Misplaced Pages

#331668

69-561: The almost perfect circle (the earth is an oblate spheroid that is wider around the equator), drawn with a line, demarcating the Eastern and Western Hemispheres must be an arbitrarily decided and published convention, unlike the Equator (an imaginary line encircling Earth , equidistant from its poles ), which divides the Northern and Southern Hemispheres. The prime meridian at 0° longitude and

138-454: 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

207-528: 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

276-621: 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

345-549: 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 ) = (

414-466: 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

483-542: 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 −

552-462: 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 (

621-425: 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

690-458: 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.

759-596: 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 }

SECTION 10

#1732837055332

828-429: 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

897-405: A = c reduces to a sphere. An oblate spheroid with c < a has surface area The oblate spheroid is generated by rotation about the z -axis of an ellipse with semi-major axis a and semi-minor axis c , therefore e may be identified as the eccentricity . (See ellipse .) A prolate spheroid with c > a has surface area The prolate spheroid is generated by rotation about

966-418: 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

1035-404: A massive body in a close orbit. The most extreme example is Jupiter's moon Io , which becomes slightly more or less prolate in its orbit due to a slight eccentricity, causing intense volcanism . The major axis of the prolate spheroid does not run through the satellite's poles in this case, but through the two points on its equator directly facing toward and away from the primary. This combines with

1104-522: 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

1173-425: A spheroid as having a major axis c , and minor axes a = b , the moments of inertia along these principal axes are C , A , and B . However, in a spheroid the minor axes are symmetrical. Therefore, our inertial terms along the major axes are: where M is the mass of the body defined as Ellipse In mathematics , an ellipse is a plane curve surrounding two focal points , such that for all points on

1242-421: A spheroid whose radius is 6,378.137 km (3,963.191 mi) at the Equator and 6,356.752 km (3,949.903 mi) at the poles . The word spheroid originally meant "an approximately spherical body", admitting irregularities even beyond the bi- or tri-axial ellipsoidal shape; that is how the term is used in some older papers on geodesy (for example, referring to truncated spherical harmonic expansions of

1311-452: 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

1380-757: Is also a shape of archaeological artifacts. The oblate spheroid is the approximate shape of rotating planets and other celestial bodies , including Earth, Saturn , Jupiter , and the quickly spinning star Altair . Saturn is the most oblate planet in the Solar System , with a flattening of 0.09796. See planetary flattening and equatorial bulge for details. Enlightenment scientist Isaac Newton , working from Jean Richer 's pendulum experiments and Christiaan Huygens 's theories for their interpretation, reasoned that Jupiter and Earth are oblate spheroids owing to their centrifugal force . Earth's diverse cartographic and geodetic systems are based on reference ellipsoids , all of which are oblate. The prolate spheroid

1449-520: 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

SECTION 20

#1732837055332

1518-559: 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

1587-447: 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

1656-473: 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

1725-641: Is not listed below due to its inclusion above, though the meridian does pass Wallis and Futuna .) With the exception of the United States (due to Wake Island , Guam and the Northern Mariana Islands ), all of them are located on just one side of the International Date Line , which curves around them. The following countries and territories lie outside Europe , Asia , Africa , and Oceania and are entirely, mostly, or partially within

1794-422: Is the longitude , and − ⁠ π / 2 ⁠ < β < + ⁠ π / 2 ⁠ and −π < λ < +π . Then, the spheroid's Gaussian curvature is and its mean curvature is Both of these curvatures are always positive, so that every point on a spheroid is elliptic. The aspect ratio of an oblate spheroid/ellipse, c  : a , is the ratio of the polar to equatorial lengths, while

1863-625: 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

1932-538: Is the approximate shape of the ball in several sports, such as in the rugby ball . Several moons of the Solar System approximate prolate spheroids in shape, though they are actually triaxial ellipsoids . Examples are Saturn 's satellites Mimas , Enceladus , and Tethys and Uranus ' satellite Miranda . In contrast to being distorted into oblate spheroids via rapid rotation, celestial objects distort slightly into prolate spheroids via tidal forces when they orbit

2001-805: 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

2070-421: The flattening (also called oblateness ) f , is the ratio of the equatorial-polar length difference to the equatorial length: The first eccentricity (usually simply eccentricity, as above) is often used instead of flattening. It is defined by: The relations between eccentricity and flattening are: All modern geodetic ellipsoids are defined by the semi-major axis plus either the semi-minor axis (giving

2139-492: The Earth's gravity geopotential model ). The equation of a tri-axial ellipsoid centred at the origin with semi-axes a , b and c aligned along the coordinate axes is The equation of a spheroid with z as the symmetry axis is given by setting a = b : The semi-axis a is the equatorial radius of the spheroid, and c is the distance from centre to pole along the symmetry axis. There are two possible cases: The case of

Eastern Hemisphere - Misplaced Pages Continue

2208-547: The Western Hemisphere 82% of humans live in the Eastern Hemisphere, and 18% in the Western Hemisphere . Oblate spheroid A spheroid , also known as an ellipsoid of revolution or rotational ellipsoid , is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters . A spheroid has circular symmetry . If

