Ellipse

In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, 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, a number ranging from e = 0 (the limiting case of a circle) to e = 1 (the limiting case of infinite elongation, no longer an ellipse but a parabola). x 2 a 2 + y 2 b 2 = 1. {displaystyle {frac {x^{2}}{a^{2}}}+{frac {y^{2}}{b^{2}}}=1.} ( x , y ) = ( a cos ⁡ ( t ) , b sin ⁡ ( t ) ) for 0 ≤ t ≤ 2 π . {displaystyle (x,y)=(acos(t),bsin(t))quad { ext{for}}quad 0leq tleq 2pi .} x 2 a 2 + y 2 b 2 = 1 , {displaystyle {frac {x^{2}}{a^{2}}}+{frac {y^{2}}{b^{2}}}=1,} e = c a = 1 − ( b a ) 2 {displaystyle e={frac {c}{a}}={sqrt {1-left({frac {b}{a}} ight)^{2}}}} , ℓ = b 2 a = a ( 1 − e 2 ) . {displaystyle ell ,=,{frac {b^{2}}{a}},=,a(1{-}e^{2}).}   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;.}   x → = ( x 1 y 1 ) + s ( − y 1 a 2       x 1 b 2 ) {displaystyle {vec {x}}={egin{pmatrix}x_{1}\y_{1}end{pmatrix}}+s;{egin{pmatrix}!-y_{1}a^{2}!\ x_{1}b^{2}!end{pmatrix}}quad } with s ∈ R   . {displaystyle quad sin mathbb {R} .} ( x − x ∘ ) 2 a 2 + ( y − y ∘ ) 2 b 2 = 1   . {displaystyle {frac {(x{-}x_{circ })^{2}}{a^{2}}}+{frac {(y{-}y_{circ })^{2}}{b^{2}}}=1 .} ( x , y ) = ( a cos ⁡ t , b sin ⁡ t ) ,   0 ≤ t < 2 π   . {displaystyle (x,y)=(acos t,bsin t), 0leq t<2pi .}   x ( u ) = a ( 1 − u 2 ) / ( u 2 + 1 ) y ( u ) = 2 b u / ( u 2 + 1 ) , − ∞ < u < ∞ , {displaystyle {egin{array}{lcl}x(u)&=&a(1-u^{2})/(u^{2}+1)\y(u)&=&2bu/(u^{2}+1)end{array}};,quad -infty <u<infty ;,} c → ± ( m ) = ( − m a 2 ± m 2 a 2 + b 2 , b 2 ± m 2 a 2 + b 2 ) , m ∈ R . {displaystyle {vec {c}}_{pm }(m)=left(-{frac {ma^{2}}{pm {sqrt {m^{2}a^{2}+b^{2}}}}};,;{frac {b^{2}}{pm {sqrt {m^{2}a^{2}+b^{2}}}}} ight),,min mathbb {R} .} y = m x ± m 2 a 2 + b 2 . {displaystyle y=mxpm {sqrt {m^{2}a^{2}+b^{2}}};.} x → = p → ( t ) = f → 0 + f → 1 cos ⁡ t + f → 2 sin ⁡ t   . {displaystyle {vec {x}},=,{vec {p}}(t),=,{vec {f}}!_{0}+{vec {f}}!_{1}cos t+{vec {f}}!_{2}sin t .} cot ⁡ ( 2 t 0 ) = f → 1 2 − f → 2 2 2 f → 1 ⋅ f → 2 . {displaystyle cot(2t_{0})={frac {{vec {f}}!_{1}^{,2}-{vec {f}}!_{2}^{,2}}{2{vec {f}}!_{1}{cdot }{vec {f}}!_{2}}}.} de La Hire's methodAnimation of the methodEllipse construction: paper strip method 1Ellipses with Tusi couple. Two examples: red and cyan.Variation of the paper strip method 1Animation of the variation of the paper strip method 1Trammel of Archimedes (principle)Ellipsograph due to Benjamin BramerVariation of the paper strip method 2 q = a 2 b 2 = 1 1 − e 2 , {displaystyle {color {blue}q}={frac {a^{2}}{b^{2}}}={frac {1}{1-e^{2}}},} ( x − x ∘ ) 2 + q ( y − y ∘ ) 2 = a 2 , {displaystyle (x{-}x_{circ })^{2}+{color {blue}q},(y{-}y_{circ })^{2}=a^{2},} EllipsoidElliptic coneElliptic cylinderHyperboloid of one sheetHyperboloid of two sheets In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, 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, a number ranging from e = 0 (the limiting case of a circle) to e = 1 (the limiting case of infinite elongation, no longer an ellipse but a parabola). Analytically, the equation of a standard ellipse centered at the origin with width 2a and height 2b is: Assuming a ≥ b, the foci are (±c, 0) for c = a 2 − b 2 {displaystyle c={sqrt {a^{2}-b^{2}}}} . The standard parametric equation is: 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 the other two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. An angled cross section of a cylinder is also an ellipse. An ellipse may also be defined in terms of one focus 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 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 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 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 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 the vertex points V 1 , V 2 {displaystyle V_{1},V_{2}} , which have distance a {displaystyle a} to 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={ frac {c}{a}}} is the eccentricity.