At first the measure is available only for chords not parallel to the y-axis, but the final formula works for any chord.

The angle θ {\displaystyle \theta } is called the true anomaly of the point. The numerator ℓ = a ( 1 − e 2 ) {\displaystyle \ell =a(1-e^{2})} is the semi-latus rectum.

The diagram shows an easy way to find the centers of curvature C 1 = ( a − b 2 a , 0 ) , C 3 = ( 0 , b − a 2 b ) {\displaystyle C_{1}=\left(a-{\tfrac {b^{2}}{a}},0\right),\,C_{3}=\left(0,b-{\tfrac {a^{2}}{b}}\right)} at vertex V 1 {\displaystyle V_{1}} and co-vertex V 3 {\displaystyle V_{3}} , respectively:

More generally, the arc length of a portion of the circumference, as a function of the angle subtended (or x coordinates of any two points on the upper half of the ellipse), is given by an incomplete elliptic integral. The upper half of an ellipse is parameterized by y = b   1 − x 2 a 2     . {\displaystyle y=b\ {\sqrt {1-{\frac {x^{2}}{a^{2}}}\ }}~.}

In the 17th century, Johannes Kepler discovered that the orbits along which the planets travel around the Sun are ellipses with the Sun [approximately] at one focus, in his first law of planetary motion. Later, Isaac Newton explained this as a corollary of his law of universal gravitation.

Two diameters d 1 , d 2 {\displaystyle d_{1},\,d_{2}} of an ellipse are conjugate if the midpoints of chords parallel to d 1 {\displaystyle d_{1}} lie on d 2   . {\displaystyle d_{2}\ .}

An ellipse defined implicitly by A x 2 + B x y + C y 2 = 1 {\displaystyle Ax^{2}+Bxy+Cy^{2}=1} has area 2 π / 4 A C − B 2 . {\displaystyle 2\pi /{\sqrt {4AC-B^{2}}}.}

Jul 8, 2018 — 2x Barlowlinse, achromatisch, 1,25" · Solid metal housing · Full multi-coated lens surfaces.

If the water's surface is disturbed at one focus of an elliptical water tank, the circular waves of that disturbance, after reflecting off the walls, converge simultaneously to a single point: the second focus. This is a consequence of the total travel length being the same along any wall-bouncing path between the two foci.

The radius is the distance between any of the three points and the center. r = ( x 1 − x ∘ ) 2 + ( y 1 − y ∘ ) 2 = ( x 2 − x ∘ ) 2 + ( y 2 − y ∘ ) 2 = ( x 3 − x ∘ ) 2 + ( y 3 − y ∘ ) 2 . {\displaystyle r={\sqrt {\left(x_{1}-x_{\circ }\right)^{2}+\left(y_{1}-y_{\circ }\right)^{2}}}={\sqrt {\left(x_{2}-x_{\circ }\right)^{2}+\left(y_{2}-y_{\circ }\right)^{2}}}={\sqrt {\left(x_{3}-x_{\circ }\right)^{2}+\left(y_{3}-y_{\circ }\right)^{2}}}.}

The parameter t (called 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).[8]

For the generation of points of the ellipse x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1} one uses the pencils at the vertices V 1 , V 2 {\displaystyle V_{1},\,V_{2}} . Let P = ( 0 , b ) {\displaystyle P=(0,\,b)} be an upper co-vertex of the ellipse and A = ( − a , 2 b ) , B = ( a , 2 b ) {\displaystyle A=(-a,\,2b),\,B=(a,\,2b)} .

Breadboardfunction

The proof for the pair F 1 , l 1 {\displaystyle F_{1},l_{1}} follows from the fact that | P F 1 | 2 = ( x − c ) 2 + y 2 ,   | P l 1 | 2 = ( x − a 2 c ) 2 {\textstyle \left|PF_{1}\right|^{2}=(x-c)^{2}+y^{2},\ \left|Pl_{1}\right|^{2}=\left(x-{\tfrac {a^{2}}{c}}\right)^{2}} and y 2 = b 2 − b 2 a 2 x 2 {\displaystyle y^{2}=b^{2}-{\tfrac {b^{2}}{a^{2}}}x^{2}} satisfy the equation | P F 1 | 2 − c 2 a 2 | P l 1 | 2 = 0 . {\displaystyle \left|PF_{1}\right|^{2}-{\frac {c^{2}}{a^{2}}}\left|Pl_{1}\right|^{2}=0\,.}

For the common parametric representation ( a cos ⁡ t , b sin ⁡ t ) {\displaystyle (a\cos t,b\sin t)} of the ellipse with equation x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1} one gets: The points

This circle is called orthoptic or director circle of the ellipse (not to be confused with the circular directrix defined above).

Such a relation between points and lines generated by a conic is called pole-polar relation or polarity. The pole is the point; the polar the line.

