White supremacy in the United States

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/White_supremacy_in_the_United_States 

History

White men pose for a photograph of the 1920 Duluth, Minnesota lynchings. Two of the black victims are still hanging while the third is on the ground. Lynchings were often public spectacles for the white community to celebrate white supremacy in the U.S., and photos were often sold as postcards.

Ku Klux Klan parade in Washington, D.C. in 1926

Early history

White supremacy was dominant in the United States both before and after the American Civil War, and it persisted for decades after the Reconstruction Era. The Virginia Slave Codes of 1705 socially segregated white colonists from black enslaved persons, making them disparate groups and hindering their ability to unite. Unity of the commoners was a perceived fear of the Virginia aristocracy, who wished to prevent repeated events such as Bacon's Rebellion, occurring 29 years prior. Prior to the Civil War, many wealthy white Americans owned slaves; they tried to justify their economic exploitation of black people by creating a "scientific" theory of white superiority and black inferiority. One such slave owner, future president Thomas Jefferson, wrote in 1785 that blacks were "inferior to the whites in the endowments of body and mind." In the antebellum South, four million slaves were denied freedom. The outbreak of the Civil War saw the desire to uphold white supremacy being cited as a cause for state secession and the formation of the Confederate States of America. In an 1890 editorial about Native Americans and the American Indian Wars, author L. Frank Baum wrote: "The Whites, by law of conquest, by justice of civilization, are masters of the American continent, and the best safety of the frontier settlements will be secured by the total annihilation of the few remaining Indians."

The Naturalization Act of 1790 limited U.S. citizenship to whites only. In some parts of the United States, many people who were considered non-white were disenfranchised, barred from government office, and prevented from holding most government jobs well into the second half of the 20th century. Professor Leland T. Saito of the University of Southern California writes: "Throughout the history of the United States, race has been used by whites for legitimizing and creating difference and social, economic and political exclusion."

20th century

The denial of social and political freedom to minorities continued into the mid-20th century, resulting in the civil rights movement. The movement was spurred by the lynching of Emmett Till, a 14-year-old boy. David Jackson writes it was the image of the "murdered child's ravaged body, that forced the world to reckon with the brutality of American racism." Vann R. Newkirk II wrote "the trial of his killers became a pageant illuminating the tyranny of white supremacy." Moved by the image of Till's body in the casket, one hundred days after his murder Rosa Parks refused to give up her seat on a bus to a white person.

Sociologist Stephen Klineberg has stated that U.S. immigration laws prior to 1965 clearly "declared that Northern Europeans are a superior subspecies of the white race". The Immigration and Nationality Act of 1965 opened entry to the U.S. to non-Germanic groups, and significantly altered the demographic mix in the U.S. as a result. With 38 U.S. states having banned interracial marriage through anti-miscegenation laws, the last 16 states had such laws in place until 1967 when they were invalidated by the Supreme Court of the United States' decision in Loving v. Virginia. These mid-century gains had a major impact on white Americans' political views; segregation and white racial superiority, which had been publicly endorsed in the 1940s, became minority views within the white community by the mid-1970s, and continued to decline in 1990s' polls to a single-digit percentage. For sociologist Howard Winant, these shifts marked the end of "monolithic white supremacy" in the United States.

After the mid-1960s, white supremacy remained an important ideology to the American far-right. According to Kathleen Belew, a historian of race and racism in the United States, white militancy shifted after the Vietnam War from supporting the existing racial order to a more radical position (self-described as "white power" or "white nationalism") committed to overthrowing the United States government and establishing a white homeland. Such anti-government militia organizations are one of three major strands of violent right-wing movements in the United States, with white-supremacist groups (such as the Ku Klux Klan, neo-Nazi organizations, and racist skinheads) and a religious fundamentalist movement (such as Christian Identity) being the other two. Howard Winant writes that, "On the far right the cornerstone of white identity is belief in an ineluctable, unalterable racialized difference between whites and nonwhites." In the view of philosopher Jason Stanley, white supremacy in the United States is an example of the fascist politics of hierarchy, in that it "demands and implies a perpetual hierarchy" in which whites dominate and control non-whites.

21st century

The presidential campaign of Donald Trump led to a surge of interest in white supremacy and white nationalism in the United States, bringing increased media attention and new members to their movement; his campaign enjoyed their widespread support.

Some academics argue that outcomes from the 2016 United States Presidential Election reflect ongoing challenges with white supremacy. Psychologist Janet Helms suggested that the normalizing behaviors of social institutions of education, government, and healthcare are organized around the "birthright of...the power to control society's resources and determine the rules for [those resources]". Educators, literary theorists, and other political experts have raised similar questions, connecting the scapegoating of disenfranchised populations to white superiority.

As of 2018, there were over 600 white-supremacist organizations recorded in the U.S. On July 23, 2019, Christopher A. Wray, the head of the FBI, said at a Senate Judiciary Committee hearing that the agency had made around 100 domestic terrorism arrests since October 1, 2018, and that the majority of them were connected in some way with white supremacy. Wray said that the Bureau was "aggressively pursuing [domestic terrorism] using both counterterrorism resources and criminal investigative resources and partnering closely with our state and local partners," but said that it was focused on the violence itself and not on its ideological basis. A similar number of arrests had been made for instances of international terrorism. In the past, Wray has said that white supremacy was a significant and "pervasive" threat to the U.S.

On September 20, 2019, the acting Secretary of Homeland Security, Kevin McAleenan, announced his department's revised strategy for counter-terrorism, which included a new emphasis on the dangers inherent in the white-supremacy movement. McAleenan called white supremacy one of the most "potent ideologies" behind domestic terrorism-related violent acts. In a speech at the Brookings Institution, McAleenan cited a series of high-profile shooting incidents, and said "In our modern age, the continued menace of racially based violent extremism, particularly white supremacist extremism, is an abhorrent affront to the nation, the struggle and unity of its diverse population." The new strategy will include better tracking and analysis of threats, sharing information with local officials, training local law enforcement on how to deal with shooting events, discouraging the hosting of hate sites online, and encouraging counter-messages.

