In mathematics , a spiral is a curve which emanates from a point, moving further away as it revolves around the point. It is a subtype of whorled patterns, a broad group that also includes concentric objects .
89-508: Two major definitions of "spiral" in the American Heritage Dictionary are: The first definition describes a planar curve, that extends in both of the perpendicular directions within its plane; the groove on one side of a gramophone record closely approximates a plane spiral (and it is by the finite width and depth of the groove, but not by the wider spacing between than within tracks, that it falls short of being
178-414: A φ n {\displaystyle r(\varphi )=a\varphi ^{n}} (Archimedean, hyperbolic, Fermat's, lituus spirals) and the logarithmic spiral r = a e k φ {\displaystyle r=ae^{k\varphi }} . The angle α {\displaystyle \alpha } between the spiral tangent and the corresponding polar circle (see diagram)
267-570: A φ {\displaystyle \;r(\varphi )=a\varphi \;} one gets the conical spiral (see diagram) Any cylindrical map projection can be used as the basis for a spherical spiral : draw a straight line on the map and find its inverse projection on the sphere, a kind of spherical curve . One of the most basic families of spherical spirals is the Clelia curves , which project to straight lines on an equirectangular projection . These are curves for which longitude and colatitude are in
356-462: A ( φ 2 + 1 ) 3 / 2 {\displaystyle \kappa ={\tfrac {\varphi ^{2}+2}{a(\varphi ^{2}+1)^{3/2}}}} . Only for − 1 < n < 0 {\displaystyle -1<n<0} the spiral has an inflection point . The curvature of a logarithmic spiral r = a e k φ {\displaystyle \;r=ae^{k\varphi }\;}
445-404: A ( arctan ( k φ ) + π / 2 ) {\displaystyle \;r=a(\arctan(k\varphi )+\pi /2)\;} and k = 0.2 , a = 2 , − ∞ < φ < ∞ {\displaystyle \;k=0.2,a=2,\;-\infty <\varphi <\infty \;} one gets a spiral, that approaches
534-406: 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 : Because of the circular nature of
623-483: A logarithmic spiral r = a e k φ {\displaystyle \;r=ae^{k\varphi }\;} is L = k 2 + 1 k ( r ( φ 2 ) − r ( φ 1 ) ) . {\displaystyle \ L={\tfrac {\sqrt {k^{2}+1}}{k}}{\big (}r(\varphi _{2})-r(\varphi _{1}){\big )}\ .} The inversion at
712-453: A logarithmic spiral r = a e k φ {\displaystyle \;r=ae^{k\varphi }\;} is A = r ( φ 2 ) 2 − r ( φ 1 ) 2 ) 4 k . {\displaystyle \ A={\tfrac {r(\varphi _{2})^{2}-r(\varphi _{1})^{2})}{4k}}\ .} The length of an arc of
801-434: 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 and spherical coordinate systems. The concepts of angle and radius were already used by ancient peoples of
890-508: A circle with a center at ( r 0 , γ ) {\displaystyle (r_{0},\gamma )} and radius a is r 2 − 2 r r 0 cos ( φ − γ ) + r 0 2 = a 2 . {\displaystyle r^{2}-2rr_{0}\cos(\varphi -\gamma )+r_{0}^{2}=a^{2}.} This can be simplified in various ways, to conform to more specific cases, such as
979-456: A circle-inversion (see below). The name logarithmic spiral is due to the equation φ = 1 k ⋅ ln r a {\displaystyle \varphi ={\tfrac {1}{k}}\cdot \ln {\tfrac {r}{a}}} . Approximations of this are found in nature. Spirals which do not fit into this scheme of the first 5 examples: A Cornu spiral has two asymptotic points. The spiral of Theodorus
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#17327723036701068-458: A closed curve around a fixed axis: the shape of the curve remains fixed, but its size grows in a geometric progression . In some shells, such as Nautilus and ammonites , the generating curve revolves in a plane perpendicular to the axis and the shell will form a planar discoid shape. In others it follows a skew path forming a helico -spiral pattern. Thompson also studied spirals occurring in horns , teeth , claws and plants . A model for
1157-450: A curve best defined by a polar equation. 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: r = ℓ 1 − e cos φ {\displaystyle r={\ell \over {1-e\cos \varphi }}} where e is the eccentricity and ℓ {\displaystyle \ell }