Removing 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:[3] 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 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)}}.}

where the sign in the denominator is negative if the reference direction θ = 0 {\displaystyle \theta =0} points towards the center (as illustrated on the right), and positive if that direction points away from the center.

breadboard中文

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:[4] ℓ = b 2 a = a ( 1 − e 2 ) . {\displaystyle \ell ={\frac {b^{2}}{a}}=a\left(1-e^{2}\right).}

If e < 1 {\displaystyle e<1} , introduce new parameters a , b {\displaystyle a,\,b} so that 1 − e 2 = b 2 a 2 ,  and    p = b 2 a {\displaystyle 1-e^{2}={\tfrac {b^{2}}{a^{2}}},{\text{ and }}\ p={\tfrac {b^{2}}{a}}} , and then the equation above becomes ( x − a ) 2 a 2 + y 2 b 2 = 1 , {\displaystyle {\frac {(x-a)^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1\,,}

If 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\ .}

For four points P i = ( x i , y i ) ,   i = 1 , 2 , 3 , 4 , {\displaystyle P_{i}=\left(x_{i},\,y_{i}\right),\ i=1,\,2,\,3,\,4,\,} no three of them on a line, we have the following (see diagram):

where a {\displaystyle a} and b {\displaystyle b} are the lengths of the semi-major and semi-minor axes, respectively. The area formula π a b {\displaystyle \pi ab} is intuitive: start with a circle of radius b {\displaystyle b} (so its area is π b 2 {\displaystyle \pi b^{2}} ) and stretch it by a factor a / b {\displaystyle a/b} to make an ellipse. This scales the area by the same factor: π b 2 ( a / b ) = π a b . {\displaystyle \pi b^{2}(a/b)=\pi ab.} [18] However, using the same approach for the circumference would be fallacious – compare the integrals ∫ f ( x ) d x {\textstyle \int f(x)\,dx} and ∫ 1 + f ′ 2 ( x ) d x {\textstyle \int {\sqrt {1+f'^{2}(x)}}\,dx} . It is also easy to rigorously prove the area formula using integration as follows. Equation (1) can be rewritten as y ( x ) = b 1 − x 2 / a 2 . {\textstyle y(x)=b{\sqrt {1-x^{2}/a^{2}}}.} For x ∈ [ − a , a ] , {\displaystyle x\in [-a,a],} this curve is the top half of the ellipse. So twice the integral of y ( x ) {\displaystyle y(x)} over the interval [ − a , a ] {\displaystyle [-a,a]} will be the area of the ellipse: A ellipse = ∫ − a a 2 b 1 − x 2 a 2 d x = b a ∫ − a a 2 a 2 − x 2 d x . {\displaystyle {\begin{aligned}A_{\text{ellipse}}&=\int _{-a}^{a}2b{\sqrt {1-{\frac {x^{2}}{a^{2}}}}}\,dx\\&={\frac {b}{a}}\int _{-a}^{a}2{\sqrt {a^{2}-x^{2}}}\,dx.\end{aligned}}}

The four vertices of the ellipse are p → ( t 0 ) , p → ( t 0 ± π 2 ) , p → ( t 0 + π ) {\displaystyle {\vec {p}}(t_{0}),\;{\vec {p}}\left(t_{0}\pm {\tfrac {\pi }{2}}\right),\;{\vec {p}}\left(t_{0}+\pi \right)} , for a parameter t = t 0 {\displaystyle t=t_{0}} defined by: 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}}}.}

One marks the point, which divides the strip into two substrips of length b {\displaystyle b} and a − b {\displaystyle a-b} . The strip is positioned onto the axes as described in the diagram. Then the free end of the strip traces an ellipse, while the strip is moved. For the proof, one recognizes that the tracing point can be described parametrically by ( a cos ⁡ t , b sin ⁡ t ) {\displaystyle (a\cos t,\,b\sin t)} , where parameter t {\displaystyle t} is the angle of slope of the paper strip.

It follows from the equation that the ellipse is symmetric with respect to the coordinate axes and hence with respect to the origin.

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 general solution for a harmonic oscillator in two or more dimensions is also an ellipse. Such is the case, for instance, of a long pendulum that is free to move in two dimensions; of a mass attached to a fixed point by a perfectly elastic spring; or of any object that moves under influence of an attractive force that is directly proportional to its distance from a fixed attractor. Unlike Keplerian orbits, however, these "harmonic orbits" have the center of attraction at the geometric center of the ellipse, and have fairly simple equations of motion.

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}}}

Steiner generation can also be defined for hyperbolas and parabolas. It is sometimes called a parallelogram method because one can use other points rather than the vertices, which starts with a parallelogram instead of a rectangle.

In 1970 Danny Cohen presented at the "Computer Graphics 1970" conference in England a linear algorithm for drawing ellipses and circles. In 1971, L. B. Smith published similar algorithms for all conic sections and proved them to have good properties.[36] These algorithms need only a few multiplications and additions to calculate each vector.

Some lower and upper bounds on the circumference of the canonical ellipse   x 2 / a 2 + y 2 / b 2 = 1   {\displaystyle \ x^{2}/a^{2}+y^{2}/b^{2}=1\ } with   a ≥ b   {\displaystyle \ a\geq b\ } are[25] 2 π b ≤ C ≤ 2 π a   , π ( a + b ) ≤ C ≤ 4 ( a + b )   , 4 a 2 + b 2   ≤ C ≤ 2   π a 2 + b 2     . {\displaystyle {\begin{aligned}2\pi b&\leq C\leq 2\pi a\ ,\\\pi (a+b)&\leq C\leq 4(a+b)\ ,\\4{\sqrt {a^{2}+b^{2}\ }}&\leq C\leq {\sqrt {2\ }}\pi {\sqrt {a^{2}+b^{2}\ }}~.\end{aligned}}}

Two non-circular gears with the same elliptical outline, each pivoting around one focus and positioned at the proper angle, turn smoothly while maintaining contact at all times. Alternatively, they can be connected by a link chain or timing belt, or in the case of a bicycle the main chainring may be elliptical, or an ovoid similar to an ellipse in form. Such elliptical gears may be used in mechanical equipment to produce variable angular speed or torque from a constant rotation of the driving axle, or in the case of a bicycle to allow a varying crank rotation speed with inversely varying mechanical advantage.

The case F 1 = F 2 {\displaystyle F_{1}=F_{2}} yields a circle and is included as a special type of ellipse.

