A chord of a circle is a geometric line segment whose endpoints both lie on the circumference of the circle. A secant or a secant line is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, such as but not limited to an ellipse. A chord that passes through the circle's center point is the circle's diameter.
(as is the diameter line AB).
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Chords of a circle [link]
Among properties of chords of a circle are the following:
- Chords are equidistant from the center only if their lengths are equal.
- A chord's perpendicular bisector passes through the centre.
- If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths satisfy AP·PB = CP·PD (power of a point theorem).
The area that a circular chord "cuts off" is called a circular segment.
Chords of an ellipse [link]
The midpoints of a set of parallel chords of an ellipse are collinear.[1]:p.147
Chords in trigonometry [link]
Chords were used extensively in the early development of trigonometry. The first known trigonometric table, compiled by Hipparchus, tabulated the value of the chord function for every 7.5 degrees. Ptolemy of Alexandria compiled a more extensive table of chords in his book on astronomy, giving the value of the chord for angles ranging from 1/2 degree to 180 degrees by increments of half a degree.
The chord function is defined geometrically as in the picture to the left. The chord of an angle is the length of the chord between two points on a unit circle separated by that angle. The chord function can be related to the modern sine function, by taking one of the points to be (1,0), and the other point to be (cos Failed to parse (Missing texvc executable; please see math/README to configure.): \theta , sin Failed to parse (Missing texvc executable; please see math/README to configure.): \theta ), and then using the Pythagorean theorem to calculate the chord length:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{crd}\ \theta = \sqrt{(1-\cos \theta)^2+\sin^2 \theta} = \sqrt{2-2\cos \theta} = 2 \sqrt{\frac{1-\cos \theta}{2}} = 2 \sin \frac{\theta}{2}.
The last step uses the half-angle formula. Much as modern trigonometry is built on the sine function, ancient trigonometry was built on the chord function. Hipparchus is purported to have written a twelve volume work on chords, all now lost, so presumably a great deal was known about them. The chord function satisfies many identities analogous to well-known modern ones:
| Name | Sine-based | Chord-based |
|---|---|---|
| Pythagorean | Failed to parse (Missing texvc executable; please see math/README to configure.): \sin^2 \theta + \cos^2 \theta = 1 \, | Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{crd}^2 \theta + \mathrm{crd}^2 (180^\circ - \theta) = 4 \, |
| Half-angle | Failed to parse (Missing texvc executable; please see math/README to configure.): \sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos \theta}{2}} \, | Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{crd}\ \frac{\theta}{2} = \pm \sqrt{2-\mathrm{crd}(180^\circ - \theta)} \, |
The half-angle identity greatly expedites the creation of chord tables. Ancient chord tables typically used a large value for the radius of the circle, and reported the chords for this circle. It was then a simple matter of scaling to determine the necessary chord for any circle. According to G. J. Toomer, Hipparchus used a circle of radius 3438' (= 3438/60 = 57.3). This value is extremely close to Failed to parse (Missing texvc executable; please see math/README to configure.): 180/\pi
(= 57.29577951...). One advantage of this choice of radius was that he could very accurately approximate the chord of a small angle as the angle itself. In modern terms, it allowed a simple linear approximation:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{3438}{60} \mathrm{crd}\ \theta = 2 \frac{3438}{60} \sin \frac{\theta}{2} \approx 2 \frac{3438}{60} \frac{\pi}{180} \frac{\theta}{2} = \left(\frac{3438}{60} \frac{\pi}{180}\right) \theta \approx \theta.
Calculating circular chords [link]
The chord of a circle can be calculated using other information:[2]
-
Initial data Radius (r) Diameter (D) Sagitta (s) Failed to parse (Missing texvc executable; please see math/README to configure.): c = 2 \sqrt {s (2 r - s)} Failed to parse (Missing texvc executable; please see math/README to configure.): c = 2 \sqrt {s (D - s)} Apothem (a) Failed to parse (Missing texvc executable; please see math/README to configure.): c=2 \sqrt{r^2- a^2} Failed to parse (Missing texvc executable; please see math/README to configure.): c=\sqrt{D ^2-4 a^2} Angle (θ) Failed to parse (Missing texvc executable; please see math/README to configure.): c=2 r \sin \left(\frac{\theta }{2}\right) Failed to parse (Missing texvc executable; please see math/README to configure.): c=D \sin \left(\frac{\theta }{2}\right)
See also [link]
References [link]
External links [link]
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Wikimedia Commons has media related to: Chord (geometry) |
- History of Trigonometry Outline
- Trigonometric functions, focusing on history
- Chord (of a circle) With interactive animation
https://wn.com/Chord_(geometry)
Chord (astronomy)
In the field of astronomy the term chord typically refers to a line crossing an object which is formed during an occultation event. By taking accurate measurements of the start and end times of the event, in conjunction with the known location of the observer and the object's orbit, the length of the chord can be determined giving an indication of the size of the occulting object. By combining observations made from several different locations, multiple chords crossing the occulting object can be determined giving a more accurate shape and size model. This technique of using multiple observers during the same event has been used to derive more sophisticated shape models for asteroids, whose shape can be highly irregular. A notable example of this occurred in 2002 when the asteroid 345 Tercidina underwent a stellar occultation of a very bright star as seen from Europe. During this event a team of at least 105 observers recorded 75 chords across the asteroid's surface allowing for a very accurate size and shape determination.