In a 2020 article in The New York Times titled "How White Women Use Themselves as Instruments of Terror", columnist Charles M. Blow wrote:

We often like to make white supremacy a testosterone-fueled masculine expression, but it is just as likely to wear heels as a hood. Indeed, untold numbers of lynchings were executed because white women had claimed that a black man raped, assaulted, talked to or glanced at them. The Tulsa race massacre, the destruction of Black Wall Street, was spurred by an incident between a white female elevator operator and a black man. As the Oklahoma Historical Society points out, the most common explanation is that he stepped on her toe. As many as 300 people were killed because of it. The torture and murder of 14-year-old Emmett Till in 1955, a lynching actually, occurred because a white woman said that he "grabbed her and was menacing and sexually crude toward her". This practice, this exercise in racial extremism has been dragged into the modern era through the weaponizing of 9-1-1, often by white women, to invoke the power and force of the police who they are fully aware are hostile to black men. This was again evident when a white woman in New York's Central Park told a black man, a bird-watcher, that she was going to call the police and tell them that he was threatening her life.

Patterns of influence

Political violence

The Tuskegee Institute has estimated that 3,446 blacks were the victims of lynchings in the United States between 1882 and 1968, with the peak occurring in the 1890s at a time of economic stress in the South and increasing political suppression of blacks. If 1,297 whites were also lynched during this period, blacks were disproportionally targeted, representing 72.7% of all people lynched. According to scholar Amy L. Wood, "lynching photographs constructed and perpetuated white supremacist ideology by creating permanent images of a controlled white citizenry juxtaposed to images of helpless and powerless black men."

School curriculum

Main article: White supremacy in U.S. school curriculum

White supremacy has also played a part in U.S. school curriculum. Over the course of the 19th, 20th, and 21st centuries, material across the spectrum of academic disciplines has been taught with a heavy emphasis on white culture, contributions, and experiences, and a lack of representation of non-white groups' perspectives and accomplishments. In the 19th century, Geography lessons contained teachings on a fixed racial hierarchy, which white people topped. Mills (1994) writes that history as it is taught is really the history of white people, and it is taught in a way that favors white Americans and white people in general. He states that the language used to tell history minimizes the violent acts committed by white people over the centuries, citing the use of the words, for example, "discovery," "colonization," and "New World" when describing what was ultimately a European conquest of the Western Hemisphere and its indigenous peoples. Swartz (1992) seconds this reading of modern history narratives when it comes to the experiences, resistances, and accomplishments of black Americans throughout the Middle Passage, slavery, Reconstruction, Jim Crow, and the civil rights movement. In an analysis of American history textbooks, she highlights word choices that repetitively "normalize" slavery and the inhumane treatment of black people. She also notes the frequent showcasing of white abolitionists and actual exclusion of black abolitionists and the fact that black Americans had been mobilizing for abolition for centuries before the major white American push for abolition in the 19th century. She ultimately asserts the presence of a masternarrative that centers Europe and its associated peoples (white people) in school curriculum, particularly as it pertains to history. She writes that this masternarrative condenses history into only history that is relevant to, and to some extent beneficial for, white Americans.

Elson (1964) provides detailed information about the historic dissemination of simplistic and negative ideas about non-white races. Native Americans, who were subjected to attempts of cultural genocide by the U.S. government through the use of American Indian boarding schools, were characterized as homogenously "cruel," a violent menace toward white Americans, and lacking civilization or societal complexity (p. 74). For example, in the 19th century, black Americans were consistently portrayed as lazy, immature, and intellectually and morally inferior to white Americans, and in many ways not deserving of equal participation in U.S. society. For example, a math problem in a 19th-century textbook read, "If 5 white men can do as much work as 7 negroes..." implying that white men are more industrious and competent than black men (p. 99). In addition, little to nothing was taught about black Americans' contributions, or their histories before being brought to U.S. soil as slaves. According to Wayne (1972), this approach was taken especially much after the Civil War to maintain whites' hegemony over emancipated black Americans. Other racial groups have received oppressive treatment, including Mexican Americans, who temporarily were prevented from learning the same curriculum as white Americans because they supposedly were intellectually inferior, and Asian Americans, some of whom were prevented from learning much about their ancestral lands because they were deemed a threat to "American" culture, i.e. white culture, at the turn of the 20th century.

Role of the Internet

With the emergence of Twitter in 2006, and platforms such as Stormfront, which was launched in 1996, an alt-right portal for white supremacists with similar beliefs, both adults and children, was provided in which they were given a way to connect. Jessie Daniels, of CUNY-Hunter College, discussed the emergence of other social media outlets such as 4chan and Reddit, which meant that the "spread of white nationalist symbols and ideas could be accelerated and amplified." Sociologist Kathleen Blee notes that the anonymity which the Internet provides can make it difficult to track the extent of white-supremacist activity in the country, but nevertheless she and other experts see an increase in the number of hate crimes and amount of white-supremacist violence. In the latest wave of white supremacy, in the age of the Internet, Blee sees the movement as having become primarily a virtual one, in which divisions between groups become blurred: "[A]ll these various groups that get jumbled together as the alt-right and people who have come in from the more traditional neo-Nazi world. We're in a very different world now."

David Duke, a former Grand Wizard of the Ku Klux Klan, wrote in 1999 that the Internet was going to create a "chain reaction of racial enlightenment that will shake the world." Daniels documents that racist groups see the Internet as a way to spread their ideologies, influence others and gain supporters. Legal scholar Richard Hasen describes a "dark side" of social media:

There certainly were hate groups before the Internet and social media. [But with social media] it just becomes easier to organize, to spread the word, for people to know where to go. It could be to raise money, or it could be to engage in attacks on social media. Some of the activity is virtual. Some of it is in a physical place. Social media has lowered the collective-action problems that individuals who might want to be in a hate group would face. You can see that there are people out there like you. That's the dark side of social media.