Keplerian elliptical orbits are the result of any radially directed attraction force whose strength is inversely proportional to the square of the distance. Thus, in principle, the motion of two oppositely charged particles in empty space would also be an ellipse. (However, this conclusion ignores losses due to electromagnetic radiation and quantum effects, which become significant when the particles are moving at high speed.)

For u ∈ [ 0 , 1 ] , {\displaystyle u\in [0,\,1],} this formula represents 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)\;.}

For any method described below, knowledge of the axes and the semi-axes is necessary (or equivalently: the foci and the semi-major axis). If this presumption is not fulfilled one has to know at least two conjugate diameters. With help of Rytz's construction the axes and semi-axes can be retrieved.

Breadboardsimulator

To distinguish 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}).}

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. 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 a parabola).

Radius of curvature at the two co-vertices ( 0 , ± b ) {\displaystyle (0,\pm b)} and the centers of curvature: ρ 1 = a 2 b   , ( 0 | ± c 2 b )   . {\displaystyle \rho _{1}={\frac {a^{2}}{b}}\ ,\qquad \left(0\,{\bigg |}\,\pm {\frac {c^{2}}{b}}\right)\ .} The locus of all the centers of curvature is called an evolute. In the case of an ellipse, the evolute is an astroid.

(If f → 1 ⋅ f → 2 = 0 {\displaystyle {\vec {f}}\!_{1}\cdot {\vec {f}}\!_{2}=0} , then t 0 = 0 {\displaystyle t_{0}=0} .) This is derived as follows. The tangent vector at point p → ( t ) {\displaystyle {\vec {p}}(t)} is: p → ′ ( t ) = − f → 1 sin ⁡ t + f → 2 cos ⁡ t   . {\displaystyle {\vec {p}}\,'(t)=-{\vec {f}}\!_{1}\sin t+{\vec {f}}\!_{2}\cos t\ .}

In case of a circle the last equation collapses to x 1 x 2 + y 1 y 2 = 0   . {\displaystyle x_{1}x_{2}+y_{1}y_{2}=0\ .}

For the ellipse x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1} the intersection points of orthogonal tangents lie on the circle x 2 + y 2 = a 2 + b 2 {\displaystyle x^{2}+y^{2}=a^{2}+b^{2}} .

At a vertex parameter t = t 0 {\displaystyle t=t_{0}} , the tangent is perpendicular to the major/minor axes, so: 0 = p → ′ ( t ) ⋅ ( p → ( t ) − f → 0 ) = ( − f → 1 sin ⁡ t + f → 2 cos ⁡ t ) ⋅ ( f → 1 cos ⁡ t + f → 2 sin ⁡ t ) . {\displaystyle 0={\vec {p}}'(t)\cdot \left({\vec {p}}(t)-{\vec {f}}\!_{0}\right)=\left(-{\vec {f}}\!_{1}\sin t+{\vec {f}}\!_{2}\cos t\right)\cdot \left({\vec {f}}\!_{1}\cos t+{\vec {f}}\!_{2}\sin t\right).}

The circumference of the ellipse may be evaluated in terms of E ( e ) {\displaystyle E(e)} using Gauss's arithmetic-geometric mean;[19] this is a quadratically converging iterative method (see here for details).

Analogously to the circle case, the equation can be written more clearly using vectors: ( x → − x → 1 ) ∗ ( x → − x → 2 ) det ( x → − x → 1 , x → − x → 2 ) = ( x → 3 − x → 1 ) ∗ ( x → 3 − x → 2 ) det ( x → 3 − x → 1 , x → 3 − x → 2 ) , {\displaystyle {\frac {\left({\color {red}{\vec {x}}}-{\vec {x}}_{1}\right)*\left({\color {red}{\vec {x}}}-{\vec {x}}_{2}\right)}{\det \left({\color {red}{\vec {x}}}-{\vec {x}}_{1},{\color {red}{\vec {x}}}-{\vec {x}}_{2}\right)}}={\frac {\left({\vec {x}}_{3}-{\vec {x}}_{1}\right)*\left({\vec {x}}_{3}-{\vec {x}}_{2}\right)}{\det \left({\vec {x}}_{3}-{\vec {x}}_{1},{\vec {x}}_{3}-{\vec {x}}_{2}\right)}},}

Sound waves are reflected in a similar way, so in a large elliptical room a person standing at one focus can hear a person standing at the other focus remarkably well. The effect is even more evident under a vaulted roof shaped as a section of a prolate spheroid. Such a room is called a whisper chamber. The same effect can be demonstrated with two reflectors shaped like the end caps of such a spheroid, placed facing each other at the proper distance. Examples are the National Statuary Hall at the United States Capitol (where John Quincy Adams is said to have used this property for eavesdropping on political matters); the Mormon Tabernacle at Temple Square in Salt Lake City, Utah; at an exhibit on sound at the Museum of Science and Industry in Chicago; in front of the University of Illinois at Urbana–Champaign Foellinger Auditorium; and also at a side chamber of the Palace of Charles V, in the Alhambra.

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 two vertices 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={\tfrac {c}{a}}} is the eccentricity.

