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55-546: Half Light may refer to: Effective radius or half-light radius Half Light (film) , a 2006 British mystery-horror drama Half-Light (album) , by Rostam, 2017 "Half Light" (Athlete song) , 2005 "Half Light" (Wilkinson song) , 2014 Half-light: Collected Poems 1965–2016 , by Frank Bidart, 2017 "Half Light", a song by Porcupine Tree from Deadwing "Half Light I" and Half Light II (No Celebration)", songs by Arcade Fire from The Suburbs Topics referred to by

110-454: A a 2 − x 2 = ± ( a 2 − x 2 ) ( 1 − e 2 ) . {\displaystyle y=\pm {\frac {b}{a}}{\sqrt {a^{2}-x^{2}}}=\pm {\sqrt {\left(a^{2}-x^{2}\right)\left(1-e^{2}\right)}}.} The width and height parameters a , b {\displaystyle a,\;b} are called

165-528: A 2 x 1 b 2 ) {\displaystyle {\begin{pmatrix}-y_{1}a^{2}&x_{1}b^{2}\end{pmatrix}}} is a tangent vector at point ( x 1 , y 1 ) {\displaystyle (x_{1},\,y_{1})} , which proves the vector equation. If ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} and ( u , v ) {\displaystyle (u,v)} are two points of

220-621: A 2 + y 1 v b 2 ) + s 2 ( u 2 a 2 + v 2 b 2 ) = 0   . {\displaystyle {\frac {\left(x_{1}+su\right)^{2}}{a^{2}}}+{\frac {\left(y_{1}+sv\right)^{2}}{b^{2}}}=1\ \quad \Longrightarrow \quad 2s\left({\frac {x_{1}u}{a^{2}}}+{\frac {y_{1}v}{b^{2}}}\right)+s^{2}\left({\frac {u^{2}}{a^{2}}}+{\frac {v^{2}}{b^{2}}}\right)=0\ .} There are then cases: Using (1) one finds that ( − y 1

275-549: A 2 + y 2 b 2 = 1. {\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1.} Assuming a ≥ b {\displaystyle a\geq b} , the foci are ( ± c , 0 ) {\displaystyle (\pm c,0)} for c = a 2 − b 2 {\textstyle c={\sqrt {a^{2}-b^{2}}}} . The standard parametric equation is: ( x , y ) = (

330-466: A 2 b 2 . {\displaystyle {\begin{aligned}A&=a^{2}\sin ^{2}\theta +b^{2}\cos ^{2}\theta &B&=2\left(b^{2}-a^{2}\right)\sin \theta \cos \theta \\[1ex]C&=a^{2}\cos ^{2}\theta +b^{2}\sin ^{2}\theta &D&=-2Ax_{\circ }-By_{\circ }\\[1ex]E&=-Bx_{\circ }-2Cy_{\circ }&F&=Ax_{\circ }^{2}+Bx_{\circ }y_{\circ }+Cy_{\circ }^{2}-a^{2}b^{2}.\end{aligned}}} These expressions can be derived from

385-542: A 2 cos 2 ⁡ θ + b 2 sin 2 ⁡ θ D = − 2 A x ∘ − B y ∘ E = − B x ∘ − 2 C y ∘ F = A x ∘ 2 + B x ∘ y ∘ + C y ∘ 2 −

440-462: A 2 − b 2 {\displaystyle c={\sqrt {a^{2}-b^{2}}}} . The eccentricity can be expressed as: e = c a = 1 − ( b a ) 2 , {\displaystyle e={\frac {c}{a}}={\sqrt {1-\left({\frac {b}{a}}\right)^{2}}},} assuming a > b . {\displaystyle a>b.} An ellipse with equal axes (

495-425: A ≥ b > 0   . {\displaystyle a\geq b>0\ .} In principle, the canonical ellipse equation x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1} may have a < b {\displaystyle a<b} (and hence

550-458: A ⋅ b {\displaystyle R_{\text{mean}}={\sqrt {a\cdot b}}} . For a circle, where a = b {\displaystyle a=b} , this simplifies to R mean = a {\displaystyle R_{\text{mean}}=a} . Applications: The volume of a sphere of radius R is 4 3 π R 3 {\displaystyle {\frac {4}{3}}\pi R^{3}} . Given

605-422: A ⋅ b ⋅ c 3 {\displaystyle R_{\text{mean}}={\sqrt[{3}]{a\cdot b\cdot c}}} . The formula for a rotational ellipsoid is the special case where a = b {\displaystyle a=b} . Likewise, an oblate spheroid or rotational ellipsoid with axes a {\displaystyle a} and c {\displaystyle c} has

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660-458: A + e x {\displaystyle a+ex} and a − e x {\displaystyle a-ex} . It follows from the equation that the ellipse is symmetric with respect to the coordinate axes and hence with respect to the origin. Throughout this article, the semi-major and semi-minor axes are denoted a {\displaystyle a} and b {\displaystyle b} , respectively, i.e.

