to have introduced the concept of polar coordinate system almost simultaneously and independently.Saint Vincent discussed it in a private manuscript in 1625 and published it in 1647, while Cavalieri published it in 1635 and then corrected it in 1653.Kavalieri first used polar coordinate system to solve a problem aboutArchimedes spiralArea problem inside.Blaise Pascal Then use polar coordinate system to calculateparabolaLength of.
In the book Flow Numerology and Infinite Series, written in 1671 and published in 1736,Isaac NewtonThe first applies the polar coordinate system to any point on the representation plane.Newton verified the transformation relationship between polar coordinates and nine other coordinate systems in his book.In the book Learned Bulletin published in 1691Jacob Bernoulli The fixed point and a ray from the fixed point are officially used. The fixed point is called the pole, and the ray is called the polar axis.The coordinates of any point in the plane are expressed by the distance between the point and the fixed point and the angle between the point and the polar axis.Bernoulli studied the radius of curvature of the curve through the polar coordinate system.
In fact, the term "polar coordinates" is applied byGregorio FontanaIt was first used by Italian mathematicians in the 18th century.The term is defined byGeorge Peacock Translated in 1816Sives Lacroix"Differential Calculus and Integral Calculus" was translated into English.
As with all2D coordinate system, polar coordinate system also has two coordinate axes:
(radius coordinates) and
(Angular coordinates, polar angles orazimuth, sometimes expressed as
or
。
The coordinates represent the distance from the pole,
The coordinate indicates that the coordinate distance is 0 ° in the counterclockwise directionradial(sometimes referred to as polar axis)Rectangular coordinate systemThe positive x-axis direction in the.
For example, (3, 60 °) in polar coordinates represents a point with a distance of 3 units from the pole and an angle of 60 ° with the polar axis.(− 3, 240 °) and (3, 60 °) represent the same point, because the radius of the point is 3 units of length from the pole on the reverse extension line of the included angle ray (240 ° − 180 °=60 °).
An important characteristic of the polar coordinate system is that any point in the plane rectangular coordinate can have unlimited expressions in the polar coordinate system.Generally, the point (r, θ) can be arbitrarily expressed as（r, θ ±n× 360 °) or (−r, θ ± (2n+1) 180 °), herenIs any integer.If therIf the coordinate is 0, no matter what value θ takes, the position of this point will fall on the pole.
Use radian units
Angle in polar coordinate system is usually expressed as angle orradian, using the formula 2 π * rad=360 °.The specific way to use is basically determined by the use occasion.navigationAspects often use angles to measure, andphysicsThe ratio of radius and circumference is widely used in some fields of the, so the physical aspect prefers to use radians.
Transformation between polar coordinate system and plane rectangular coordinate system
It can also be transformed into polar coordinates:
This equation gives
In the range
The radian of.Use angle units instead, and the range is
。These equations assume that the pole is the origin of a rectangular coordinate system
The polar axis is the x-axis, while the radian in the y-axis direction is
, angle is
。
Most common programming languages will set a function specifically from
and
Coordinate Calculate the correct angular coordinate.For example, inC languageThe function is marked as atan2 (y, x) in theCommon LispIn, it is marked as (atan y x).For these two cases, the calculation results are in the range
When both are equal to zero, the spoke angle has no defined value;For this case, the spoke angle is set to zero for convenience.
If necessary, set the angular coordinates
In the range
The value of plus
, you can get
Value of.[3]
Polar coordinate system equation
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Functions: described in polar coordinatescurveThe equation is called polar coordinate equation and is usually expressed asrIs the independent variable θfunction。
Symmetry: polar coordinate equations often show differentsymmetricForm, ifr(−θ) =r(θ) , then the curve is symmetric about the pole (0 °/180 °). If r (π − θ)=r (θ), then the curve is symmetric about the pole (90 °/270 °). If r (θ − α)=r (θ), then the curve is equivalent to from the polecounter-clockwise directionrotateα°。
circular
In polar coordinates, the center of the circle is（rzero, φ) with a radius ofaThe general equation of the circle of is:
Specific cases: such as equation:
It indicates that the radius with the pole as the center isaThe circle of.
Guidance:
Let the radius of the circle be
, the polar coordinate of the circle center is
, and transform to rectangular coordinates:
。Then the rectangular coordinate system equation of the point on the circle is:
Let the polar coordinates of the points on the circle be
, then
Therefore:
Simplify to:
straight line
Ray equation passing pole:
Where φ is the inclination angle of the ray.If m isRectangular coordinate systemRaySlope, then φ=arctanm。Any straight line that does not pass through the pole will be associated with a rayvertical。These are at point（rzeroThe line at, φ) is perpendicular to the ray θ=φ, and its equation is:
Rose line
Figure 2. A rose line with equation r (θ)=2sin4 θ
The rose line of polar coordinates is a very famous curve in mathematical curves, which looks like a petal. It can only be described by the polar coordinate equation, which is as follows:
IfkIs an integer whenkIf it is odd, then the curve will be k petals, whenkIf it is even, the curve will have 2k petals.IfkIs a non integer, which will generatediskShape, and the number of petals is also a non integer.Note: this equation cannot generate a multiple of 4 plus 2 (such as 2, 6, 10...) petals.variableaRepresents the length of rose petals.
Archimedes spiral
Archimedes spiralThe following equation is used in polar coordinates:
Change parametersaWill change the shape of the spiral,bControls the distance between spirals, which is usually constant.Archimedes spiral has two spirals, one with θ>0 and the other with θ<0.The two spirals are smoothly connected at the poles.Turn one of them 90 °/270 ° to get its image, which is another spiral.
amongeexpressEccentricity，pexpressDistance from focus to guide line。
Other curves
Since the coordinate system is based on a circle, polar coordinates are much simpler than rectangular coordinates (Cartesian form) in many equations related to curves.such asBernoulli double newcastle line，Clam thread, andCardioid。[3]
application
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Positioning and navigation
Polar coordinates are commonly usedNavigation, as the destination or direction of travel, can be taken as the distance and angle from the object under consideration.For example,aircraftNavigate using a slightly modified version of polar coordinates.This system is generally used to navigate any kind of system. In 0 ° ray, it is generally called heading 360, and the angle isClockwiseContinue, not counterclockwise, as in a mathematical system.Course 360 correspondingmagnetic north , while heading 90180 and 270 correspond to Cidong, South and West respectively.Therefore, an aircraft sails due east and upwards 5in the seaWill be heading 90（air traffic control(090).
modeling
The system with radial symmetry provides the natural setting of polar coordinate system, and the center point acts as the pole.A typical example of this usage is in the case of radial symmetrywellAtGroundwater flow equation。Polar coordinate systems are also suitable for systems with radial forces.These systems include complianceInverse square lawOfgravitational field, and systems with point sources, such asradioantenna。
Kepler's law of planetary motion
Polar coordinates provide an expression ingravitational fieldThe natural number method of Kepler's planetary motion law.Kepler's first lawIt is believed that the orbit of a planet orbiting a star forms aellipse, one focus of the ellipse iscentroidOn.The equation of conic part given above can be used to express this ellipse.Kepler's second law, i.eIsodomain lawIt is believed that the area drawn by the line connecting the planet and the star it surrounds at equal time intervals is equal, that is