Given an ellipse whose axes are drawn, we can construct the endpoints of a particular elliptic arc whose length is one eighth of the ellipse's circumference using only straightedge and compass in a finite number of steps; for some specific shapes of ellipses, such as when the axes have a length ratio of ⁠ 2 : 1 {\displaystyle {\sqrt {2}}:1} ⁠, it is additionally possible to construct the endpoints of a particular arc whose length is one twelfth of the circumference.[26] (The vertices and co-vertices are already endpoints of arcs whose length is one half or one quarter of the ellipse's circumference.) However, the general theory of straightedge-and-compass elliptic division appears to be unknown, unlike in the case of the circle and the lemniscate. The division in special cases has been investigated by Legendre in his classical treatise.[27]

So far we have dealt with erect ellipses, whose major and minor axes are parallel to the x {\displaystyle x} and y {\displaystyle y} axes. However, some applications require tilted ellipses. In charged-particle beam optics, for instance, the enclosed area of an erect or tilted ellipse is an important property of the beam, its emittance. In this case a simple formula still applies, namely

Composite Bézier curves may also be used to draw an ellipse to sufficient accuracy, since any ellipse may be construed as an affine transformation of a circle. The spline methods used to draw a circle may be used to draw an ellipse, since the constituent Bézier curves behave appropriately under such transformations.

How to usebreadboard

It is beneficial to use a parametric formulation in computer graphics because the density of points is greatest where there is the most curvature. Thus, the change in slope between each successive point is small, reducing the apparent "jaggedness" of the approximation.

With help of trigonometric formulae one obtains: cos ⁡ t = cot ⁡ t ± 1 + cot 2 ⁡ t = − m a ± m 2 a 2 + b 2   , sin ⁡ t = 1 ± 1 + cot 2 ⁡ t = b ± m 2 a 2 + b 2 . {\displaystyle \cos t={\frac {\cot t}{\pm {\sqrt {1+\cot ^{2}t}}}}={\frac {-ma}{\pm {\sqrt {m^{2}a^{2}+b^{2}}}}}\ ,\quad \quad \sin t={\frac {1}{\pm {\sqrt {1+\cot ^{2}t}}}}={\frac {b}{\pm {\sqrt {m^{2}a^{2}+b^{2}}}}}.}

This section considers the family of ellipses defined by equations ( x − x ∘ ) 2 a 2 + ( y − y ∘ ) 2 b 2 = 1 {\displaystyle {\tfrac {\left(x-x_{\circ }\right)^{2}}{a^{2}}}+{\tfrac {\left(y-y_{\circ }\right)^{2}}{b^{2}}}=1} with a fixed eccentricity e {\displaystyle e} . It is convenient to use the parameter: q = a 2 b 2 = 1 1 − e 2 , {\displaystyle {\color {blue}q}={\frac {a^{2}}{b^{2}}}={\frac {1}{1-e^{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}}},}

A variation of the paper strip method 1 uses the observation that the midpoint N {\displaystyle N} of the paper strip is moving on the circle with center M {\displaystyle M} (of the ellipse) and radius a + b 2 {\displaystyle {\tfrac {a+b}{2}}} . Hence, the paperstrip can be cut at point N {\displaystyle N} into halves, connected again by a joint at N {\displaystyle N} and the sliding end K {\displaystyle K} fixed at the center M {\displaystyle M} (see diagram). After this operation the movement of the unchanged half of the paperstrip is unchanged.[15] This variation requires only one sliding shoe.

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

Because the tangent line is perpendicular to the normal, an equivalent statement is that the tangent is the external angle bisector of the lines to the foci (see diagram). Let L {\displaystyle L} be the point on the line P F 2 ¯ {\displaystyle {\overline {PF_{2}}}} with distance 2 a {\displaystyle 2a} to the focus F 2 {\displaystyle F_{2}} , where a {\displaystyle a} is the semi-major axis of the ellipse. Let line w {\displaystyle w} be the external angle bisector of the lines P F 1 ¯ {\displaystyle {\overline {PF_{1}}}} and P F 2 ¯ . {\displaystyle {\overline {PF_{2}}}.} Take any other point Q {\displaystyle Q} on w . {\displaystyle w.} By the triangle inequality and the angle bisector theorem, 2 a = | L F 2 | < {\displaystyle 2a=\left|LF_{2}\right|<{}} | Q F 2 | + | Q L | = {\displaystyle \left|QF_{2}\right|+\left|QL\right|={}} | Q F 2 | + | Q F 1 | , {\displaystyle \left|QF_{2}\right|+\left|QF_{1}\right|,} therefore Q {\displaystyle Q} must be outside the ellipse. As this is true for every choice of Q , {\displaystyle Q,} w {\displaystyle w} only intersects the ellipse at the single point P {\displaystyle P} so must be the tangent line.

Here f → 0 {\displaystyle {\vec {f}}\!_{0}} is the center and f → 1 , f → 2 {\displaystyle {\vec {f}}\!_{1},\;{\vec {f}}\!_{2}} are the directions of two conjugate diameters, in general not perpendicular.

The exact infinite series is: C = 2 π a [ 1 − ( 1 2 ) 2 e 2 − ( 1 ⋅ 3 2 ⋅ 4 ) 2 e 4 3 − ( 1 ⋅ 3 ⋅ 5 2 ⋅ 4 ⋅ 6 ) 2 e 6 5 − ⋯ ] = 2 π a [ 1 − ∑ n = 1 ∞ ( ( 2 n − 1 ) ! ! ( 2 n ) ! ! ) 2 e 2 n 2 n − 1 ] = − 2 π a ∑ n = 0 ∞ ( ( 2 n − 1 ) ! ! ( 2 n ) ! ! ) 2 e 2 n 2 n − 1 , {\displaystyle {\begin{aligned}C&=2\pi a\left[{1-\left({\frac {1}{2}}\right)^{2}e^{2}-\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}{\frac {e^{4}}{3}}-\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right)^{2}{\frac {e^{6}}{5}}-\cdots }\right]\\&=2\pi a\left[1-\sum _{n=1}^{\infty }\left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}{\frac {e^{2n}}{2n-1}}\right]\\&=-2\pi a\sum _{n=0}^{\infty }\left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}{\frac {e^{2n}}{2n-1}},\end{aligned}}} where n ! ! {\displaystyle n!!} is the double factorial (extended to negative odd integers in the usual way, giving ( − 1 ) ! ! = 1 {\displaystyle (-1)!!=1} and ( − 3 ) ! ! = − 1 {\displaystyle (-3)!!=-1} ).