A series on YouTube hosted by the grandson of Thomas Robb, the national director of the Knights of the Ku Klux Klan, "presents the Klan's ideology in a format aimed at kids — more specifically, white kids." The short episodes inveigh against race-mixing, and extol other white-supremacist ideologies. A short documentary published by TRT describes Imran Garda's experience, a journalist of Indian descent, who met with Thomas Robb and a traditional KKK group. A sign that greets people who enter the town states "Diversity is a code for white genocide." The KKK group interviewed in the documentary summarizes its ideals, principles, and beliefs, which are emblematic of white supremacists in the United States. The comic book super hero Captain America was used for dog whistle politics by the alt-right in college campus recruitment in 2017, an ironic co-opting because Captain America battled against Nazis in the comics, and was created by Jewish cartoonists.

Polar coordinate system

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Polar_coordinate_system 

Points in the polar coordinate system with pole O and polar axis L. In green, the point with radial coordinate 3 and angular coordinate 60 degrees or (3, 60°). In blue, the point (4, 210°).

In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are

• the point's distance from a reference point called the pole, and
• the point's direction from the pole relative to the direction of the polar axis, a ray drawn from the pole.

The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. The pole is analogous to the origin in a Cartesian coordinate system.

Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as spirals. Planar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more intuitive to model using polar coordinates.

The polar coordinate system is extended to three dimensions in two ways: the cylindrical coordinate system adds a second distance coordinate, and the spherical coordinate system adds a second angular coordinate.

Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the system's concepts in the mid-17th century, though the actual term polar coordinates has been attributed to Gregorio Fontana in the 18th century. The initial motivation for introducing the polar system was the study of circular and orbital motion.

History

See also: History of trigonometry

Hipparchus

The concepts of angle and radius were already used by ancient peoples of the first millennium BC. The Greek astronomer and astrologer Hipparchus (190–120 BC) created a table of chord functions giving the length of the chord for each angle, and there are references to his using polar coordinates in establishing stellar positions. In On Spirals, Archimedes describes the Archimedean spiral, a function whose radius depends on the angle. The Greek work, however, did not extend to a full coordinate system.

From the 8th century AD onward, astronomers developed methods for approximating and calculating the direction to Mecca (qibla)—and its distance—from any location on the Earth. From the 9th century onward they were using spherical trigonometry and map projection methods to determine these quantities accurately. The calculation is essentially the conversion of the equatorial polar coordinates of Mecca (i.e. its longitude and latitude) to its polar coordinates (i.e. its qibla and distance) relative to a system whose reference meridian is the great circle through the given location and the Earth's poles and whose polar axis is the line through the location and its antipodal point.

There are various accounts of the introduction of polar coordinates as part of a formal coordinate system. The full history of the subject is described in Harvard professor Julian Lowell Coolidge's Origin of Polar Coordinates. Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-seventeenth century. Saint-Vincent wrote about them privately in 1625 and published his work in 1647, while Cavalieri published his in 1635 with a corrected version appearing in 1653. Cavalieri first used polar coordinates to solve a problem relating to the area within an Archimedean spiral. Blaise Pascal subsequently used polar coordinates to calculate the length of parabolic arcs.

In Method of Fluxions (written 1671, published 1736), Sir Isaac Newton examined the transformations between polar coordinates, which he referred to as the "Seventh Manner; For Spirals", and nine other coordinate systems. In the journal Acta Eruditorum (1691), Jacob Bernoulli used a system with a point on a line, called the pole and polar axis respectively. Coordinates were specified by the distance from the pole and the angle from the polar axis. Bernoulli's work extended to finding the radius of curvature of curves expressed in these coordinates.

The actual term polar coordinates has been attributed to Gregorio Fontana and was used by 18th-century Italian writers. The term appeared in English in George Peacock's 1816 translation of Lacroix's Differential and Integral Calculus. Alexis Clairaut was the first to think of polar coordinates in three dimensions, and Leonhard Euler was the first to actually develop them.

Conventions

A polar grid with several angles, increasing in counterclockwise orientation and labelled in degrees

The radial coordinate is often denoted by r or ρ, and the angular coordinate by φ, θ, or t. The angular coordinate is specified as φ by ISO standard 31-11, now 80000-2:2019. However, in mathematical literature the angle is often denoted by θ instead.

Angles in polar notation are generally expressed in either degrees or radians (2π rad being equal to 360°). Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics and mathematical physics.

The angle φ is defined to start at 0° from a reference direction, and to increase for rotations in either clockwise (cw) or counterclockwise (ccw) orientation. For example, in mathematics, the reference direction is usually drawn as a ray from the pole horizontally to the right, and the polar angle increases to positive angles for ccw rotations, whereas in navigation (bearing, heading) the 0°-heading is drawn vertically upwards and the angle increases for cw rotations. The polar angles decrease towards negative values for rotations in the respectively opposite orientations.

Uniqueness of polar coordinates

Adding any number of full turns (360°) to the angular coordinate does not change the corresponding direction. Similarly, any polar coordinate is identical to the coordinate with the negative radial component and the opposite direction (adding 180° to the polar angle). Therefore, the same point (r, φ) can be expressed with an infinite number of different polar coordinates (r, φ + n × 360°) and (−r, φ + 180° + n × 360°) = (−r, φ + (2n + 1) × 180°), where n is an arbitrary integer. Moreover, the pole itself can be expressed as (0, φ) for any angle φ.

Where a unique representation is needed for any point besides the pole, it is usual to limit r to positive numbers (r > 0) and φ to either the interval [0, 360°) or the interval (−180°, 180°], which in radians are [0, 2π) or (−π, π]. Another convention, in reference to the usual codomain of the arctan function, is to allow for arbitrary nonzero real values of the radial component and restrict the polar angle to (−90°, 90°]. In all cases a unique azimuth for the pole (r = 0) must be chosen, e.g., φ = 0.