1246-409: A curve with polar equation r = r ( φ ) {\displaystyle r=r(\varphi )} is For a spiral with r = a φ n {\displaystyle r=a\varphi ^{n}} one gets In case of n = 1 {\displaystyle n=1} (Archimedean spiral) κ = φ 2 + 2
1335-423: A curve with polar equation r = r ( φ ) {\displaystyle r=r(\varphi )} is For the spiral r = a φ n {\displaystyle r=a\varphi ^{n}\;} the length is Not all these integrals can be solved by a suitable table. In case of a Fermat's spiral, the integral can be expressed by elliptic integrals only. The arc length of
1424-598: 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
1513-415: 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
1602-739: 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 < 2 π . The length of L is given by the following integral L = ∫ a b [ r ( φ ) ] 2 + [ d r ( φ ) d φ ] 2 d φ {\displaystyle L=\int _{a}^{b}{\sqrt {\left[r(\varphi )\right]^{2}+\left[{\tfrac {dr(\varphi )}{d\varphi }}\right]^{2}}}d\varphi } Let R denote
1691-551: A linear relationship, analogous to Archimedean spirals in the plane; under the azimuthal equidistant projection a Clelia curve projects to a planar Archimedean spiral. If one represents a unit sphere by spherical coordinates then setting the linear dependency φ = c θ {\displaystyle \varphi =c\theta } for the angle coordinates gives a parametric curve in terms of parameter θ {\displaystyle \theta } , Another family of spherical spirals
1780-533: A loxodrome projects to a logarithmic spiral in the plane. The study of spirals in nature has a long history. Christopher Wren observed that many shells form a logarithmic spiral ; Jan Swammerdam observed the common mathematical characteristics of a wide range of shells from Helix to Spirula ; and Henry Nottidge Moseley described the mathematics of univalve shells. D’Arcy Wentworth Thompson 's On Growth and Form gives extensive treatment to these spirals. He describes how shells are formed by rotating
1869-501: A million-word, three-line citation database prepared by Brown University linguist Henry Kučera . For expert consultation on words or constructions whose usage was controversial or problematic, the American Heritage Dictionary relied on the advice of a usage panel. In its final form, the panel comprised nearly 200 prominent members of professions whose work demanded sensitivity to language. Former members of
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#17327723036701958-499: A minus sign in front of the square root gives the same curve. Radial lines (those running through the pole) are represented by the equation φ = γ , {\displaystyle \varphi =\gamma ,} where γ {\displaystyle \gamma } is the angle of elevation of the line; that is, φ = arctan m {\displaystyle \varphi =\arctan m} , where m {\displaystyle m}
2047-507: A perfect example); note that successive loops differ in diameter. In another example, the "center lines" of the arms of a spiral galaxy trace logarithmic spirals . The second definition includes two kinds of 3-dimensional relatives of spirals: In the side picture, the black curve at the bottom is an Archimedean spiral , while the green curve is a helix. The curve shown in red is a conical spiral. A two-dimensional , or plane, spiral may be easily described using polar coordinates , where
2136-408: A phase angle. 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 r ( φ ) = a + b φ . {\displaystyle r(\varphi )=a+b\varphi .} Changing the parameter a will turn the spiral, while b controls the distance between the arms, which for
2225-590: 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 z = x + i y {\displaystyle z=x+iy} where i
2314-553: 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