715-596: A = b {\displaystyle a=b} ) has zero eccentricity, and is a circle. The length of the chord through one focus, perpendicular to the major axis, is called the latus rectum . One half of it is the semi-latus rectum ℓ {\displaystyle \ell } . A calculation shows: ℓ = b 2 a = a ( 1 − e 2 ) . {\displaystyle \ell ={\frac {b^{2}}{a}}=a\left(1-e^{2}\right).} The semi-latus rectum ℓ {\displaystyle \ell }

770-429: A cos ⁡ ( t ) , b sin ⁡ ( t ) ) for 0 ≤ t ≤ 2 π . {\displaystyle (x,y)=(a\cos(t),b\sin(t))\quad {\text{for}}\quad 0\leq t\leq 2\pi .} Ellipses are the closed type of conic section : a plane curve tracing the intersection of a cone with a plane (see figure). Ellipses have many similarities with

825-418: A parabola ). An ellipse has a simple algebraic solution for its area, but for its perimeter (also known as circumference ), integration is required to obtain an exact solution. Analytically , the equation of a standard ellipse centered at the origin with width 2 a {\displaystyle 2a} and height 2 b {\displaystyle 2b} is: x 2

880-422: A mean radius of R mean = a 2 ⋅ c 3 {\displaystyle R_{\text{mean}}={\sqrt[{3}]{a^{2}\cdot c}}} . For a sphere, where a = b = c {\displaystyle a=b=c} , this simplifies to R mean = a {\displaystyle R_{\text{mean}}=a} . Applications: The authalic radius

935-404: A perimeter of 4 L {\displaystyle 4L} . Setting that perimeter to be equal to that of a circle imply that Applications: The area of a circle of radius R is π R 2 {\displaystyle \pi R^{2}} . Given the area of a non-circular object A , one can calculate its area-equivalent radius by setting or, alternatively: Often

990-522: A point on an ellipse and x → = ( x 1 y 1 ) + s ( u v ) {\textstyle {\vec {x}}={\begin{pmatrix}x_{1}\\y_{1}\end{pmatrix}}+s{\begin{pmatrix}u\\v\end{pmatrix}}} be the equation of any line g {\displaystyle g} containing ( x 1 , y 1 ) {\displaystyle (x_{1},\,y_{1})} . Inserting

1045-458: Is a constant. This constant ratio is the above-mentioned eccentricity: e = c a = 1 − b 2 a 2 . {\displaystyle e={\frac {c}{a}}={\sqrt {1-{\frac {b^{2}}{a^{2}}}}}.} Ellipses are common in physics , astronomy and engineering . For example, the orbit of each planet in the Solar System

1100-525: Is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency: a similar effect leads to elliptical polarization of light in optics . The name, ἔλλειψις ( élleipsis , "omission"), was given by Apollonius of Perga in his Conics . An ellipse can be defined geometrically as a set or locus of points in the Euclidean plane: The midpoint C {\displaystyle C} of

1155-406: Is an surface area -equivalent radius for solid figures such as an ellipsoid. The osculating circle and osculating sphere define curvature -equivalent radii at a particular point of tangency for plane figures and solid figures, respectively. Ellipse In mathematics , an ellipse is a plane curve surrounding two focal points , such that for all points on the curve, the sum of

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1210-562: Is approximately an ellipse with the Sun at one focus point (more precisely, the focus is the barycenter of the Sun–planet pair). The same is true for moons orbiting planets and all other systems of two astronomical bodies. The shapes of planets and stars are often well described by ellipsoids . A circle viewed from a side angle looks like an ellipse: that is, the ellipse is the image of a circle under parallel or perspective projection . The ellipse

1265-416: Is different from Wikidata All article disambiguation pages All disambiguation pages Effective radius The perimeter of a circle of radius R is 2 π R {\displaystyle 2\pi R} . Given the perimeter of a non-circular object P , one can calculate its perimeter-equivalent radius by setting or, alternatively: For example, a square of side L has

