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An ellipse (red) obtained as the intersection of a cone with an inclined plane
In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. As such, it is a generalization of a circle, which is a special type of an ellipse having both focal points at the same location. The shape of an ellipse (how "elongated" it is) is represented by its eccentricity, which for an ellipse can be any number from 0 (the limiting case of a circle) to arbitrarily close to but less than 1.
Analytically, an ellipse may also be defined as the set of points such that the ratio of the distance of each point on the curve from a given point (called a focus or focal point) to the distance from that same point on the curve to a given line (called the directrix) is a constant. This ratio is the above-mentioned eccentricity of the ellipse.
An ellipse may also be defined analytically as the set of points for each of which the sum of its distances to two foci is a fixed number.
The name, (élleipsis, "omission"), was given by Apollonius of Perga in his Conics, emphasizing the connection of the curve with "application of areas".
Definition of an ellipse as locus of points
Ellipse: definition with director circle
An ellipse can be defined geometrically as a set of points (locus of points) in the Euclidean plane:
An ellipse can be defined using two fixed points, , , called the foci and a distance, usually denoted . The ellipse defined with , and is the set of points such that the sum of the distances is constant and equal to . In order to omit the special case of a line segment, one assumes More formally, for a given , an ellipse is the set
The midpoint 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 called the minor axis. The major axis contains the vertices, which have distance to the center. The distance of the foci to the center is called the focal distance or linear eccentricity. The quotient is the eccentricity.
The case yields a circle and is included.
The equation can be viewed in a different way (see picture):
If is the circle with midpoint and radius , then the distance of a point to the circle equals the distance to the focus :
is called the circular directrix (related to focus ) of the ellipse. This property should not be confused with the definition of an ellipse with help of a directrix (line) below.
An arbitrary line intersects an ellipse at 0, 1 or 2 points. In the first case the line is called exterior line, in the second case tangent and secant in the third case. Through any point of an ellipse there is exactly one tangent.
The tangent at a point of the ellipse has the coordinate equation
A vector equation of the tangent is
Let be an ellipse point and the vector equation of a line (containing ). Inserting the line's equation into the ellipse equation and respecting yields:
In case of line and the ellipse have only point in common and is a tangent. The tangent direction is orthogonal to vector which is then a normal vector of the tangent and the tangent has the equation with a still unknown . Because is on the tangent and on the ellipse, one gets .
In case of line has a second point with the ellipse in common.
With help of (1) one easily checks, that is a tangent vector at point , which proves the vector equation.
Remark: If and are two points of the ellipse, such that
holds, then the points lie on two conjugate diameters of the Ellipse (see below). In case of the ellipse is a circle and "conjugate" means "orthogonal".
Equation of a shifted ellipse
If the ellipse is shifted such that its center is the equation is
The axes are still parallel to the x- and y-axes.
The construction of points based on the parametric equation and the interpretation of parameter t, which is due to de la Hire
Ellipse points calculated by the rational representation with equal spaced parameters ()
Standard parametric representation:
Using the sine and cosine functions, a parametric representation of the ellipse can be obtained, :
Parameter t can be taken as shown in the diagram and is due to de la Hire.
The parameter t (called the eccentric anomaly in astronomy) is not the angle of with the x-axis (see diagram at right). For other interpretations of parameter t see section Drawing ellipses.
With the substitution and trigonometric formulae one gets
and the rational parametric equation of an ellipse
which covers any point of the ellipse except the left vertex .
For this formula represents the right upper quarter of the ellipse moving counter-clockwise with increasing The left vertex is the limit
Rational representations of conic sections are popular with Computer Aided Design (see Bezier curve).
Tangent slope as parameter:
A parametric representation, which uses the slope of the tangent at a point of the ellipse
can be obtained from the derivative of the standard representation :
Replacing und of the standard representation one yields
Where is the slope of the tangent at the corresponding ellipse point, is the upper and the lower half of the ellipse. The points with vertical tangents (vertices) are not covered by the representation.
The equation of the tangent at point has the form . The still unknown can be determined by inserting the coordinates of the corresponding ellipse point :
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, which omits differential calculus and trigonometric formulae.
A shifted ellipse with center can be described by
The parameters and represent the lengths of line segments and are therefore non-negative real numbers. Throughout this article is the semi-major axis, i.e., In general the canonical ellipse equation may have (and hence the ellipse would be taller than it is wide); in this form the semi-major axis would be . This form can be converted to the form assumed in the remainder of this article simply by transposing the variable names and and the parameter names and
Definition of an ellipse by the directrix property
Ellipse: directrix property
The two lines at distance and parallel to the minor axis are called directrices of the ellipse (see diagram).