2403-675: A unique azimuth for the pole ( r = 0) must be chosen, e.g., φ = 0. The polar coordinates r and φ can be converted to the Cartesian coordinates x and y by using the trigonometric functions sine and cosine: x = r cos φ , y = r sin φ . {\displaystyle {\begin{aligned}x&=r\cos \varphi ,\\y&=r\sin \varphi .\end{aligned}}} The Cartesian coordinates x and y can be converted to polar coordinates r and φ with r ≥ 0 and φ in
2492-494: Is κ = 1 r 1 + k 2 . {\displaystyle \;\kappa ={\tfrac {1}{r{\sqrt {1+k^{2}}}}}\;.} The area of a sector of a curve (see diagram) with polar equation r = r ( φ ) {\displaystyle r=r(\varphi )} is For a spiral with equation r = a φ n {\displaystyle r=a\varphi ^{n}\;} one gets The formula for
2581-517: 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 : z = r c i s φ = r ∠ φ . {\displaystyle z=r\operatorname {\mathrm {cis} } \varphi =r\angle \varphi .} For
2670-411: Is a "comprehensive update" of the 2011 edition, containing "... [t]housands of revisions to definitions and etymologies, 150 new words and senses, and new usage advice ...." The various printings of the 5th edition are available in hardcover and, with reduced print size and smaller page count, trade paperback form. The 5th edition dropped several of the supplementary features of the fourth edition, and
2759-895: 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 2 r θ π {\displaystyle {\frac {2r\theta }{\pi }}} , the curve is given by ρ ( θ ) = 2 r θ π sin θ {\displaystyle \rho (\theta )={\frac {2r\theta }{\pi \sin \theta }}} . The graphs of two polar functions r = f ( θ ) {\displaystyle r=f(\theta )} and r = g ( θ ) {\displaystyle r=g(\theta )} have possible intersections of three types: Calculus can be applied to equations expressed in polar coordinates. The angular coordinate φ
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2848-428: Is a polygon. The Fibonacci Spiral consists of a sequence of circle arcs. The involute of a circle looks like an Archimedean, but is not: see Involute#Examples . The following considerations are dealing with spirals, which can be described by a polar equation r = r ( φ ) {\displaystyle r=r(\varphi )} , especially for the cases r ( φ ) =
2937-510: Is also the name given to spiral shaped fingerprints . A spiral like form has been found in Mezine , Ukraine , as part of a decorative object dated to 10,000 BCE. Spiral and triple spiral motifs served as Neolithic symbols in Europe ( Megalithic Temples of Malta ). The Celtic triple-spiral is in fact a pre-Celtic symbol. It is carved into the rock of a stone lozenge near the main entrance of
3026-406: Is also used for a series of American history books. Polar coordinate system#Vector calculus In mathematics , the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian coordinate system )
3115-467: Is an integer, these equations will produce a k -petaled rose if k is odd , or a 2 k -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 γ 0 can be regarded as
3204-798: Is calculated first as above, then this formula for φ may be stated more simply using the arccosine function: φ = { arccos ( x r ) if y ≥ 0 and r ≠ 0 − arccos ( x r ) if y < 0 undefined if r = 0. {\displaystyle \varphi ={\begin{cases}\arccos \left({\frac {x}{r}}\right)&{\mbox{if }}y\geq 0{\mbox{ and }}r\neq 0\\-\arccos \left({\frac {x}{r}}\right)&{\mbox{if }}y<0\\{\text{undefined}}&{\mbox{if }}r=0.\end{cases}}} Every complex number can be represented as
3293-448: Is called angle of the polar slope and tan α {\displaystyle \tan \alpha } the polar slope . From vector calculus in polar coordinates one gets the formula Hence the slope of the spiral r = a φ n {\displaystyle \;r=a\varphi ^{n}\;} is In case of an Archimedean spiral ( n = 1 {\displaystyle n=1} )
3382-512: Is called the pole , and the ray from the pole in the reference direction is the polar axis . 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 . Angles in polar notation are generally expressed in either degrees or radians ( π rad being equal to 180° and 2 π rad being equal to 360°). Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced
3471-401: 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
3560-432: 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. 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
3649-1595: Is expressed in radians throughout this section, which is the conventional choice when doing 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 r d u d r = r ∂ u ∂ x cos φ + r ∂ u ∂ y sin φ = x ∂ u ∂ x + y ∂ u ∂ y , d u d φ = − ∂ u ∂ x r sin φ + ∂ u ∂ y r cos φ = − y ∂ u ∂ x + x ∂ u ∂ y . {\displaystyle {\begin{aligned}r{\frac {du}{dr}}&=r{\frac {\partial u}{\partial x}}\cos \varphi +r{\frac {\partial u}{\partial y}}\sin \varphi =x{\frac {\partial u}{\partial x}}+y{\frac {\partial u}{\partial y}},\\[2pt]{\frac {du}{d\varphi }}&=-{\frac {\partial u}{\partial x}}r\sin \varphi +{\frac {\partial u}{\partial y}}r\cos \varphi =-y{\frac {\partial u}{\partial x}}+x{\frac {\partial u}{\partial y}}.\end{aligned}}} Hence, we have
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3738-1139: Is not available with a disc-based electronic version. The university-student version was renamed The American Heritage College Writer's Dictionary in 2013, and stripped of biographical and geographical entries to make room for more vocabulary while simultaneously reducing the number of pages compared to the fourth college edition. The AHD inserts minor revisions (such as a biographical entry, with photograph, for each newly elected U.S. president) in successive printings of any given edition. Supporting volumes have been issued, including The American Heritage Book of English Usage , The American Heritage Dictionary of Indo-European Roots , The American Heritage Abbreviations Dictionary , The American Heritage Dictionary of Idioms , The American Heritage Thesaurus in various sizes; usage dictionaries of special vocabulary such as The American Heritage Science Dictionary , The American Heritage Medical Dictionary and The American Heritage Dictionary of Business Terms ; plus special dictionary editions for children, high-school students, and English-language learners. The American Heritage brand
3827-1547: Is the Pythagorean sum and atan2 is a common variation on the arctangent function defined as atan2 ( y , x ) = { arctan ( y x ) if x > 0 arctan ( y x ) + π if x < 0 and y ≥ 0 arctan ( y x ) − π if x < 0 and y < 0 π 2 if x = 0 and y > 0 − π 2 if x = 0 and y < 0 undefined if x = 0 and y = 0. {\displaystyle \operatorname {atan2} (y,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&{\mbox{if }}x>0\\\arctan \left({\frac {y}{x}}\right)+\pi &{\mbox{if }}x<0{\mbox{ and }}y\geq 0\\\arctan \left({\frac {y}{x}}\right)-\pi &{\mbox{if }}x<0{\mbox{ and }}y<0\\{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y>0\\-{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y<0\\{\text{undefined}}&{\mbox{if }}x=0{\mbox{ and }}y=0.\end{cases}}} If r
3916-468: Is the imaginary unit , or can alternatively be written in polar form as z = r ( cos φ + i sin φ ) {\displaystyle z=r(\cos \varphi +i\sin \varphi )} and from there, by Euler's formula , as z = r e i φ = r exp i φ . {\displaystyle z=re^{i\varphi }=r\exp i\varphi .} where e
4005-456: Is the rhumb lines or loxodromes, that project to straight lines on the Mercator projection . These are the trajectories traced by a ship traveling with constant bearing . Any loxodrome (except for the meridians and parallels) spirals infinitely around either pole, closer and closer each time, unlike a Clelia curve which maintains uniform spacing in colatitude. Under stereographic projection ,
4094-419: 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 ℓ {\displaystyle \ell } . A quadratrix in the first quadrant ( x, y )