1320-447: Is equal to the radius of curvature at the vertices (see section curvature ). An arbitrary line g {\displaystyle g} intersects an ellipse at 0, 1, or 2 points, respectively called an exterior line , tangent and secant . Through any point of an ellipse there is a unique tangent. The tangent at a point ( x 1 , y 1 ) {\displaystyle (x_{1},\,y_{1})} of

1375-473: Is included as a special type of ellipse. The equation | P F 2 | + | P F 1 | = 2 a {\displaystyle \left|PF_{2}\right|+\left|PF_{1}\right|=2a} can be viewed in a different way (see figure): c 2 {\displaystyle c_{2}} is called the circular directrix (related to focus F 2 {\displaystyle F_{2}} ) of

1430-625: Is the 2-argument arctangent function. Using trigonometric functions , a parametric representation of the standard ellipse x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1} is: ( x , y ) = ( a cos ⁡ t , b sin ⁡ t ) ,   0 ≤ t < 2 π . {\displaystyle (x,\,y)=(a\cos t,\,b\sin t),\ 0\leq t<2\pi \,.} The parameter t (called

1485-805: The eccentric anomaly in astronomy) is not the angle of ( x ( t ) , y ( t ) ) {\displaystyle (x(t),y(t))} with the x -axis, but has a geometric meaning due to Philippe de La Hire (see § Drawing ellipses below). With the substitution u = tan ⁡ ( t 2 ) {\textstyle u=\tan \left({\frac {t}{2}}\right)} and trigonometric formulae one obtains cos ⁡ t = 1 − u 2 1 + u 2   , sin ⁡ t = 2 u 1 + u 2 {\displaystyle \cos t={\frac {1-u^{2}}{1+u^{2}}}\ ,\quad \sin t={\frac {2u}{1+u^{2}}}} and

1540-786: The degenerate cases from the non-degenerate case, let ∆ be the determinant Δ = | A 1 2 B 1 2 D 1 2 B C 1 2 E 1 2 D 1 2 E F | = A C F + 1 4 B D E − 1 4 ( A E 2 + C D 2 + F B 2 ) . {\displaystyle \Delta ={\begin{vmatrix}A&{\frac {1}{2}}B&{\frac {1}{2}}D\\{\frac {1}{2}}B&C&{\frac {1}{2}}E\\{\frac {1}{2}}D&{\frac {1}{2}}E&F\end{vmatrix}}=ACF+{\tfrac {1}{4}}BDE-{\tfrac {1}{4}}(AE^{2}+CD^{2}+FB^{2}).} Then

1595-491: The radicals by suitable squarings and using b 2 = a 2 − c 2 {\displaystyle b^{2}=a^{2}-c^{2}} (see diagram) produces the standard equation of the ellipse: x 2 a 2 + y 2 b 2 = 1 , {\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1,} or, solved for y : y = ± b

1650-557: The rational parametric equation of an ellipse { x ( u ) = a 1 − u 2 1 + u 2 y ( u ) = b 2 u 1 + u 2 − ∞ < u < ∞ {\displaystyle {\begin{cases}x(u)=a\,{\dfrac {1-u^{2}}{1+u^{2}}}\\[10mu]y(u)=b\,{\dfrac {2u}{1+u^{2}}}\\[10mu]-\infty <u<\infty \end{cases}}} which covers any point of

1705-423: The semi-major and semi-minor axes . The top and bottom points V 3 = ( 0 , b ) , V 4 = ( 0 , − b ) {\displaystyle V_{3}=(0,\,b),\;V_{4}=(0,\,-b)} are the co-vertices . The distances from a point ( x , y ) {\displaystyle (x,\,y)} on the ellipse to the left and right foci are

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1760-653: The x - and y -axes. In analytic geometry , the ellipse is defined as a quadric : the set of points ( x , y ) {\displaystyle (x,\,y)} of the Cartesian plane that, in non-degenerate cases, satisfy the implicit equation A x 2 + B x y + C y 2 + D x + E y + F = 0 {\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0} provided B 2 − 4 A C < 0. {\displaystyle B^{2}-4AC<0.} To distinguish

1815-442: The area considered is that of a cross section . For example, a square of side length L has an area of L 2 {\displaystyle L^{2}} . Setting that area to be equal that of a circle imply that Similarly, an ellipse with semi-major axis a {\displaystyle a} and semi-minor axis b {\displaystyle b} has mean radius R mean =