Converting between polar and Cartesian coordinates

A diagram illustrating the relationship between polar and Cartesian coordinates.

The polar coordinates r and φ can be converted to the Cartesian coordinates x and y by using the trigonometric functions sine and cosine:

$${\displaystyle \begin{array}{rl}x& =r\cos \phi ,\\ y& =r\sin \phi .\end{array}}$$

The Cartesian coordinates x and y can be converted to polar coordinates r and φ with r ≥ 0 and φ in the interval (−π, π] by:$${\displaystyle \begin{array}{rl}r& =\sqrt{x^2+y^2}=\mathrm{hypot}(x,y)\\ \phi & =\mathrm{atan2}(y,x),\end{array}}$$ where hypot is the Pythagorean sum and atan2 is a common variation on the arctangent function defined as $${\displaystyle \mathrm{atan2}(y,x)=\{\begin{array}{ll}\arctan \left(\frac{y}{x}\right)& \text{if }x>0\\ \arctan \left(\frac{y}{x}\right)+\pi & \text{if }x<0\text{ and }y\ge 0\\ \arctan \left(\frac{y}{x}\right)-\pi & \text{if }x<0\text{ and }y<0\\ \frac{\pi }{2}& \text{if }x=0\text{ and }y>0\\ -\frac{\pi }{2}& \text{if }x=0\text{ and }y<0\\ \text{undefined}& \text{if }x=0\text{ and }y=0.\end{array}}$$

If r is calculated first as above, then this formula for φ may be stated more simply using the arccosine function: $${\displaystyle \phi =\{\begin{array}{ll}\arccos \left(\frac{x}{r}\right)& \text{if }y\ge 0\text{ and }r\ne 0\\ -\arccos \left(\frac{x}{r}\right)& \text{if }y<0\\ \text{undefined}& \text{if }r=0.\end{array}}$$

Complex numbers

An illustration of a complex number z plotted on the complex plane

An illustration of a complex number plotted on the complex plane using Euler's formula

Every complex number can be represented as a point in the complex plane, and can therefore be expressed by specifying either the point's Cartesian coordinates (called rectangular or Cartesian form) or the point's polar coordinates (called polar form).

In polar form, the distance and angle coordinates are often referred to as the number's magnitude and argument respectively. Two complex numbers can be multiplied by adding their arguments and multiplying their magnitudes.

The complex number z can be represented in rectangular form as $${\displaystyle z=x+iy}$$ where i is the imaginary unit, or can alternatively be written in polar form as $${\displaystyle z=r(\cos \phi +i\sin \phi )}$$ and from there, by Euler's formula, as $${\displaystyle z=r{e}^{i\phi }=r\exp i\phi .}$$ where e is Euler's number, and φ, expressed in radians, is the principal value of the complex number function arg applied to x + iy. To convert between the rectangular and polar forms of a complex number, the conversion formulae given above can be used. Equivalent are the cis and angle notations: $${\displaystyle z=r\mathrm{c}\mathrm{i}\mathrm{s}\phi =r\mathrm{\angle }\phi .}$$

For the operations of multiplication, division, exponentiation, and root extraction of complex numbers, it is generally much simpler to work with complex numbers expressed in polar form rather than rectangular form. From the laws of exponentiation:

Multiplication

${\displaystyle r_0{e}^{i{\phi }_0}{r}_1{e}^{i{\phi }_1}=r_0{r}_1{e}^{i\left({\phi }_0+{\phi }_1\right)}}$

Division

${\displaystyle \frac{r_0{e}^{i{\phi }_0}}{r_1{e}^{i{\phi }_1}}=\frac{r_0}{r_1}{e}^{i({\phi }_0-{\phi }_1)}}$

Exponentiation (De Moivre's formula)

${\displaystyle {\left(r{e}^{i\phi }\right)}^n=r^n{e}^{in\phi }}$

Root Extraction (Principal root)

${\displaystyle \sqrt[n]{r{e}^{i\phi }}=\sqrt[n]{r}{e}^{\frac{i\phi }{n}}}$

Polar equation of a curve

A curve on the Cartesian plane can be mapped into polar coordinates. In this animation, ${\displaystyle y=\sin (6\cdot x)+2}$ is mapped onto ${\displaystyle r=\sin (6\cdot \theta )+2}$. Click on image for details.

The equation defining a plane curve expressed in polar coordinates is known as a polar equation. In many cases, such an equation can simply be specified by defining r as a function of φ. The resulting curve then consists of points of the form (r(φ), φ) and can be regarded as the graph of the polar function r. Note that, in contrast to Cartesian coordinates, the independent variable φ is the second entry in the ordered pair.

Different forms of symmetry can be deduced from the equation of a polar function r:

• If r(−φ) = r(φ) the curve will be symmetrical about the horizontal (0°/180°) ray;
• If r(π − φ) = r(φ) it will be symmetric about the vertical (90°/270°) ray:
• If r(φ − α) = r(φ) it will be rotationally symmetric by α clockwise and counterclockwise about the pole.

Because of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate. Among the best known of these curves are the polar rose, Archimedean spiral, lemniscate, limaçon, and cardioid.

For the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve.

Circle

A circle with equation r(φ) = 1

The general equation for a circle with a center at ${\displaystyle (r_0,\gamma )}$ and radius a is $${\displaystyle r^2-2r{r}_0\cos (\phi -\gamma )+r_0^2=a^2.}$$

This can be simplified in various ways, to conform to more specific cases, such as the equation $${\displaystyle r(\phi )=a}$$ for a circle with a center at the pole and radius a.

When r₀ = a or the origin lies on the circle, the equation becomes $${\displaystyle r=2a\cos (\phi -\gamma ).}$$

In the general case, the equation can be solved for r, giving $${\displaystyle r=r_0\cos (\phi -\gamma )+\sqrt{a^2-r_0^2{\sin }^2(\phi -\gamma )}}$$ The solution with a minus sign in front of the square root gives the same curve.