4183-663: Is the slope of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line φ = γ {\displaystyle \varphi =\gamma } perpendicularly at the point ( r 0 , γ ) {\displaystyle (r_{0},\gamma )} has the equation r ( φ ) = r 0 sec ( φ − γ ) . {\displaystyle r(\varphi )=r_{0}\sec(\varphi -\gamma ).} Otherwise stated ( r 0 , γ ) {\displaystyle (r_{0},\gamma )}
4272-530: Is the point in which the tangent intersects the imaginary circle of radius r 0 {\displaystyle r_{0}} A polar rose is a mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation, r ( φ ) = a cos ( k φ + γ 0 ) {\displaystyle r(\varphi )=a\cos \left(k\varphi +\gamma _{0}\right)} for any constant γ 0 (including 0). If k
4361-738: The x {\displaystyle x} - y {\displaystyle y} -plane a spiral with parametric representation is given, then there can be added a third coordinate z ( φ ) {\displaystyle z(\varphi )} , such that the now space curve lies on the cone with equation m ( x 2 + y 2 ) = ( z − z 0 ) 2 , m > 0 {\displaystyle \;m(x^{2}+y^{2})=(z-z_{0})^{2}\ ,\ m>0\;} : Spirals based on this procedure are called conical spirals . Starting with an archimedean spiral r ( φ ) =
4450-554: The Webster's Third New International Dictionary (1961). A college dictionary followed several years later. The main dictionary became the flagship title as the brand grew into a family of various dictionaries, a dictionary-thesaurus combination, and a usage guide. James Parton (1912–2001) was a grandson of the English-born American biographer James Parton (1822–1891). He was the founder, publisher and co-owner of
4539-496: The AHD that are understood to have evolved from them. These entries might be called "reverse etymologies": the ag- entry there, for instance, lists 49 terms derived from it, words as diverse as agent , essay , purge , stratagem , ambassador , axiom , and pellagra , along with information about varying routes through intermediate transformations on the way to the contemporary words. A compacted American Heritage College Dictionary
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#17327723036704628-588: The G. and C. Merriam Company so that he could undo the changes. When that failed, he contracted with Houghton to publish a new dictionary. The AHD was edited by William Morris and relied on a usage panel of 105 writers, speakers, and eminent persons chosen for their well-known conservatism in the use of language. However, Morris made inconsistent use of the panels, often ignoring their advice and inserting his own opinions. The AHD broke ground among dictionaries by using corpus linguistics for compiling word frequencies and other information. Citations were based on
4717-523: The heraldic emblem on warriors' shields depicted on Greek pottery. American Heritage Dictionary The American Heritage Dictionary of the English Language ( AHD ) is a dictionary of American English published by HarperCollins . It is currently in its fifth edition (since 2011). Before HarperCollins acquired certain business lines from Houghton Mifflin Harcourt in 2022,
4806-402: The radius r {\displaystyle r} is a monotonic continuous function of angle φ {\displaystyle \varphi } : The circle would be regarded as a degenerate case (the function not being strictly monotonic, but rather constant ). In x {\displaystyle x} - y {\displaystyle y} -coordinates
4895-538: 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
4984-458: The style guide The Sense of Style , was its final chair. The members of the panel were sent regular ballots asking about matters of usage; the completed ballots were returned and tabulated, and the results formed the basis for special usage notes appended to the relevant dictionary entries. In many cases, these notes not only reported the percentage of panelists who considered a given usage or construction to be acceptable, but would also report