A technical realization of the motion of the paper strip can be achieved by a Tusi couple (see animation). The device is able to draw any ellipse with a fixed sum a + b {\displaystyle a+b} , which is the radius of the large circle. This restriction may be a disadvantage in real life. More flexible is the second paper strip method.

The two following methods rely on the parametric representation (see § Standard parametric representation, above): ( a cos ⁡ t , b sin ⁡ t ) {\displaystyle (a\cos t,\,b\sin t)}

The definition of an ellipse in this section gives a parametric representation of an arbitrary ellipse, even in space, if one allows f → 0 , f → 1 , f → 2 {\displaystyle {\vec {f}}\!_{0},{\vec {f}}\!_{1},{\vec {f}}\!_{2}} to be vectors in space.

Additionally, because of the focus-to-focus reflection property of ellipses, if the rays are allowed to continue propagating, reflected rays will eventually align closely with the major axis.

where again a {\displaystyle a} is the length of the semi-major axis, e = 1 − b 2 / a 2 {\textstyle e={\sqrt {1-b^{2}/a^{2}}}} is the eccentricity, and the function E {\displaystyle E} is the complete elliptic integral of the second kind, E ( e ) = ∫ 0 π / 2 1 − e 2 sin 2 ⁡ θ   d θ {\displaystyle E(e)\,=\,\int _{0}^{\pi /2}{\sqrt {1-e^{2}\sin ^{2}\theta }}\ d\theta } which is in general not an elementary function.

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 ) = ( 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 .}

and to write the ellipse equation as: ( x − x ∘ ) 2 + q ( y − y ∘ ) 2 = a 2 , {\displaystyle \left(x-x_{\circ }\right)^{2}+{\color {blue}q}\,\left(y-y_{\circ }\right)^{2}=a^{2},}

For example, for P 1 = ( 2 , 0 ) , P 2 = ( 0 , 1 ) , P 3 = ( 0 , 0 ) {\displaystyle P_{1}=(2,\,0),\;P_{2}=(0,\,1),\;P_{3}=(0,\,0)} and q = 4 {\displaystyle q=4} one obtains the three-point form

If there is no ellipsograph available, one can draw an ellipse using an approximation by the four osculating circles at the vertices.

Jumper wires

Let F = ( f , 0 ) ,   e > 0 {\displaystyle F=(f,\,0),\ e>0} , and assume ( 0 , 0 ) {\displaystyle (0,\,0)} is a point on the curve. The directrix l {\displaystyle l} has equation x = − f e {\displaystyle x=-{\tfrac {f}{e}}} . With P = ( x , y ) {\displaystyle P=(x,\,y)} , the relation | P F | 2 = e 2 | P l | 2 {\displaystyle |PF|^{2}=e^{2}|Pl|^{2}} produces the equations

The Rayleigh criterion defines the limit of resolution in a diffraction-limited system, in other words, when two points of light are distinguishable or resolved ...

The width and height parameters a , b {\displaystyle a,\;b} are called 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 a + e x {\displaystyle a+ex} and a − e x {\displaystyle a-ex} .

If ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} and ( u , v ) {\displaystyle (u,v)} are two points of 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".)

where q is fixed and x ∘ , y ∘ , a {\displaystyle x_{\circ },\,y_{\circ },\,a} vary over the real numbers. (Such ellipses have their axes parallel to the coordinate axes: if q < 1 {\displaystyle q<1} , the major axis is parallel to the x-axis; if q > 1 {\displaystyle q>1} , it is parallel to the y-axis.)

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which covers any point of 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)} .

c 2 {\displaystyle c_{2}} is called the circular directrix (related to focus F 2 {\displaystyle F_{2}} ) of the ellipse.[1][2] This property should not be confused with the definition of an ellipse using a directrix line below.

The semi-latus rectum ℓ {\displaystyle \ell } is equal to the radius of curvature at the vertices (see section curvature).

and the radius of curvature, ρ = 1/κ, at point ( x , y ) {\displaystyle (x,y)} : ρ = a 2 b 2 ( x 2 a 4 + y 2 b 4 ) 3 2 = 1 a 4 b 4 ( a 4 y 2 + b 4 x 2 ) 3   . {\displaystyle \rho =a^{2}b^{2}\left({\frac {x^{2}}{a^{4}}}+{\frac {y^{2}}{b^{4}}}\right)^{\frac {3}{2}}={\frac {1}{a^{4}b^{4}}}{\sqrt {\left(a^{4}y^{2}+b^{4}x^{2}\right)^{3}}}\ .} The radius of curvature of an ellipse, as a function of angle θ from the center, is: R ( θ ) = a 2 b ( 1 − e 2 ( 2 − e 2 ) ( cos ⁡ θ ) 2 ) 1 − e 2 ( cos ⁡ θ ) 2 ) 3 / 2 , {\displaystyle R(\theta )={\frac {a^{2}}{b}}{\biggl (}{\frac {1-e^{2}(2-e^{2})(\cos \theta )^{2})}{1-e^{2}(\cos \theta )^{2}}}{\biggr )}^{3/2}\,,} where e is the eccentricity.

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.

