Polar coordinate systems
Polar coordinate systems are a method of defining points in a plane using a fixed point known as the "pole" and an angle from a designated reference direction called the "polar axis." This system contrasts with the Cartesian coordinate system, which relies on perpendicular axes to specify locations. Polar coordinates are particularly useful in navigation, as they allow for straightforward representation of direction and distance, making them essential for applications like radar and air traffic control. The historical roots of polar coordinates can be traced back to ancient mathematicians like Hipparchus and Archimedes, with significant developments occurring in the 17th century by figures such as Jacob Bernoulli and Isaac Newton.
In contemporary settings, polar coordinates are employed in various practical applications, including weather radar systems and irrigation planning. They help convert radar data into actionable information displayed on maps for public use. Additionally, the concept of polar coordinates extends into acoustics, where different microphone types utilize principles of polar geometry to capture sound from specific directions. Overall, the polar coordinate system serves as a vital tool in both theoretical mathematics and practical applications across diverse fields.
Polar coordinate systems
Summary: Polar coordinate systems were developed in the seventeenth century and have numerous modern applications.
The polar coordinate system is a coordinate system for the plane in which each point is determined by a distance from a fixed point, called the “pole,” and an angle from a fixed direction, called the “polar axis.” In normal usage, the pole is analogous to the origin in the Cartesian coordinate system, named for René Descartes. Both polar and rectangular (Cartesian) coordinates require two bits of data to place a point in the plane. While the Cartesian coordinate system requires knowing and placing two chosen lines to serve as axes, polar coordinates requires knowing one fixed point and one fixed ray. This characteristic makes polar coordinates useful in navigation. Students in twenty-first-century high schools are introduced to polar coordinate systems and the topic is further developed in college mathematics and physics classrooms.
History
The concept of using an angle and a radius may be dated to the first millennium b.c.e. There are references to Hipparchus of Rhodes (c. second century b.c.e.) using a type of polar coordinates to establish the positions of the stars that he studied. Archimedes of Syracuse describes his namesake spiral in the book On Spirals, as where the distance from a given point depends on the angle from a given radius.
In a number of articles about the development of polar coordinates, most notably the 1952 article “Origin of Polar Coordinates” by Julian Lowell Coolidge, further development of polar coordinates was generated by studying the Archimedean spiral. According to Coolidge’s history, the first mention should go to Bonaventura Cavalieri in his 1635 treatise Geometria indivisilibus continuorum in which he studies the spiral of Archimedes. Cavalieri studies the area inside the spiral and relates it to other known areas.
Like all good stories in the history of mathematics, this assertion is not without disagreement. In 1647, Grégoire de Saint-Vincent in his work Opus Geometricum claimed that he was familiar with the method and had sent his work to Christopher Grienberger in 1625. Grienberger had died in 1636, and the priority of the work was the subject of an article by Moritz Cantor in 1900.
Spiral curves were of interest to many mathematicians, including Gilles Personne de Roberval, James Gregory, Descartes, and Pierre Varignon. Gregory, Descartes, and Varignon all used a type of transformation of coordinates that heralded the complete development of polar coordinates. It appears to be Jacob Bernoulli and Isaac Newton who most completely developed these transformations. Bernoulli worked on the lemniscate and introduced the terms “pole” and “polar axis.” Newton investigated transformations between coordinate systems, including polar coordinates, in his work Method of Fluxions, which was written in 1671 but not published until 1736.
Applications
Polar coordinates are the basis for navigation and radar, since the direction of travel can be given as an angle and distance from the origin. The radar screen that is used in air traffic control uses the location of the radar transmitter/receiver as the pole and magnetic north as the polar ray, zero degrees. This aspect and the fact that the angles continue in a clockwise direction instead of a counterclockwise direction are the major differences between a navigational use and the mathematical system. This same radar is the basis for all weather radar that is available for viewing either on television or from the Internet. Each radar location (there are 178 National Weather Service Doppler weather radar locations that cover the United States) sets a pole and covers a specific area. Storms are located and their paths are computed using the overlaps. This information must be transformed from the polar system (how far from the radar site and at what angle) into GIS coordinate system and then placed on a map to go to television or to the Internet. One well-known measuring device is the polar planimeter, created by mathematician and physicist Jacob Amsler in the nineteenth century. It measured the area enclosed by a curve. Amsler switched careers to focus on mathematical instruments, and he produced thousands of Amsler planimeters.
Other examples of the use of polar coordinates are very simplified uses in planning sprinkler systems in a building, as well as in irrigation systems in landscape and farming. Each of the sprinkler heads serves as a pole, and different walls, boundary lines and such serve as polar axes.
Different microphones have different recording patterns depending on the specific purpose. The omni-directional microphone is used when sound from all directions is to be recorded, such as a choir or a large group. A cardioid microphone is a unidirectional microphone, which would be used to record a performer but not the crowd. Bidirectional microphones are used in an interview situation where the voices of both the interviewer and interviewee need to be recorded. The pattern of sounds that are picked up by the microphone are a lemniscate the figure studied by Bernoulli.
Bibliography
Boyer, C. B. “Newton as an Originator of Polar Coordinates.” The American Mathematical Monthly 56, no. 2 (1949).
Coolidge, J. L. “The Origin of Polar Coordinates.” The American Mathematical Monthly 59, no. 2 (1952).
“A Periodic Shift in Polar Roses for Valentines Day.” http://www.nikolasschiller.com/blog/index.php/archives/category/renderings/quilt/polar-coordinates.