5073-1213: 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 . x = r ( φ ) cos φ y = r ( φ ) sin φ {\displaystyle {\begin{aligned}x&=r(\varphi )\cos \varphi \\y&=r(\varphi )\sin \varphi \end{aligned}}} Differentiating both equations with respect to φ yields d x d φ = r ′ ( φ ) cos φ − r ( φ ) sin φ d y d φ = r ′ ( φ ) sin φ + r ( φ ) cos φ . {\displaystyle {\begin{aligned}{\frac {dx}{d\varphi }}&=r'(\varphi )\cos \varphi -r(\varphi )\sin \varphi \\[2pt]{\frac {dy}{d\varphi }}&=r'(\varphi )\sin \varphi +r(\varphi )\cos \varphi .\end{aligned}}} Dividing
5162-410: The choice k = 0.1 , a = 4 , φ ≥ 0 {\displaystyle \;k=0.1,a=4,\;\varphi \geq 0\;} gives a spiral, that starts at the origin (like an Archimedean spiral) and approaches the circle with radius r = a π / 2 {\displaystyle \;r=a\pi /2\;} (diagram, left). For r =
5251-399: The 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 the introduction of the polar system was the study of circular and orbital motion . Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from
5340-503: 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
5429-408: The curve has the parametric representation: Some of the most important sorts of two-dimensional spirals include: An Archimedean spiral is, for example, generated while coiling a carpet. A hyperbolic spiral appears as image of a helix with a special central projection (see diagram). A hyperbolic spiral is some times called reciproke spiral, because it is the image of an Archimedean spiral with
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#17327723036705518-402: The equation r ( φ ) = a {\displaystyle r(\varphi )=a} for a circle with a center at the pole and radius a . When r 0 = a or the origin lies on the circle, the equation becomes r = 2 a cos ( φ − γ ) . {\displaystyle r=2a\cos(\varphi -\gamma ).} In
5607-399: The family of American Heritage dictionaries had long been published by Houghton Mifflin Harcourt and its predecessor Houghton Mifflin. The first edition appeared in 1969, an outgrowth of the editorial effort for Houghton Mifflin's American Heritage brand of history books and journals. The dictionary's creation was spurred by the controversy during the 1960s over the perceived permissiveness of
5696-484: 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
5785-625: The following formula: r d d r = x ∂ ∂ x + y ∂ ∂ y d d φ = − y ∂ ∂ x + x ∂ ∂ y . {\displaystyle {\begin{aligned}r{\frac {d}{dr}}&=x{\frac {\partial }{\partial x}}+y{\frac {\partial }{\partial y}}\\[2pt]{\frac {d}{d\varphi }}&=-y{\frac {\partial }{\partial x}}+x{\frac {\partial }{\partial y}}.\end{aligned}}} Using
5874-855: The following formulae: d d x = cos φ ∂ ∂ r − 1 r sin φ ∂ ∂ φ d d y = sin φ ∂ ∂ r + 1 r cos φ ∂ ∂ φ . {\displaystyle {\begin{aligned}{\frac {d}{dx}}&=\cos \varphi {\frac {\partial }{\partial r}}-{\frac {1}{r}}\sin \varphi {\frac {\partial }{\partial \varphi }}\\[2pt]{\frac {d}{dy}}&=\sin \varphi {\frac {\partial }{\partial r}}+{\frac {1}{r}}\cos \varphi {\frac {\partial }{\partial \varphi }}.\end{aligned}}} To find
5963-451: The general case, the equation can be solved for r , giving r = r 0 cos ( φ − γ ) + a 2 − r 0 2 sin 2 ( φ − γ ) {\displaystyle r=r_{0}\cos(\varphi -\gamma )+{\sqrt {a^{2}-r_{0}^{2}\sin ^{2}(\varphi -\gamma )}}} The solution with
6052-431: The interval (− π , π ] by: r = x 2 + y 2 = hypot ( x , y ) φ = atan2 ( y , x ) , {\displaystyle {\begin{aligned}r&={\sqrt {x^{2}+y^{2}}}=\operatorname {hypot} (x,y)\\\varphi &=\operatorname {atan2} (y,x),\end{aligned}}} where hypot