2277-400: The actinide and lanthanide elements are shaped like prolate spheroids. In anatomy, near-spheroid organs such as testis may be measured by their long and short axes . Many submarines have a shape which can be described as prolate spheroid. For a spheroid having uniform density, the moment of inertia is that of an ellipsoid with an additional axis of symmetry. Given a description of

2346-646: The antimeridian , at 180° longitude, are the conventionally accepted boundaries, since they divide eastern longitudes from western longitudes. This convention was established in 1884 at the International Meridian Conference held in Washington, D.C. where the standard time concepts of Canadian railroad engineer Sir Sandford Fleming were adopted. The Hemispheres agreed do not correspond with exact continents. Portions of Western Europe , West Africa , Oceania , and extreme northeastern Russia are in

2415-786: 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

2484-491: 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

2553-557: 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

2622-423: 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

2691-648: 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

2760-402: The z -axis of an ellipse with semi-major axis c and semi-minor axis a ; therefore, e may again be identified as the eccentricity . (See ellipse .) These formulas are identical in the sense that the formula for S oblate can be used to calculate the surface area of a prolate spheroid and vice versa. However, e then becomes imaginary and can no longer directly be identified with

2829-601: The Eastern Hemisphere is larger than that of the Western Hemisphere and has a wide variety of habitats . Below is a list of the sovereign states in both the Western and Eastern hemispheres on the IERS Reference Meridian , in order from north to south: Below is a list of additional sovereign states which are in both the Western and Eastern hemispheres along the 180th meridian , in order from north to south. (France

Eastern Hemisphere - Misplaced Pages Continue

2898-670: The Western Hemisphere, divorcing it from the continents which form the touchstone for most geopolitical constructs of "the East" and "the West". Consequently, meridians of 20°W and the diametrically opposed 160°E are often used outside of matters of physics and navigation, which includes all of the European and African mainlands, but also includes a small portion of northeast Greenland (typically reckoned as part of North America ) and excludes more of eastern Russia and Oceania (e.g., New Zealand ). Prior to

2967-432: The aspect ratio), the flattening, or the first eccentricity. While these definitions are mathematically interchangeable, real-world calculations must lose some precision. To avoid confusion, an ellipsoidal definition considers its own values to be exact in the form it gives. The most common shapes for the density distribution of protons and neutrons in an atomic nucleus are spherical , prolate, and oblate spheroidal, where

3036-1060: 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,

3105-1385: 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

3174-399: 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

3243-454: The combined effects of gravity and rotation , the figure of the Earth (and of all planets ) is not quite a sphere, but instead is slightly flattened in the direction of its axis of rotation. For that reason, in cartography and geodesy the Earth is often approximated by an oblate spheroid, known as the reference ellipsoid , instead of a sphere. The current World Geodetic System model uses

3312-535: The curve, the sum of 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

3381-415: The eccentricity. Both of these results may be cast into many other forms using standard mathematical identities and relations between parameters of the ellipse. The volume inside a spheroid (of any kind) is If A = 2 a is the equatorial diameter, and C = 2 c is the polar diameter, the volume is Let a spheroid be parameterized as where β is the reduced latitude or parametric latitude , λ

3450-456: 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

3519-479: 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

SECTION 50

#1732837055332

3588-620: 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

3657-453: The ellipse is rotated about its major axis , the result is a prolate spheroid , elongated like a rugby ball . The American football is similar but has a pointier end than a spheroid could. If the ellipse is rotated about its minor axis , the result is an oblate spheroid , flattened like a lentil or a plain M&;M . If the generating ellipse is a circle, the result is a sphere . Due to

3726-427: 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

3795-418: 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 =

3864-543: 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

3933-464: 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

4002-613: The global adoption of standard time, numerous prime meridians were decreed by various countries where time was defined by local noon (thereby, local). The centre of the Eastern Hemisphere is located in the Indian Ocean at the intersection of the equator and the 90th meridian east , 910 km west of Indonesia in the Ninety East Ridge . The nearest land is Simeulue Island at 2°35′N 96°05′E  /  2.583°N 96.083°E  / 2.583; 96.083 . The land mass of

4071-420: 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

4140-558: 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

4209-412: 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

SECTION 60

#1732837055332

4278-737: 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 ] ↦ ( −

4347-399: 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

4416-509: The polar axis is assumed to be the spin axis (or direction of the spin angular momentum vector). Deformed nuclear shapes occur as a result of the competition between electromagnetic repulsion between protons, surface tension and quantum shell effects . Spheroids are common in 3D cell cultures . Rotating equilibrium spheroids include the Maclaurin spheroid and the Jacobi ellipsoid . Spheroid

4485-429: 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 =

4554-432: 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

4623-486: The smaller oblate distortion from the synchronous rotation to cause the body to become triaxial. The term is also used to describe the shape of some nebulae such as the Crab Nebula . Fresnel zones , used to analyze wave propagation and interference in space, are a series of concentric prolate spheroids with principal axes aligned along the direct line-of-sight between a transmitter and a receiver. The atomic nuclei of

4692-618: 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

4761-606: 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

#331668