Misplaced Pages

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52-511: From which the solution x = y = 0 is excluded. In polar coordinates , the Kampyle has the equation Equivalently, it has a parametric representation as This quartic curve was studied by the Greek astronomer and mathematician Eudoxus of Cnidus (c. 408 BC – c.347 BC) in relation to the classical problem of doubling the cube . The Kampyle is symmetric about both the x - and y -axes. It crosses

104-406: 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

156-508: 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

208-417: A circle's center point is the circle's diameter . Among properties of chords of a circle are the following: The midpoints of a set of parallel chords of a conic are collinear ( midpoint theorem for conics ). Chords were used extensively in the early development of trigonometry . The first known trigonometric table, compiled by Hipparchus in the 2nd century BC, is no longer extant but tabulated

260-450: 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 }

312-594: 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

364-415: 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

416-739: 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

468-499: 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}

520-408: 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

572-416: 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

SECTION 10

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624-590: 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

676-548: 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

728-670: 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

780-517: 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

832-895: 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

884-467: 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

936-798: 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

988-401: 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

1040-432: 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

1092-1595: 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

SECTION 20

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1144-1541: 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

1196-468: 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

1248-419: 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 )

1300-663: 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 )}

1352-530: 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

1404-400: 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 ) is called the pole , and the ray from the pole in the reference direction is the polar axis . The distance from

1456-526: 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

1508-507: The x -axis at (± a ,0). It has inflection points at (four inflections, one in each quadrant). The top half of the curve is asymptotic to x 2 / a − a / 2 {\displaystyle x^{2}/a-a/2} as x → ∞ {\displaystyle x\to \infty } , and in fact can be written as where is the n {\displaystyle n} th Catalan number . Polar coordinates In mathematics ,

1560-1213: 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

1612-533: The Latin chorda , meaning " bowstring ") of a circle is a straight line segment whose endpoints both lie on a circular arc . If a chord were to be extended infinitely on both directions into a line , the object is a secant line . The perpendicular line passing through the chord's midpoint is called sagitta (Latin for "arrow"). More generally, a chord is a line segment joining two points on any curve , for instance, on an ellipse . A chord that passes through

Kampyle of Eudoxus - Misplaced Pages Continue

1664-503: 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

1716-402: 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

1768-478: 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

1820-625: 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

1872-855: 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

1924-451: 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

1976-406: The integer part. The chord function is defined geometrically as shown in the picture. The chord of an angle is the length of the chord between two points on a unit circle separated by that central angle . The angle θ is taken in the positive sense and must lie in the interval 0 < θ ≤ π (radian measure). The chord function can be related to the modern sine function, by taking one of

2028-431: 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

2080-2863: 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

2132-401: 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 a center point in

Kampyle of Eudoxus - Misplaced Pages Continue

2184-430: 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

2236-405: The points to be (1,0), and the other point to be ( cos θ , sin θ ), and then using the Pythagorean theorem to calculate the chord length: 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,

2288-407: 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

2340-426: 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

2392-412: 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 the concepts in

2444-448: 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

2496-422: The region enclosed by a curve r ( φ ) and the rays φ = a and φ = b , where 0 < b − a ≤ 2 π . Then, the area of R is 1 2 ∫ a b [ r ( φ ) ] 2 d φ . {\displaystyle {\frac {1}{2}}\int _{a}^{b}\left[r(\varphi )\right]^{2}\,d\varphi .} Chord (geometry) A chord (from

2548-809: 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

2600-420: 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

2652-459: The value of the chord function for every ⁠7 + 1 / 2 ⁠ degrees . In the 2nd century AD, Ptolemy compiled a more extensive table of chords in his book on astronomy , giving the value of the chord for angles ranging from ⁠ 1 / 2 ⁠ to 180 degrees by increments of ⁠ 1 / 2 ⁠ degree. Ptolemy used a circle of diameter 120, and gave chord lengths accurate to two sexagesimal (base sixty) digits after

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2704-601: 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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