6141-2863: The inverse coordinates transformation, an analogous reciprocal relationship can be derived between the derivatives. Given a function u ( r , φ ), it follows that d u d x = ∂ u ∂ r ∂ r ∂ x + ∂ u ∂ φ ∂ φ ∂ x , d u d y = ∂ u ∂ r ∂ r ∂ y + ∂ u ∂ φ ∂ φ ∂ y , {\displaystyle {\begin{aligned}{\frac {du}{dx}}&={\frac {\partial u}{\partial r}}{\frac {\partial r}{\partial x}}+{\frac {\partial u}{\partial \varphi }}{\frac {\partial \varphi }{\partial x}},\\[2pt]{\frac {du}{dy}}&={\frac {\partial u}{\partial r}}{\frac {\partial r}{\partial y}}+{\frac {\partial u}{\partial \varphi }}{\frac {\partial \varphi }{\partial y}},\end{aligned}}} or d u d x = ∂ u ∂ r x x 2 + y 2 − ∂ u ∂ φ y x 2 + y 2 = cos φ ∂ u ∂ r − 1 r sin φ ∂ u ∂ φ , d u d y = ∂ u ∂ r y x 2 + y 2 + ∂ u ∂ φ x x 2 + y 2 = sin φ ∂ u ∂ r + 1 r cos φ ∂ u ∂ φ . {\displaystyle {\begin{aligned}{\frac {du}{dx}}&={\frac {\partial u}{\partial r}}{\frac {x}{\sqrt {x^{2}+y^{2}}}}-{\frac {\partial u}{\partial \varphi }}{\frac {y}{x^{2}+y^{2}}}\\[2pt]&=\cos \varphi {\frac {\partial u}{\partial r}}-{\frac {1}{r}}\sin \varphi {\frac {\partial u}{\partial \varphi }},\\[2pt]{\frac {du}{dy}}&={\frac {\partial u}{\partial r}}{\frac {y}{\sqrt {x^{2}+y^{2}}}}+{\frac {\partial u}{\partial \varphi }}{\frac {x}{x^{2}+y^{2}}}\\[2pt]&=\sin \varphi {\frac {\partial u}{\partial r}}+{\frac {1}{r}}\cos \varphi {\frac {\partial u}{\partial \varphi }}.\end{aligned}}} Hence, we have
6230-449: The magazines American Heritage and Horizon , and was appalled by the perceived permissiveness of Webster's Third , published in 1961. (Webster's Third presented all entries without labeling any as nonstandard or informal, or so it was widely claimed. In fact, the dictionary did apply the labels slang , substandard and nonstandard , but in the view of critics, not often enough and with insufficient disapproval.) Parton tried to buy
6319-534: The normally expected etymologies, which for instance trace the word ambiguous to a Proto-Indo-European root ag- , meaning "to drive", the appendices included a seven-page article by Professor Calvert Watkins entitled "Indo-European and the Indo-Europeans" and "Indo-European Roots", 46 pages of entries that are each organized around one of some thousand Proto-Indo-European roots and the English words of
6408-430: 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: 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
6497-466: The origin (like a hyperbolic spiral) and approaches the circle with radius r = a π {\displaystyle \;r=a\pi \;} (diagram, right). Two well-known spiral space curves are conical spirals and spherical spirals , defined below. Another instance of space spirals is the toroidal spiral . A spiral wound around a helix, also known as double-twisted helix , represents objects such as coiled coil filaments . If in
6586-408: The pattern of florets in the head of a sunflower was proposed by H. Vogel. This has the form where n is the index number of the floret and c is a constant scaling factor, and is a form of Fermat's spiral . The angle 137.5° is the golden angle which is related to the golden ratio and gives a close packing of florets. Spirals in plants and animals are frequently described as whorls . This
6675-407: 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 + (2 n + 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
6764-426: 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. The general equation for
6853-425: The polar slope is tan α = 1 φ . {\displaystyle \;\tan \alpha ={\tfrac {1}{\varphi }}\ .} In a logarithmic spiral , tan α = k {\displaystyle \ \tan \alpha =k\ } is constant. The curvature κ {\displaystyle \kappa } of
6942-448: 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
7031-631: The prehistoric Newgrange monument in County Meath , Ireland . Newgrange was built around 3200 BCE, predating the Celts; triple spirals were carved at least 2,500 years before the Celts reached Ireland, but have long since become part of Celtic culture. The triskelion symbol, consisting of three interlocked spirals or three bent human legs, appears in many early cultures: examples include Mycenaean vessels, coinage from Lycia , staters of Pamphylia (at Aspendos , 370–333 BC) and Pisidia , as well as