Line

Radial lines (those running through the pole) are represented by the equation $${\displaystyle \phi =\gamma ,}$$ where ${\displaystyle \gamma }$ is the angle of elevation of the line; that is, ${\displaystyle \phi =\arctan m}$, where ${\displaystyle m}$ is the slope of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line ${\displaystyle \phi =\gamma }$ perpendicularly at the point ${\displaystyle (r_0,\gamma )}$ has the equation $${\displaystyle r(\phi )=r_0\sec (\phi -\gamma ).}$$

Otherwise stated ${\displaystyle (r_0,\gamma )}$ is the point in which the tangent intersects the imaginary circle of radius ${\displaystyle r_0}$

Polar rose

A polar rose with equation r(φ) = 2 sin 4φ

A polar rose is a mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation, $${\displaystyle r(\phi )=a\cos \left(k\phi +{\gamma }_0\right)}$$

for any constant γ₀ (including 0). If k is an integer, these equations will produce a k-petaled rose if k is odd, or a 2k-petaled rose if k is even. If k is rational, but not an integer, a rose-like shape may form but with overlapping petals. Note that these equations never define a rose with 2, 6, 10, 14, etc. petals. The variable a directly represents the length or amplitude of the petals of the rose, while k relates to their spatial frequency. The constant γ₀ can be regarded as a phase angle.

Archimedean spiral

One arm of an Archimedean spiral with equation r(φ) = φ / 2π for 0 < φ < 6π

The Archimedean spiral is a spiral discovered by Archimedes which can also be expressed as a simple polar equation. It is represented by the equation $${\displaystyle r(\phi )=a+b\phi .}$$ Changing the parameter a will turn the spiral, while b controls the distance between the arms, which for a given spiral is always constant. The Archimedean spiral has two arms, one for φ > 0 and one for φ < 0. The two arms are smoothly connected at the pole. If a = 0, taking the mirror image of one arm across the 90°/270° line will yield the other arm. This curve is notable as one of the first curves, after the conic sections, to be described in a mathematical treatise, and as a prime example of a curve best defined by a polar equation.

Ellipse, showing semi-latus rectum

Conic sections

A conic section with one focus on the pole and the other somewhere on the 0° ray (so that the conic's major axis lies along the polar axis) is given by: $${\displaystyle r=\frac{\ell }{1-e\cos \phi }}$$ where e is the eccentricity and ${\displaystyle \ell }$ is the semi-latus rectum (the perpendicular distance at a focus from the major axis to the curve). If e > 1, this equation defines a hyperbola; if e = 1, it defines a parabola; and if e < 1, it defines an ellipse. The special case e = 0 of the latter results in a circle of the radius ${\displaystyle \ell }$.

Quadratrix

Main article: Quadratrix of Hippias

A quadratrix in the first quadrant (x, y) is a curve with y = ρ sin θ equal to the fraction of the quarter circle with radius r determined by the radius through the curve point. Since this fraction is ${\displaystyle \frac{2r\theta }{\pi }}$, the curve is given by ${\displaystyle \rho (\theta )=\frac{2r\theta }{\pi \sin \theta }}$.

Intersection of two polar curves

The graphs of two polar functions ${\displaystyle r=f(\theta )}$ and ${\displaystyle r=g(\theta )}$ have possible intersections of three types:

1. In the origin, if the equations ${\displaystyle f(\theta )=0}$ and ${\displaystyle g(\theta )=0}$ have at least one solution each.
2. All the points ${\displaystyle [g({\theta }_i),{\theta }_i]}$ where ${\displaystyle {\theta }_i}$ are solutions to the equation ${\displaystyle f(\theta +2k\pi )=g(\theta )}$ where ${\displaystyle k}$ is an integer.
3. All the points ${\displaystyle [g({\theta }_i),{\theta }_i]}$ where ${\displaystyle {\theta }_i}$ are solutions to the equation ${\displaystyle f(\theta +(2k+1)\pi )=-g(\theta )}$ where ${\displaystyle k}$ is an integer.

Calculus

Calculus can be applied to equations expressed in polar coordinates.

The angular coordinate φ is expressed in radians throughout this section, which is the conventional choice when doing calculus.

Differential calculus

Using x = r cos φ and y = r sin φ, one can derive a relationship between derivatives in Cartesian and polar coordinates. For a given function, u(x,y), it follows that (by computing its total derivatives) or $${\displaystyle \begin{array}{rl}r\frac{du}{dr}& =r\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\cos \phi +r\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\sin \phi =x\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+y\frac{\mathrm{\partial }u}{\mathrm{\partial }y},\\ \frac{du}{d\phi }& =-\frac{\mathrm{\partial }u}{\mathrm{\partial }x}r\sin \phi +\frac{\mathrm{\partial }u}{\mathrm{\partial }y}r\cos \phi =-y\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+x\frac{\mathrm{\partial }u}{\mathrm{\partial }y}.\end{array}}$$

Hence, we have the following formula: $${\displaystyle \begin{array}{rl}r\frac{d}{dr}& =x\frac{\mathrm{\partial }}{\mathrm{\partial }x}+y\frac{\mathrm{\partial }}{\mathrm{\partial }y}\\ \frac{d}{d\phi }& =-y\frac{\mathrm{\partial }}{\mathrm{\partial }x}+x\frac{\mathrm{\partial }}{\mathrm{\partial }y}.\end{array}}$$