This is the equation of an ellipse ( e < 1 {\displaystyle e<1} ), or a parabola ( e = 1 {\displaystyle e=1} ), or a hyperbola ( e > 1 {\displaystyle e>1} ). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram).

It is sometimes useful to find the minimum bounding ellipse on a set of points. The ellipsoid method is quite useful for solving this problem.

In the parametric equation for a general ellipse given above, 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,}

and from the diagram it can be seen that the area of the parallelogram is 8 times that of A Δ {\displaystyle A_{\Delta }} . Hence Area 12 = 4 a b . {\displaystyle {\text{Area}}_{12}=4ab\,.}

Alternately, if 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).}

This is equivalent to s = b   [ E ( z | 1 − a 2 b 2 ) ] z   =   arccos ⁡ x 2 a arccos ⁡ x 1 a {\displaystyle s=b\ \left[\;E\left(z\;{\Biggl |}\;1-{\frac {a^{2}}{b^{2}}}\right)\;\right]_{z\ =\ \arccos {\frac {x_{2}}{a}}}^{\arccos {\frac {x_{1}}{a}}}}

Using (1) one finds that ( − y 1 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.

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 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\ .}

A parametric representation, which uses the slope m {\displaystyle m} of the tangent at a point of the ellipse can be obtained from the derivative of the standard representation x → ( t ) = ( a cos ⁡ t , b sin ⁡ t ) T {\displaystyle {\vec {x}}(t)=(a\cos t,\,b\sin t)^{\mathsf {T}}} : x → ′ ( t ) = ( − a sin ⁡ t , b cos ⁡ t ) T → m = − b a cot ⁡ t → cot ⁡ t = − m a b . {\displaystyle {\vec {x}}'(t)=(-a\sin t,\,b\cos t)^{\mathsf {T}}\quad \rightarrow \quad m=-{\frac {b}{a}}\cot t\quad \rightarrow \quad \cot t=-{\frac {ma}{b}}.}

Using vectors, dot products and determinants this formula can be arranged more clearly, letting x → = ( x , y ) {\displaystyle {\vec {x}}=(x,\,y)} : ( x → − x → 1 ) ⋅ ( x → − x → 2 ) det ( x → − x → 1 , x → − x → 2 ) = ( x → 3 − x → 1 ) ⋅ ( x → 3 − x → 2 ) det ( x → 3 − x → 1 , x → 3 − x → 2 ) . {\displaystyle {\frac {\left({\color {red}{\vec {x}}}-{\vec {x}}_{1}\right)\cdot \left({\color {red}{\vec {x}}}-{\vec {x}}_{2}\right)}{\det \left({\color {red}{\vec {x}}}-{\vec {x}}_{1},{\color {red}{\vec {x}}}-{\vec {x}}_{2}\right)}}={\frac {\left({\vec {x}}_{3}-{\vec {x}}_{1}\right)\cdot \left({\vec {x}}_{3}-{\vec {x}}_{2}\right)}{\det \left({\vec {x}}_{3}-{\vec {x}}_{1},{\vec {x}}_{3}-{\vec {x}}_{2}\right)}}.}

In electronics, the relative phase of two sinusoidal signals can be compared by feeding them to the vertical and horizontal inputs of an oscilloscope. If the Lissajous figure display is an ellipse, rather than a straight line, the two signals are out of phase.

These expressions can be derived from 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}}}

The second integral is the area of a circle of radius a , {\displaystyle a,} that is, π a 2 . {\displaystyle \pi a^{2}.} So A ellipse = b a π a 2 = π a b . {\displaystyle A_{\text{ellipse}}={\frac {b}{a}}\pi a^{2}=\pi ab.}

Here m {\displaystyle m} is the slope of the tangent at the corresponding ellipse point, c → + {\displaystyle {\vec {c}}_{+}} is the upper and c → − {\displaystyle {\vec {c}}_{-}} the lower half of the ellipse. The vertices ( ± a , 0 ) {\displaystyle (\pm a,\,0)} , having vertical tangents, are not covered by the representation.

Inductor

An affine transformation preserves parallelism and midpoints of line segments, so this property is true for any ellipse. (Note that the parallel chords and the diameter are no longer orthogonal.)

A vector parametric equation of 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 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 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 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:

Srinivasa Ramanujan gave two close approximations for the circumference in §16 of "Modular Equations and Approximations to π {\displaystyle \pi } ";[24] they are C ≈ π [ 3 ( a + b ) − ( 3 a + b ) ( a + 3 b ) ] = π [ 3 ( a + b ) − 3 ( a + b ) 2 + 4 a b ] {\displaystyle C\approx \pi {\biggl [}3(a+b)-{\sqrt {(3a+b)(a+3b)}}{\biggr ]}=\pi {\biggl [}3(a+b)-{\sqrt {3(a+b)^{2}+4ab}}{\biggr ]}} and C ≈ π ( a + b ) ( 1 + 3 h 10 + 4 − 3 h ) , {\displaystyle C\approx \pi \left(a+b\right)\left(1+{\frac {3h}{10+{\sqrt {4-3h}}}}\right),} where h {\displaystyle h} takes on the same meaning as above. The errors in these approximations, which were obtained empirically, are of order h 3 {\displaystyle h^{3}} and h 5 , {\displaystyle h^{5},} respectively.

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Because of c ⋅ a 2 c = a 2 {\displaystyle c\cdot {\tfrac {a^{2}}{c}}=a^{2}} point L 1 {\displaystyle L_{1}} of directrix l 1 {\displaystyle l_{1}} (see diagram) and focus F 1 {\displaystyle F_{1}} are inverse with respect to the circle inversion at circle x 2 + y 2 = a 2 {\displaystyle x^{2}+y^{2}=a^{2}} (in diagram green). Hence L 1 {\displaystyle L_{1}} can be constructed as shown in the diagram. Directrix l 1 {\displaystyle l_{1}} is the perpendicular to the main axis at point L 1 {\displaystyle L_{1}} .