1870-1060: The canonical equation X 2 a 2 + Y 2 b 2 = 1 {\displaystyle {\frac {X^{2}}{a^{2}}}+{\frac {Y^{2}}{b^{2}}}=1} by a Euclidean transformation of the coordinates ( X , Y ) {\displaystyle (X,\,Y)} : X = ( x − x ∘ ) cos ⁡ θ + ( y − y ∘ ) sin ⁡ θ , Y = − ( x − x ∘ ) sin ⁡ θ + ( y − y ∘ ) cos ⁡ θ . {\displaystyle {\begin{aligned}X&=\left(x-x_{\circ }\right)\cos \theta +\left(y-y_{\circ }\right)\sin \theta ,\\Y&=-\left(x-x_{\circ }\right)\sin \theta +\left(y-y_{\circ }\right)\cos \theta .\end{aligned}}} Conversely,

1925-1385: The canonical form parameters can be obtained from the general-form coefficients by the equations: a , b = − 2 ( A E 2 + C D 2 − B D E + ( B 2 − 4 A C ) F ) ( ( A + C ) ± ( A − C ) 2 + B 2 ) B 2 − 4 A C , x ∘ = 2 C D − B E B 2 − 4 A C , y ∘ = 2 A E − B D B 2 − 4 A C , θ = 1 2 atan2 ⁡ ( − B , C − A ) , {\displaystyle {\begin{aligned}a,b&={\frac {-{\sqrt {2{\big (}AE^{2}+CD^{2}-BDE+(B^{2}-4AC)F{\big )}{\big (}(A+C)\pm {\sqrt {(A-C)^{2}+B^{2}}}{\big )}}}}{B^{2}-4AC}},\\x_{\circ }&={\frac {2CD-BE}{B^{2}-4AC}},\\[5mu]y_{\circ }&={\frac {2AE-BD}{B^{2}-4AC}},\\[5mu]\theta &={\tfrac {1}{2}}\operatorname {atan2} (-B,\,C-A),\end{aligned}}} where atan2

1980-399: The center. The distance c {\displaystyle c} of the foci to the center is called the focal distance or linear eccentricity. The quotient e = c a {\displaystyle e={\tfrac {c}{a}}} is the eccentricity . The case F 1 = F 2 {\displaystyle F_{1}=F_{2}} yields a circle and

2035-456: The ellipse x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1} except the left vertex ( − a , 0 ) {\displaystyle (-a,\,0)} . For u ∈ [ 0 , 1 ] , {\displaystyle u\in [0,\,1],} this formula represents

2090-479: The ellipse x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1} has the coordinate equation: x 1 a 2 x + y 1 b 2 y = 1. {\displaystyle {\frac {x_{1}}{a^{2}}}x+{\frac {y_{1}}{b^{2}}}y=1.} A vector parametric equation of

2145-620: The ellipse is a non-degenerate real ellipse if and only if C∆ < 0. If C∆ > 0, we have an imaginary ellipse, and if ∆ = 0, we have a point ellipse. The general equation's coefficients can be obtained from known semi-major axis a {\displaystyle a} , semi-minor axis b {\displaystyle b} , center coordinates ( x ∘ , y ∘ ) {\displaystyle \left(x_{\circ },\,y_{\circ }\right)} , and rotation angle θ {\displaystyle \theta } (the angle from

2200-427: The ellipse such that x 1 u a 2 + y 1 v b 2 = 0 {\textstyle {\frac {x_{1}u}{a^{2}}}+{\tfrac {y_{1}v}{b^{2}}}=0} , then the points lie on two conjugate diameters (see below ). (If a = b {\displaystyle a=b} , the ellipse is a circle and "conjugate" means "orthogonal".) If

2255-418: The ellipse would be taller than it is wide). This form can be converted to the standard form by transposing the variable names x {\displaystyle x} and y {\displaystyle y} and the parameter names a {\displaystyle a} and b . {\displaystyle b.} This is the distance from the center to a focus: c =

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2310-543: The ellipse, the x -axis is the major axis, and: For an arbitrary point ( x , y ) {\displaystyle (x,y)} the distance to the focus ( c , 0 ) {\displaystyle (c,0)} is ( x − c ) 2 + y 2 {\textstyle {\sqrt {(x-c)^{2}+y^{2}}}} and to the other focus ( x + c ) 2 + y 2 {\textstyle {\sqrt {(x+c)^{2}+y^{2}}}} . Hence

2365-469: The ellipse. This property should not be confused with the definition of an ellipse using a directrix line below. Using Dandelin spheres , one can prove that any section of a cone with a plane is an ellipse, assuming the plane does not contain the apex and has slope less than that of the lines on the cone. The standard form of an ellipse in Cartesian coordinates assumes that the origin is the center of