For an arbitrary point of the ellipse, the quotient of the distance to one focus and to the corresponding directrix (see diagram) is equal to the eccentricity:
The proof for the pair follows from the fact that and satisfy the equation
The second case is proven analogously.
The inverse statement is also true and can be used to define an ellipse (in a manner similar to the definition of a parabola):
For any point (focus), any line (directrix) not through , and any real number with the locus of points for which the quotient of the distances to the point and to the line is that is,
is an ellipse.
The choice , which is the eccentricity of a circle, is in this context not allowed. One may consider the directrix of a circle to be the line at infinity.
Pencil of conics with a common vertex and common semi-latus rectum
Let , and assume is a point on the curve.
The directrix has equation . With , the relation produces the equations
The substitution yields
This is the equation of an ellipse (), or a parabola (), or a hyperbola (). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram).
If , introduce new parameters so that
, and then the equation above becomes
which is the equation of an ellipse with center , the x-axis as major axis, and
the major/minor semi axis .
If the focus is and the directrix , one obtains the equation
(The right side of the equation uses the Hesse normal form of a line to calculate the distance .)
The normal bisects the angle between the lines to the foci
Ellipse: the tangent bisects the supplementary angle of the angle between the lines to the foci.
Rays from one focus reflect off the ellipse to pass through the other focus.
For an ellipse the following statement is true:
The normal at a point bisects the angle between the lines .
Because the tangent is perpendicular to the normal, the statement is true for the tangent and the supplementary angle of the angle between the lines to the foci (see diagram), too.
Let be the point on the line with the distance to the focus , is the semi-major axis of the ellipse. Let line be the bisector of the supplementary angle to the angle between the lines . In order to prove that is the tangent line at point , one checks that any point on line which is different from cannot be on the ellipse. Hence has only point in common with the ellipse and is, therefore, the tangent at point .
From the diagram and the triangle inequality one recognizes that holds, which means: . But if is a point of the ellipse, the sum should be .
The rays from one focus are reflected by the ellipse to the second focus. This property has optical and acoustic applications similar to the reflective property of a parabola (see whispering gallery).
Ellipse as an affine image of the unit circle x²+y²=1
Any ellipse is the affine image of the unit circle with equation .
An affine transformation of the Euclidean plane has the form , where is a regular matrix (its determinant is not 0) and is an arbitrary vector. If are the column vectors of the matrix , the unit circle is mapped onto the Ellipse
is the center, are the directions of two conjugate diameters of the ellipse. In general the vectors are not perpendicular. That means, in general and are not the vertices of the ellipse.
The tangent vector at point is
Because at a vertex the tangent is perpendicular to the major/minor axis (diameters) of the ellipse one gets the parameter of a vertex from the equation
(The formulae were used.)
If , then .
The four vertices of the ellipse are
The advantage of this definition is that one gets a simple parametric representation of an arbitrary ellipse, even in the space, if the vectors are vectors of the Euclidean space.
Conjugate diameters and the midpoints of parallel chords
Orthogonal diameters of a circle with a square of tangents, midpoints of parallel chords and an affine image, which is an ellipse with conjugate diameters, a parallelogram of tangents and midpoints of chords
For a circle, the property (M) holds:
(M) The midpoints of parallel chords lie on a diameter.
The diameter and the parallel chords are orthogonal. An affine transformation in general does not preserve orthogonality but does preserve parallelism and midpoints of line segments. Hence: property (M) (which omits the term orthogonal) is true for any ellipse.
Two diameters of an ellipse are conjugate if the midpoints of chords parallel to lie on
From the diagram one finds:
(T) Two diameters , of an ellipse are conjugate, if the tangents at and are parallel to and visa versa.
The term conjugate diameters is a kind of generalization of orthogonal.
of an ellipse, any pair of points belong to a diameter and the pair belongs to its conjugate diameter.
Ellipse with its orthoptic
For the ellipse
the intersection points of orthogonal tangents lie on the circle .
This circle is called orthoptic of the given ellipse.