In polar coordinates, with the origin at the center of the ellipse and with the angular coordinate θ {\displaystyle \theta } measured from the major axis, the ellipse's equation is[7]: 75  r ( θ ) = a b ( b cos ⁡ θ ) 2 + ( a sin ⁡ θ ) 2 = b 1 − ( e cos ⁡ θ ) 2 {\displaystyle r(\theta )={\frac {ab}{\sqrt {(b\cos \theta )^{2}+(a\sin \theta )^{2}}}}={\frac {b}{\sqrt {1-(e\cos \theta )^{2}}}}} where e {\displaystyle e} is the eccentricity, not Euler's number.

Throughout this article, the semi-major and semi-minor axes are denoted a {\displaystyle a} and b {\displaystyle b} , respectively, i.e. a ≥ b > 0   . {\displaystyle a\geq b>0\ .}

The following construction of single points of an ellipse is due to de La Hire.[12] It is based on the standard parametric representation ( a cos ⁡ t , b sin ⁡ t ) {\displaystyle (a\cos t,\,b\sin t)} of an ellipse:

which is the equation of an ellipse with center ( a , 0 ) {\displaystyle (a,\,0)} , the x-axis as major axis, and the major/minor semi axis a , b {\displaystyle a,\,b} .

The center of the circle ( x ∘ , y ∘ ) {\displaystyle \left(x_{\circ },\,y_{\circ }\right)} satisfies: [ 1 y 1 − y 2 x 1 − x 2 x 1 − x 3 y 1 − y 3 1 ] [ x ∘ y ∘ ] = [ x 1 2 − x 2 2 + y 1 2 − y 2 2 2 ( x 1 − x 2 ) y 1 2 − y 3 2 + x 1 2 − x 3 2 2 ( y 1 − y 3 ) ] . {\displaystyle {\begin{bmatrix}1&{\dfrac {y_{1}-y_{2}}{x_{1}-x_{2}}}\\[2ex]{\dfrac {x_{1}-x_{3}}{y_{1}-y_{3}}}&1\end{bmatrix}}{\begin{bmatrix}x_{\circ }\\[1ex]y_{\circ }\end{bmatrix}}={\begin{bmatrix}{\dfrac {x_{1}^{2}-x_{2}^{2}+y_{1}^{2}-y_{2}^{2}}{2(x_{1}-x_{2})}}\\[2ex]{\dfrac {y_{1}^{2}-y_{3}^{2}+x_{1}^{2}-x_{3}^{2}}{2(y_{1}-y_{3})}}\end{bmatrix}}.}

P {\displaystyle P} is the center of the rectangle V 1 , V 2 , B , A {\displaystyle V_{1},\,V_{2},\,B,\,A} . The side A B ¯ {\displaystyle {\overline {AB}}} of the rectangle is divided into n equal spaced line segments and this division is projected parallel with the diagonal A V 2 {\displaystyle AV_{2}} as direction onto the line segment V 1 B ¯ {\displaystyle {\overline {V_{1}B}}} and assign the division as shown in the diagram. The parallel projection together with the reverse of the orientation is part of the projective mapping between the pencils at V 1 {\displaystyle V_{1}} and V 2 {\displaystyle V_{2}} needed. The intersection points of any two related lines V 1 B i {\displaystyle V_{1}B_{i}} and V 2 A i {\displaystyle V_{2}A_{i}} are points of the uniquely defined ellipse. With help of the points C 1 , … {\displaystyle C_{1},\,\dotsc } the points of the second quarter of the ellipse can be determined. Analogously one obtains the points of the lower half of the ellipse.

The standard form of an ellipse in Cartesian coordinates assumes that the origin is the center of the ellipse, the x-axis is the major axis, and:

Replacing cos ⁡ t {\displaystyle \cos t} and sin ⁡ t {\displaystyle \sin t} of the standard representation yields: 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}}}}}\right),\,m\in \mathbb {R} .}

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The substitution p = f ( 1 + e ) {\displaystyle p=f(1+e)} yields x 2 ( e 2 − 1 ) + 2 p x − y 2 = 0. {\displaystyle x^{2}\left(e^{2}-1\right)+2px-y^{2}=0.}

For this family of ellipses, one introduces the following q-analog angle measure, which is not a function of the usual angle measure θ:[16][17]

At first the measure is available only for chords which are not parallel to the y-axis. But the final formula works for any chord. The proof follows from a straightforward calculation. For the direction of proof given that the points are on an ellipse, one can assume that the center of the ellipse is the origin.

An affine transformation of the Euclidean plane has the form x → ↦ f → 0 + A x → {\displaystyle {\vec {x}}\mapsto {\vec {f}}\!_{0}+A{\vec {x}}} , where A {\displaystyle A} is a regular matrix (with non-zero determinant) and f → 0 {\displaystyle {\vec {f}}\!_{0}} is an arbitrary vector. If f → 1 , f → 2 {\displaystyle {\vec {f}}\!_{1},{\vec {f}}\!_{2}} are the column vectors of the matrix A {\displaystyle A} , the unit circle ( cos ⁡ ( t ) , sin ⁡ ( t ) ) {\displaystyle (\cos(t),\sin(t))} , 0 ≤ t ≤ 2 π {\displaystyle 0\leq t\leq 2\pi } , is mapped onto the ellipse: 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\,.}

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 a 2 + y 2 b 2 = 1. {\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1.}

where E ( z ∣ m ) {\displaystyle E(z\mid m)} is the incomplete elliptic integral of the second kind with parameter m = k 2 . {\displaystyle m=k^{2}.}

This description of the tangents of an ellipse is an essential tool for the determination of the orthoptic of an ellipse. The orthoptic article contains another proof, without differential calculus and trigonometric formulae.