2420-420: The line segment joining the foci is called the center of the ellipse. The line through the foci is called the major axis , and the line perpendicular to it through the center is the minor axis . The major axis intersects the ellipse at two vertices V 1 , V 2 {\displaystyle V_{1},V_{2}} , which have distance a {\displaystyle a} to

2475-558: The line's equation into the ellipse equation and respecting x 1 2 a 2 + y 1 2 b 2 = 1 {\textstyle {\frac {x_{1}^{2}}{a^{2}}}+{\frac {y_{1}^{2}}{b^{2}}}=1} yields: ( x 1 + s u ) 2 a 2 + ( y 1 + s v ) 2 b 2 = 1   ⟹ 2 s ( x 1 u

2530-412: The other two forms of conic sections, parabolas and hyperbolas , both of which are open and unbounded . An angled cross section of a right circular cylinder is also an ellipse. An ellipse may also be defined in terms of one focal point and a line outside the ellipse called the directrix : for all points on the ellipse, the ratio between the distance to the focus and the distance to the directrix

2585-737: The parameter [ u : v ] {\displaystyle [u:v]} is considered to be a point on the real projective line P ( R ) {\textstyle \mathbf {P} (\mathbf {R} )} , then the corresponding rational parametrization is [ u : v ] ↦ ( a v 2 − u 2 v 2 + u 2 , b 2 u v v 2 + u 2 ) . {\displaystyle [u:v]\mapsto \left(a{\frac {v^{2}-u^{2}}{v^{2}+u^{2}}},b{\frac {2uv}{v^{2}+u^{2}}}\right).} Then [ 1 : 0 ] ↦ ( −

2640-399: The point ( x , y ) {\displaystyle (x,\,y)} is on the ellipse whenever: ( x − c ) 2 + y 2 + ( x + c ) 2 + y 2 = 2 a   . {\displaystyle {\sqrt {(x-c)^{2}+y^{2}}}+{\sqrt {(x+c)^{2}+y^{2}}}=2a\ .} Removing

2695-429: The positive horizontal axis to the ellipse's major axis) using the formulae: A = a 2 sin 2 ⁡ θ + b 2 cos 2 ⁡ θ B = 2 ( b 2 − a 2 ) sin ⁡ θ cos ⁡ θ C =

2750-432: The right upper quarter of the ellipse moving counter-clockwise with increasing u . {\displaystyle u.} The left vertex is the limit lim u → ± ∞ ( x ( u ) , y ( u ) ) = ( − a , 0 ) . {\textstyle \lim _{u\to \pm \infty }(x(u),\,y(u))=(-a,\,0)\;.} Alternately, if

2805-416: The same term [REDACTED] This disambiguation page lists articles associated with the title Half Light . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Half_Light&oldid=969344420 " Category : Disambiguation pages Hidden categories: Short description

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2860-618: The standard ellipse is shifted to have center ( x ∘ , y ∘ ) {\displaystyle \left(x_{\circ },\,y_{\circ }\right)} , its equation is ( x − x ∘ ) 2 a 2 + ( y − y ∘ ) 2 b 2 = 1   . {\displaystyle {\frac {\left(x-x_{\circ }\right)^{2}}{a^{2}}}+{\frac {\left(y-y_{\circ }\right)^{2}}{b^{2}}}=1\ .} The axes are still parallel to

2915-606: The tangent is: x → = ( x 1 y 1 ) + s ( − y 1 a 2 x 1 b 2 ) , s ∈ R . {\displaystyle {\vec {x}}={\begin{pmatrix}x_{1}\\y_{1}\end{pmatrix}}+s\left({\begin{array}{r}-y_{1}a^{2}\\x_{1}b^{2}\end{array}}\right),\quad s\in \mathbb {R} .} Proof: Let ( x 1 , y 1 ) {\displaystyle (x_{1},\,y_{1})} be

2970-513: The two distances to the focal points is a constant. It generalizes a circle , which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity e {\displaystyle e} , a number ranging from e = 0 {\displaystyle e=0} (the limiting case of a circle) to e = 1 {\displaystyle e=1} (the limiting case of infinite elongation, no longer an ellipse but

3025-543: The volume of a non-spherical object V , one can calculate its volume-equivalent radius by setting or, alternatively: For example, a cube of side length L has a volume of L 3 {\displaystyle L^{3}} . Setting that volume to be equal that of a sphere imply that Similarly, a tri-axial ellipsoid with axes a {\displaystyle a} , b {\displaystyle b} and c {\displaystyle c} has mean radius R mean =

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