Theorem of Apollonios on conjugate diameters
Ellipse: theorem of Apollonios on conjugate diameters
For an ellipse with semi-axes the following is true:
Let and be halves of two conjugate diameters (see diagram) then
(2) the triangle formed by has the constant area
(3) the parallelogram of tangents adjacent to the given conjugate diameters has the
Let the ellipse be in the canonical form with parametric equation
The two points are on conjugate diameters (see previous section). From trigonometric formulae one gets and
The area of the triangle generated by is
and from the diagram it can be seen that the area of the parallelogram is 8 times that of . Hence
Central projection of circles (gate)
Ellipses appear in descriptive geometry as images (parallel or central projection) of circles (for details: see Ellipses in DG (German)). 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 of ellipsographs were known to Greek mathematicians (Archimedes, Proklos).
the knowledge of the axes and the semi-axes is necessary (or equivalent: 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.
Ellipse: gardener's method
The characterization of an ellipse as the locus of points so that sum of the distances to the foci is constant leads to a method of drawing one using two drawing pins, a length of string, and a pencil. In this method, pins are pushed into the paper at two points, which become the ellipse's foci. A string tied at each end to the two pins and the tip of a pencil pulls the loop taut to form a triangle. The tip of the pencil then traces an ellipse if it is moved while keeping the string taut. Using two pegs and a rope, gardeners use this procedure to outline an elliptical flower bed--thus it is called the gardener's ellipse.
The two following methods rely on the parametric representation (see section parametric representation, above):
This representation can be modeled technically by two simple methods. In both cases center, the axes and semi axes have to be known.
The first method starts with
a strip of paper of length .
The point, where the semi axes meet is marked by . If the strip slides with both ends on the axes of the desired ellipse, then point P traces the ellipse. For the proof one shows that point has the parametric representation , where parameter is the angle of the slope of the paper strip.
A technichal realization of the motion of the paper strip can be achieved by a Tusi couple (s. animation). The device is able to draw any ellipse with a fixed sum , 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.
A nice application: If one stands somewhere in the middle of a ladder, which stands on a slippery ground and leans on a slippery wall, the ladder slides down and the persons feet trace an ellipse.
Ellipse construction: paper strip method 1
Ellipses with Tusi couple. Two examples: red and cyan.
A variation of the paper strip method 1 uses the observation, that the midpoint of the paper strip is moving on the circle with center (of the ellipse) and radius . Hence the paperstrip can be cut at point into halves, connected again by a joint at and the sliding end fixed at the center (see diagram). After this operation the movement of the unchanged half of the paperstrip is unchanged. The advantage of this variation is: Only one expensive sliding shoe is necessary.
One should be aware that the end, which is sliding on the minor axis, has to be changed.
Variation of the paper strip method 1
Animation of the variation of the paper strip method 1
Ellipse construction: paper strip method 2
The second method starts with
a strip of paper of length .
One marks the point, which divides the strip into two substrips of length and . 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 , where parameter is the angle of slope of the paper strip.
This method is the base for several ellipsographs (see section below).
Remark: Similar to the variation of the paper strip method 1 a variation of the paper strip method 2 can be established (see diagram) by cutting the part between the axes into halves.
Given two pencils of lines at two points (all lines containing and , respectively) and a projective but not perspective mapping of onto , then the intersection points of corresponding lines form a non-degenerate projective conic section.
For the generation of points of the ellipse one uses the pencils at the vertices . Let be an upper co-vertex of the ellipse and .
is the center of the rectangle . The side of the rectangle is divided into n equal spaced line segments and this division is projected parallel with the diagonal as direction onto the line segment 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 and needed. The intersection points of any two related lines and are points of the uniquely defined ellipse. With help of the points the points of the second quarter of the ellipse can be determined. Analogously one gets the points of the lower half of the ellipse.
The Steiner generation exists for hyperbolas and parabolas, too.
The Steiner generation 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.
Most technical instruments for drawing ellipses are based on the second paperstrip method.
where the ratio is fixed.
With the abbreviation one gets the more convenient form
Such ellipses have their axes parallel to the coordinate axes and their eccentricity fixed. Their major axes are parallel to the x-axis if and parallel to the y-axis if .
Inscribed angle theorem for an ellipse
Like a circle, such an ellipse is determined by three points not on a line.
In this more general case one introduces the following measurement of an angle,:
In order to measure an angle between two lines with equations one uses the quotient
Inscribed angle theorem for ellipses
For four points , no three of them on a line (see diagram), the following statement is true:
The four points are on an ellipse with equation , if and only if the angles at and are equal in the sense of the measurement above--that is, if
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.