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 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.}

With the abbreviations M = f → 1 2 + f → 2 2 ,   N = | det ( f → 1 , f → 2 ) | {\displaystyle \;M={\vec {f}}_{1}^{2}+{\vec {f}}_{2}^{2},\ N=\left|\det({\vec {f}}_{1},{\vec {f}}_{2})\right|} the statements of Apollonios's theorem can be written as: a 2 + b 2 = M , a b = N   . {\displaystyle a^{2}+b^{2}=M,\quad ab=N\ .} Solving this nonlinear system for a , b {\displaystyle a,b} yields the semiaxes: a = 1 2 ( M + 2 N + M − 2 N ) b = 1 2 ( M + 2 N − M − 2 N ) . {\displaystyle {\begin{aligned}a&={\frac {1}{2}}({\sqrt {M+2N}}+{\sqrt {M-2N}})\\[1ex]b&={\frac {1}{2}}({\sqrt {M+2N}}-{\sqrt {M-2N}})\,.\end{aligned}}}

Expanding and applying the identities cos 2 ⁡ t − sin 2 ⁡ t = cos ⁡ 2 t ,     2 sin ⁡ t cos ⁡ t = sin ⁡ 2 t {\displaystyle \;\cos ^{2}t-\sin ^{2}t=\cos 2t,\ \ 2\sin t\cos t=\sin 2t\;} gives the equation for t = t 0 . {\displaystyle t=t_{0}\;.}

Let the ellipse be in the canonical form with parametric equation p → ( t ) = ( a cos ⁡ t , b sin ⁡ t ) . {\displaystyle {\vec {p}}(t)=(a\cos t,\,b\sin t).}

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}}}}

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Drawing an ellipse as a graphics primitive is common in standard display libraries, such as the MacIntosh QuickDraw API, and Direct2D on Windows. Jack Bresenham at IBM is most famous for the invention of 2D drawing primitives, including line and circle drawing, using only fast integer operations such as addition and branch on carry bit. M. L. V. Pitteway extended Bresenham's algorithm for lines to conics in 1967.[34] Another efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken.[35]

The area can also be expressed in terms of eccentricity and the length of the semi-major axis as a 2 π 1 − e 2 {\displaystyle a^{2}\pi {\sqrt {1-e^{2}}}} (obtained by solving for flattening, then computing the semi-minor axis).

For f → 0 = ( 0 0 ) , f → 1 = a ( cos ⁡ θ sin ⁡ θ ) , f → 2 = b ( − sin ⁡ θ cos ⁡ θ ) {\displaystyle {\vec {f}}_{0}={0 \choose 0},\;{\vec {f}}_{1}=a{\cos \theta \choose \sin \theta },\;{\vec {f}}_{2}=b{-\sin \theta \choose \;\cos \theta }} one obtains a parametric representation of the standard ellipse rotated by angle θ {\displaystyle \theta } : x = x θ ( t ) = a cos ⁡ θ cos ⁡ t − b sin ⁡ θ sin ⁡ t , y = y θ ( t ) = a sin ⁡ θ cos ⁡ t + b cos ⁡ θ sin ⁡ t . {\displaystyle {\begin{aligned}x&=x_{\theta }(t)=a\cos \theta \cos t-b\sin \theta \sin t\,,\\y&=y_{\theta }(t)=a\sin \theta \cos t+b\cos \theta \sin t\,.\end{aligned}}}

κ = 1 a 2 b 2 ( x 2 a 4 + y 2 b 4 ) − 3 2   , {\displaystyle \kappa ={\frac {1}{a^{2}b^{2}}}\left({\frac {x^{2}}{a^{4}}}+{\frac {y^{2}}{b^{4}}}\right)^{-{\frac {3}{2}}}\ ,}

The ellipse is a special case of the hypotrochoid when R = 2 r {\displaystyle R=2r} , as shown in the adjacent image. The special case of a moving circle with radius r {\displaystyle r} inside a circle with radius R = 2 r {\displaystyle R=2r} is called a Tusi couple.

Ellipses appear in descriptive geometry as images (parallel or central projection) of circles. There exist various tools to draw an ellipse. Computers provide the fastest and most accurate method for drawing an ellipse. However, technical tools (ellipsographs) to draw an ellipse without a computer exist. The principle was known to the 5th century mathematician Proclus, and the tool now known as an elliptical trammel was invented by Leonardo da Vinci.[11]

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 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}}}}}.}

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):

where ∗ {\displaystyle *} is the modified dot product u → ∗ v → = u x v x + q u y v y . {\displaystyle {\vec {u}}*{\vec {v}}=u_{x}v_{x}+{\color {blue}q}\,u_{y}v_{y}.}

where y int {\displaystyle y_{\text{int}}} , x int {\displaystyle x_{\text{int}}} are intercepts and x max {\displaystyle x_{\text{max}}} , y max {\displaystyle y_{\text{max}}} are maximum values. It follows directly from Apollonios's theorem.

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 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 = 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 − 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}}}

Each of the two lines parallel to the minor axis, and at a distance of d = a 2 c = a e {\textstyle d={\frac {a^{2}}{c}}={\frac {a}{e}}} from it, is called a directrix of the ellipse (see diagram).