Using the inverse coordinates transformation, an analogous reciprocal relationship can be derived between the derivatives. Given a function u(r,φ), it follows that $${\displaystyle \begin{array}{rl}\frac{du}{dx}& =\frac{\mathrm{\partial }u}{\mathrm{\partial }r}\frac{\mathrm{\partial }r}{\mathrm{\partial }x}+\frac{\mathrm{\partial }u}{\mathrm{\partial }\phi }\frac{\mathrm{\partial }\phi }{\mathrm{\partial }x},\\ \frac{du}{dy}& =\frac{\mathrm{\partial }u}{\mathrm{\partial }r}\frac{\mathrm{\partial }r}{\mathrm{\partial }y}+\frac{\mathrm{\partial }u}{\mathrm{\partial }\phi }\frac{\mathrm{\partial }\phi }{\mathrm{\partial }y},\end{array}}$$ or $${\displaystyle \begin{array}{rl}\frac{du}{dx}& =\frac{\mathrm{\partial }u}{\mathrm{\partial }r}\frac{x}{\sqrt{x^2+y^2}}-\frac{\mathrm{\partial }u}{\mathrm{\partial }\phi }\frac{y}{x^2+y^2}\\ & =\cos \phi \frac{\mathrm{\partial }u}{\mathrm{\partial }r}-\frac{1}{r}\sin \phi \frac{\mathrm{\partial }u}{\mathrm{\partial }\phi },\\ \frac{du}{dy}& =\frac{\mathrm{\partial }u}{\mathrm{\partial }r}\frac{y}{\sqrt{x^2+y^2}}+\frac{\mathrm{\partial }u}{\mathrm{\partial }\phi }\frac{x}{x^2+y^2}\\ & =\sin \phi \frac{\mathrm{\partial }u}{\mathrm{\partial }r}+\frac{1}{r}\cos \phi \frac{\mathrm{\partial }u}{\mathrm{\partial }\phi }.\end{array}}$$

Hence, we have the following formulae: $${\displaystyle \begin{array}{rl}\frac{d}{dx}& =\cos \phi \frac{\mathrm{\partial }}{\mathrm{\partial }r}-\frac{1}{r}\sin \phi \frac{\mathrm{\partial }}{\mathrm{\partial }\phi }\\ \frac{d}{dy}& =\sin \phi \frac{\mathrm{\partial }}{\mathrm{\partial }r}+\frac{1}{r}\cos \phi \frac{\mathrm{\partial }}{\mathrm{\partial }\phi }.\end{array}}$$

To find the Cartesian slope of the tangent line to a polar curve r(φ) at any given point, the curve is first expressed as a system of parametric equations. $${\displaystyle \begin{array}{rl}x& =r(\phi )\cos \phi \\ y& =r(\phi )\sin \phi \end{array}}$$

Differentiating both equations with respect to φ yields $${\displaystyle \begin{array}{rl}\frac{dx}{d\phi }& =r^{\prime }(\phi )\cos \phi -r(\phi )\sin \phi \\ \frac{dy}{d\phi }& =r^{\prime }(\phi )\sin \phi +r(\phi )\cos \phi .\end{array}}$$

Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point (r(φ), φ): $${\displaystyle \frac{dy}{dx}=\frac{r^{\prime }(\phi )\sin \phi +r(\phi )\cos \phi }{r^{\prime }(\phi )\cos \phi -r(\phi )\sin \phi }.}$$

For other useful formulas including divergence, gradient, and Laplacian in polar coordinates, see curvilinear coordinates.

Integral calculus (arc length)

The arc length (length of a line segment) defined by a polar function is found by the integration over the curve r(φ). Let L denote this length along the curve starting from points A through to point B, where these points correspond to φ = a and φ = b such that 0 < b − a < 2π. The length of L is given by the following integral $${\displaystyle L={\int }_a^b\sqrt{{\left[r(\phi )\right]}^2+{\left[\frac{dr(\phi )}{d\phi }\right]}^2}d\phi }$$

Integral calculus (area)

The integration region R is bounded by the curve r(φ) and the rays φ = a and φ = b.

Let R denote the region enclosed by a curve r(φ) and the rays φ = a and φ = b, where 0 < b − a ≤ 2π. Then, the area of R is $${\displaystyle \frac{1}{2}{\int }_a^b{\left[r(\phi )\right]}^{2}d\phi .}$$

The region R is approximated by n sectors (here, n = 5).

A planimeter, which mechanically computes polar integrals

This result can be found as follows. First, the interval [a, b] is divided into n subintervals, where n is some positive integer. Thus Δφ, the angle measure of each subinterval, is equal to b − a (the total angle measure of the interval), divided by n, the number of subintervals. For each subinterval i = 1, 2, ..., n, let φ_i be the midpoint of the subinterval, and construct a sector with the center at the pole, radius r(φ_i), central angle Δφ and arc length r(φ_i)Δφ. The area of each constructed sector is therefore equal to $${\displaystyle {\left[r({\phi }_i)\right]}^2\pi \cdot \frac{\mathrm{\Delta }\phi }{2\pi }=\frac{1}{2}{\left[r({\phi }_i)\right]}^2\mathrm{\Delta }\phi .}$$ Hence, the total area of all of the sectors is $${\displaystyle \sum _{i=1}^n\frac{1}{2}r({\phi }_i{)}^2\mathrm{\Delta }\phi .}$$

As the number of subintervals n is increased, the approximation of the area improves. Taking n → ∞, the sum becomes the Riemann sum for the above integral.

A mechanical device that computes area integrals is the planimeter, which measures the area of plane figures by tracing them out: this replicates integration in polar coordinates by adding a joint so that the 2-element linkage effects Green's theorem, converting the quadratic polar integral to a linear integral.