7120-656: The results from balloting of the same question in past decades, to give a clearer sense of how the language changes over time. Houghton Mifflin dissolved the usage panel on February 1, 2018, citing the decline in demand for print dictionaries. The AHD is also somewhat innovative in its liberal use of photographic illustrations, which at the time was highly unusual for general reference dictionaries, many of which went largely or completely unillustrated. It also has an unusually large number of biographical entries for notable persons. The first edition appeared in 1969, highly praised for its Indo-European etymologies . In addition to
7209-809: The second equation by the first yields the Cartesian slope of the tangent line to the curve at the point ( r ( φ ), φ ) : d y d x = r ′ ( φ ) sin φ + r ( φ ) cos φ r ′ ( φ ) cos φ − r ( φ ) sin φ . {\displaystyle {\frac {dy}{dx}}={\frac {r'(\varphi )\sin \varphi +r(\varphi )\cos \varphi }{r'(\varphi )\cos \varphi -r(\varphi )\sin \varphi }}.} For other useful formulas including divergence, gradient, and Laplacian in polar coordinates, see curvilinear coordinates . The arc length (length of
7298-506: The standard spirals r ( φ ) {\displaystyle r(\varphi )} is either a power function or an exponential function. If one chooses for r ( φ ) {\displaystyle r(\varphi )} a bounded function, the spiral is bounded, too. A suitable bounded function is the arctan function: Setting r = a arctan ( k φ ) {\displaystyle \;r=a\arctan(k\varphi )\;} and
7387-420: 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
7476-456: The unabridged Random House Dictionary of the English Language . A lower-priced college edition, also the fourth, was issued in black-and-white printing and with fewer illustrations, in 2002 (reprinted in 2007 and 2010). The fifth and most recent full edition was published in November 2011, with new printings in 2012 and 2016 and a 50th Anniversary Printing in 2018, which the publisher states
7565-441: The unit circle has in polar coordinates the simple description: ( r , φ ) ↦ ( 1 r , φ ) {\displaystyle \ (r,\varphi )\mapsto ({\tfrac {1}{r}},\varphi )\ } . The function r ( φ ) {\displaystyle r(\varphi )} of a spiral is usually strictly monotonic, continuous and un bounded . For
7654-595: The usage panel include novelists ( Isaac Asimov , Barbara Kingsolver , David Foster Wallace and Eudora Welty ), poets ( Rita Dove , Galway Kinnell , Mary Oliver and Robert Pinsky ), playwrights ( Terrence McNally and Marsha Norman ), journalists ( Liane Hansen and Susan Stamberg ), literary critics ( Harold Bloom ), columnists and commentators ( William F. Buckley, Jr. and Robert J. Samuelson ), linguists and cognitive scientists ( Anne Curzan , Steven Pinker and Calvert Watkins ) and humorists ( Garrison Keillor , David Sedaris and Alison Bechdel ). Pinker, author of
7743-478: The use of the linguistic data for other applications, such as electronic dictionaries. The third edition included over 350,000 entries and meanings. The fourth edition (2000, reissued in 2006) added an appendix of Semitic language etymological roots, and included color illustrations, and was also available with a CD-ROM edition in some versions. This revision was larger than a typical desk dictionary but smaller than Webster's Third New International Dictionary or
7832-479: Was first released in 1974. The first edition's concise successor, The American Heritage Dictionary, Second College Edition , was published in 1982 (without a larger-format version). It omitted the Indo-European etymologies, but they were reintroduced in the third full edition, published in 1992. The third edition was also a departure for the publisher because it was developed in a database, which facilitated
7921-601: Was the first to actually develop them. 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 . However, in mathematical literature the angle is often denoted by θ instead. Angles in polar notation are generally expressed in either degrees or radians (2 π 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
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