Chord (concurrency)
A chord is a concurrency construct available in Polyphonic C♯ and Cω inspired by the join pattern of the join-calculus. A chord is a function body that is associated with multiple function headers and cannot execute until all function headers are called.
Synchronicity
Cω defines two types of functions synchronous and asynchronous. A synchronous function acts like a standard function in most Object-Oriented Language, upon invocation the function body is executed and a return value may or may not be returned to the caller. An asynchronous function acts similar to a function that returns void except that it is guaranteed to return immediately with the execution being done in a separate thread.
References
DATA (band)
DATA were an electronic music band created in the late 1970s by Georg Kajanus, creator of such bands as Eclection, Sailor and Noir (with Tim Dry of the robotic/music duo Tik and Tok). After the break-up of Sailor in the late 1970s, Kajanus decided to experiment with electronic music and formed DATA, together with vocalists Francesca ("Frankie") and Phillipa ("Phil") Boulter, daughters of British singer John Boulter.
The classically orientated title track of DATA’s first album, Opera Electronica, was used as the theme music to the short film, Towers of Babel (1981), which was directed by Jonathan Lewis and starred Anna Quayle and Ken Campbell. Towers of Babel was nominated for a BAFTA award in 1982 and won the Silver Hugo Award for Best Short Film at the Chicago International Film Festival of the same year.
DATA released two more albums, the experimental 2-Time (1983) and the Country & Western-inspired electronica album Elegant Machinery (1985). The title of the last album was the inspiration for the name of Swedish pop synth group, elegant MACHINERY, formerly known as Pole Position.
Data (word)
The word data has generated considerable controversy on if it is a singular, uncountable noun, or should be treated as the plural of the now-rarely-used datum.
Usage in English
In one sense, data is the plural form of datum. Datum actually can also be a count noun with the plural datums (see usage in datum article) that can be used with cardinal numbers (e.g. "80 datums"); data (originally a Latin plural) is not used like a normal count noun with cardinal numbers and can be plural with such plural determiners as these and many or as a singular abstract mass noun with a verb in the singular form. Even when a very small quantity of data is referenced (one number, for example) the phrase piece of data is often used, as opposed to datum. The debate over appropriate usage continues, but "data" as a singular form is far more common.
In English, the word datum is still used in the general sense of "an item given". In cartography, geography, nuclear magnetic resonance and technical drawing it is often used to refer to a single specific reference datum from which distances to all other data are measured. Any measurement or result is a datum, though data point is now far more common.
Data URI scheme
The data URI scheme is a uniform resource identifier (URI) scheme that provides a way to include data in-line in web pages as if they were external resources. It is a form of file literal or here document. This technique allows normally separate elements such as images and style sheets to be fetched in a single Hypertext Transfer Protocol (HTTP) request, which may be more efficient than multiple HTTP requests. Data URIs are sometimes referred to incorrectly as "data URLs". As of 2015, data URIs are fully supported by most major browsers, and partially supported in Internet Explorer and Microsoft Edge.
Syntax
The syntax of data URIs was defined in Request for Comments (RFC) 2397, published in August 1998, and follows the URI scheme syntax. A data URI consists of:
data. It is followed by a colon (:).text/plain.;) . A character set parameter comprises the label charset, an equals sign (=), and a value from the IANA list of official character set names. If this parameter is not present, the character set of the content is assumed to be US-ASCII (ASCII).
Wide Prairie
Wide Prairie is a posthumous compilation by Linda McCartney. The album was compiled and released in 1998 by Paul McCartney after his wife's death, after a fan wrote in enquiring about "Seaside Woman"; a reggae beat type song which Wings had recorded in 1972, under the name Suzy and the Red Stripes, featuring Linda on lead vocals. Her husband compiled all her recordings with the help of Parlophone Records and MPL Communications. Lead guitar on the song "The Light Comes from Within" is played by the McCartneys' son, musician/sculptor James McCartney. The album reached number 127 in the UK charts, while the title track made the top 75, at number 74. "The Light Comes from Within" also charted, at number 56 in the UK charts.
Track listing
All songs by Linda McCartney, except where noted.
- Recorded by Wings on 20 November 1973 in Paris, and in June 1974 in Nashville, Tennessee.
- Linda McCartney – Lead vocals, Mellotron, Backing vocals
- Paul McCartney – Bass, Piano, Rhodes, Electric organ, Backing vocals
- Jimmy McCulloch – Electric guitar
- Denny Laine – Acoustic guitar
- Davey Lutton – Drums
- Vassar Clements and Johnny Gimble – Fiddles
- Thaddeus Richard – Alto saxophone
- Hewlett Quillen – Trombone
- William Puett – Tenor saxophone
- George Tidwell and Barry McDonald – Trumpets
- Norman Ray – Baritone saxophone