Generalization

Using Cartesian coordinates, an infinitesimal area element can be calculated as dA = dx dy. The substitution rule for multiple integrals states that, when using other coordinates, the Jacobian determinant of the coordinate conversion formula has to be considered: $${\displaystyle J=det\frac{\mathrm{\partial }(x,y)}{\mathrm{\partial }(r,\phi )}=\left|\begin{array}{cc}\frac{\mathrm{\partial }x}{\mathrm{\partial }r}& \frac{\mathrm{\partial }x}{\mathrm{\partial }\phi }\\ \frac{\mathrm{\partial }y}{\mathrm{\partial }r}& \frac{\mathrm{\partial }y}{\mathrm{\partial }\phi }\end{array}\right|=\left|\begin{array}{cc}\cos \phi & -r\sin \phi \\ \sin \phi & r\cos \phi \end{array}\right|=r{\cos }^2\phi +r{\sin }^2\phi =r.}$$

Hence, an area element in polar coordinates can be written as $${\displaystyle dA=dxdy=Jdrd\phi =rdrd\phi .}$$

Now, a function, that is given in polar coordinates, can be integrated as follows: $${\displaystyle {\iint }_{R}f(x,y)dA={\int }_a^b{\int }_0^{r(\phi )}f(r,\phi )rdrd\phi .}$$

Here, R is the same region as above, namely, the region enclosed by a curve r(φ) and the rays φ = a and φ = b. The formula for the area of R is retrieved by taking f identically equal to 1.

A graph of ${\displaystyle f(x)=e^{-x^2}}$ and the area between the function and the ${\displaystyle x}$-axis, which is equal to ${\displaystyle \sqrt{\pi }}$.

A more surprising application of this result yields the Gaussian integral: $${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2}dx=\sqrt{\pi }.}$$

Vector calculus

Vector calculus can also be applied to polar coordinates. For a planar motion, let ${\displaystyle \mathbf{r}}$ be the position vector (r cos(φ), r sin(φ)), with r and φ depending on time t.

We define an orthonormal basis with three unit vectors: radial, transverse, and normal directions. The radial direction is defined by normalizing ${\displaystyle \mathbf{r}}$: $${\displaystyle \hat{\mathbf{r}}=(\cos (\phi ),\sin (\phi ))}$$ Radial and velocity directions span the plane of the motion, whose normal direction is denoted ${\displaystyle \hat{\mathbf{k}}}$: $${\displaystyle \hat{\mathbf{k}}=\hat{\mathbf{v}}\times \hat{\mathbf{r}}.}$$ The transverse direction is perpendicular to both radial and normal directions: $${\displaystyle \hat{\mathit{\phi }}=(-\sin (\phi ),\cos (\phi ))=\hat{\mathbf{k}}\times \hat{\mathbf{r}},}$$

Then $${\displaystyle \begin{array}{rl}\mathbf{r}& =(x,y)=r(\cos \phi ,\sin \phi )=r\hat{\mathbf{r}},\\ \dot{\mathbf{r}}& =\left(\dot{x},\dot{y}\right)=\dot{r}(\cos \phi ,\sin \phi )+r\dot{\phi }(-\sin \phi ,\cos \phi )=\dot{r}\hat{\mathbf{r}}+r\dot{\phi }\hat{\mathit{\phi }},\\ \ddot{\mathbf{r}}& =\left(\ddot{x},\ddot{y}\right)\\ & =\ddot{r}(\cos \phi ,\sin \phi )+2\dot{r}\dot{\phi }(-\sin \phi ,\cos \phi )+r\ddot{\phi }(-\sin \phi ,\cos \phi )-r{\dot{\phi }}^2(\cos \phi ,\sin \phi )\\ & =\left(\ddot{r}-r{\dot{\phi }}^2\right)\hat{\mathbf{r}}+\left(r\ddot{\phi }+2\dot{r}\dot{\phi }\right)\hat{\mathit{\phi }}\\ & =\left(\ddot{r}-r{\dot{\phi }}^2\right)\hat{\mathbf{r}}+\frac{1}{r}\frac{d}{dt}\left(r^2\dot{\phi }\right)\hat{\mathit{\phi }}.\end{array}}$$

This equation can be obtained by taking derivative of the function and derivatives of the unit basis vectors.

For a curve in 2D where the parameter is ${\displaystyle \theta }$ the previous equations simplify to: $${\displaystyle \begin{array}{rl}\mathbf{r}& =r(\theta ){\hat{\mathbf{e}}}_r\\ \frac{d\mathbf{r}}{d\theta }& =\frac{dr}{d\theta }{\hat{\mathbf{e}}}_r+r{\hat{\mathbf{e}}}_{\theta }\\ \frac{d^2\mathbf{r}}{d{\theta }^2}& =\left(\frac{d^{2}r}{d{\theta }^2}-r\right){\hat{\mathbf{e}}}_r+\frac{dr}{d\theta }{\hat{\mathbf{e}}}_{\theta }\end{array}}$$

Centrifugal and Coriolis terms

See also: Centrifugal force

 

Position vector r, always points radially from the origin.

 

Velocity vector v, always tangent to the path of motion.

 

Acceleration vector a, not parallel to the radial motion but offset by the angular and Coriolis accelerations, nor tangent to the path but offset by the centripetal and radial accelerations.

Kinematic vectors in plane polar coordinates. Notice the setup is not restricted to 2d space, but a plane in any higher dimension.

The term ${\displaystyle r{\dot{\phi }}^2}$ is sometimes referred to as the centripetal acceleration, and the term ${\displaystyle 2\dot{r}\dot{\phi }}$ as the Coriolis acceleration. For example, see Shankar.

Note: these terms, that appear when acceleration is expressed in polar coordinates, are a mathematical consequence of differentiation; they appear whenever polar coordinates are used. In planar particle dynamics these accelerations appear when setting up Newton's second law of motion in a rotating frame of reference. Here these extra terms are often called fictitious forces; fictitious because they are simply a result of a change in coordinate frame. That does not mean they do not exist, rather they exist only in the rotating frame.

Inertial frame of reference S and instantaneous non-inertial co-rotating frame of reference S′. The co-rotating frame rotates at angular rate Ω equal to the rate of rotation of the particle about the origin of S′ at the particular moment t. Particle is located at vector position r(t) and unit vectors are shown in the radial direction to the particle from the origin, and also in the direction of increasing angle ϕ normal to the radial direction. These unit vectors need not be related to the tangent and normal to the path. Also, the radial distance r need not be related to the radius of curvature of the path.

Co-rotating frame

For a particle in planar motion, one approach to attaching physical significance to these terms is based on the concept of an instantaneous co-rotating frame of reference. To define a co-rotating frame, first an origin is selected from which the distance r(t) to the particle is defined. An axis of rotation is set up that is perpendicular to the plane of motion of the particle, and passing through this origin. Then, at the selected moment t, the rate of rotation of the co-rotating frame Ω is made to match the rate of rotation of the particle about this axis, dφ/dt. Next, the terms in the acceleration in the inertial frame are related to those in the co-rotating frame. Let the location of the particle in the inertial frame be (r(t), φ(t)), and in the co-rotating frame be (r′(t), φ′(t)). Because the co-rotating frame rotates at the same rate as the particle, dφ′/dt = 0. The fictitious centrifugal force in the co-rotating frame is mrΩ², radially outward. The velocity of the particle in the co-rotating frame also is radially outward, because dφ′/dt = 0. The fictitious Coriolis force therefore has a value −2m(dr/dt)Ω, pointed in the direction of increasing φ only. Thus, using these forces in Newton's second law we find: $${\displaystyle \mathbf{F}+{\mathbf{F}}_{\text{cf}}+{\mathbf{F}}_{\text{Cor}}=m\ddot{\mathbf{r}},}$$ where over dots represent derivatives with respect to time, and F is the net real force (as opposed to the fictitious forces). In terms of components, this vector equation becomes: $${\displaystyle \begin{array}{rl}{F}_r+mr{\mathrm{\Omega }}^2& =m\ddot{r}\\ F_{\phi }-2m\dot{r}\mathrm{\Omega }& =mr\ddot{\phi },\end{array}}$$ which can be compared to the equations for the inertial frame: $${\displaystyle \begin{array}{rl}{F}_r& =m\ddot{r}-mr{\dot{\phi }}^2\\ F_{\phi }& =mr\ddot{\phi }+2m\dot{r}\dot{\phi }.\end{array}}$$

This comparison, plus the recognition that by the definition of the co-rotating frame at time t it has a rate of rotation Ω = dφ/dt, shows that we can interpret the terms in the acceleration (multiplied by the mass of the particle) as found in the inertial frame as the negative of the centrifugal and Coriolis forces that would be seen in the instantaneous, non-inertial co-rotating frame.

For general motion of a particle (as opposed to simple circular motion), the centrifugal and Coriolis forces in a particle's frame of reference commonly are referred to the instantaneous osculating circle of its motion, not to a fixed center of polar coordinates. For more detail, see centripetal force.

Differential geometry

In the modern terminology of differential geometry, polar coordinates provide coordinate charts for the differentiable manifold R² \ {(0,0)}, the plane minus the origin. In these coordinates, the Euclidean metric tensor is given by$${\displaystyle d{s}^2=d{r}^2+r^{2}d{\theta }^2.}$$This can be seen via the change of variables formula for the metric tensor, or by computing the differential forms dx, dy via the exterior derivative of the 0-forms x = r cos(θ), y = r sin(θ) and substituting them in the Euclidean metric tensor ds² = dx² + dy².

An elementary proof of the formula

An orthonormal frame with respect to this metric is given by$${\displaystyle e_r=\frac{\mathrm{\partial }}{\mathrm{\partial }r},e_{\theta }=\frac{1}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }\theta },}$$with dual coframe$${\displaystyle e^r=dr,e^{\theta }=rd\theta .}$$The connection form relative to this frame and the Levi-Civita connection is given by the skew-symmetric matrix of 1-forms$${\displaystyle {{\omega }^i}_j=\left(\begin{array}{cc}0& -d\theta \\ d\theta & 0\end{array}\right)}$$and hence the curvature form Ω = dω + ω∧ω vanishes. Therefore, as expected, the punctured plane is a flat manifold.

Extensions in 3D

The polar coordinate system is extended into three dimensions with two different coordinate systems, the cylindrical and spherical coordinate system.

Applications

Polar coordinates are two-dimensional and thus they can be used only where point positions lie on a single two-dimensional plane. They are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point. For instance, the examples above show how elementary polar equations suffice to define curves—such as the Archimedean spiral—whose equation in the Cartesian coordinate system would be much more intricate. Moreover, many physical systems—such as those concerned with bodies moving around a central point or with phenomena originating from a central point—are simpler and more intuitive to model using polar coordinates. The initial motivation for the introduction of the polar system was the study of circular and orbital motion.

Position and navigation

Polar coordinates are used often in navigation as the destination or direction of travel can be given as an angle and distance from the object being considered. For instance, aircraft use a slightly modified version of the polar coordinates for navigation. In this system, the one generally used for any sort of navigation, the 0° ray is generally called heading 360, and the angles continue in a clockwise direction, rather than counterclockwise, as in the mathematical system. Heading 360 corresponds to magnetic north, while headings 90, 180, and 270 correspond to magnetic east, south, and west, respectively. Thus, an aircraft traveling 5 nautical miles due east will be traveling 5 units at heading 90 (read zero-niner-zero by air traffic control).

Modeling

Systems displaying radial symmetry provide natural settings for the polar coordinate system, with the central point acting as the pole. A prime example of this usage is the groundwater flow equation when applied to radially symmetric wells. Systems with a radial force are also good candidates for the use of the polar coordinate system. These systems include gravitational fields, which obey the inverse-square law, as well as systems with point sources, such as radio antennas.

Radially asymmetric systems may also be modeled with polar coordinates. For example, a microphone's pickup pattern illustrates its proportional response to an incoming sound from a given direction, and these patterns can be represented as polar curves. The curve for a standard cardioid microphone, the most common unidirectional microphone, can be represented as r = 0.5 + 0.5sin(ϕ) at its target design frequency. The pattern shifts toward omnidirectionality at lower frequencies.