A consequence of the inscribed angle theorem for ellipses is the
3-point-form of an ellipse's equation:
One gets the equation of the ellipse determined by 3 points not on a line by a conversion of the equation
Analogously to the circle case this formula can be written more clearly using vectors:
Any ellipse can be described in a suitable coordinate system by an equation . The equation of the tangent at a point of the ellipse is If one allows point to be an arbitrary point different from the origin, then
point is mapped onto the line , not through the center of the ellipse.
This relation between points and lines is a bijection.
where and are the lengths of the semi-major and semi-minor axes, respectively. The area formula is intuitive: start with a circle of radius (so its area is ) and stretch it by a factor to make an ellipse. This scales the area by the same factor: It is also easy to rigorously prove the area formula using integration as follows. Equation (1) can be rewritten as For this curve is the top half of the ellipse. So twice the integral of over the interval will be the area of the ellipse:
The second integral is the area of a circle of radius that is, So
Here are the point coordinates in the canonical system, whose origin is the center of the ellipse, whose -axis is the unit vector coinciding with the major axis, and whose -axis is the perpendicular vector coinciding with the minor axis. That is, and .
In this system, the center is the origin and the foci are and for eccentricity e.
Any ellipse can be obtained by rotation and translation of a canonical ellipse with the proper semi-diameters. The expression of an ellipse centered at is
Moreover, any canonical ellipse can be obtained by scaling the unit circle of , defined by the equation
by factors a and b along the two axes.
For an ellipse in canonical form, we have
The distances from a point on the ellipse to the left and right foci are and , respectively.
The canonical form coefficients can be obtained from the general form coefficients using the following equations:
where is the angle from the positive horizontal axis to the ellipse's major axis.
Polar form relative to center
Polar coordinates centered at the center
In polar coordinates, with the origin at the center of the ellipse and with the angular coordinate measured from the major axis, the ellipse's equation is:p. 75
Polar form relative to focus
Polar coordinates centered at focus
If instead we use polar coordinates with the origin at one focus, with the angular coordinate still measured from the major axis, the ellipse's equation is
where the sign in the denominator is negative if the reference direction points towards the center (as illustrated on the right), and positive if that direction points away from the center.
In the slightly more general case of an ellipse with one focus at the origin and the other focus at angular coordinate , the polar form is
The angle in these formulas is called the true anomaly of the point. The numerator of these formulas is the semi-latus rectum of the ellipse, usually denoted . It is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular to the major axis.
Ellipse as hypotrochoid
An ellipse (in red) as a special case of the hypotrochoid with R = 2r
The ellipse is a special case of the hypotrochoid when R = 2r, as shown in the adjacent image. The special case of a moving circle with radius inside a circle with radius is called a Tusi couple.
Ellipses as plane sections of quadrics
Ellipses appear as plane sections of the following quadrics:
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.
Similarly, if a light source is placed at one focus of an elliptic mirror, all light rays on the plane of the ellipse are reflected to the second focus. Since no other smooth curve has such a property, it can be used as an alternative definition of an ellipse. (In the special case of a circle with a source at its center all light would be reflected back to the center.) If the ellipse is rotated along its major axis to produce an ellipsoidal mirror (specifically, a prolate spheroid), this property holds for all rays out of the source. Alternatively, a cylindrical mirror with elliptical cross-section can be used to focus light from a linear fluorescent lamp along a line of the paper; such mirrors are used in some document scanners.
More generally, in the gravitational two-body problem, if the two bodies are bound to each other (that is, the total energy is negative), their orbits are similar ellipses with the common barycenter being one of the foci of each ellipse. The other focus of either ellipse has no known physical significance. The orbit of either body in the reference frame of the other is also an ellipse, with the other body at the same focus.
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.)
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.
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 display is an ellipse, rather than a straight line, the two signals are out of phase.
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.
Elliptical bicycle gears make it easier for the chain to slide off the cog when changing gears.
An example gear application would be a device that winds thread onto a conical bobbin on a spinning machine. The bobbin would need to wind faster when the thread is near the apex than when it is near the base.
In lamp-pumped solid-state lasers, elliptical cylinder-shaped reflectors have been used to direct light from the pump lamp (coaxial with one ellipse focal axis) to the active medium rod (coaxial with the second focal axis).
In laser-plasma produced EUV light sources used in microchip lithography, EUV light is generated by plasma positioned in the primary focus of an ellipsoid mirror and is collected in the secondary focus at the input of the lithography machine.
Statistics and finance
In statistics, a bivariate random vector (X, Y) is jointly elliptically distributed if its iso-density contours--loci of equal values of the density function--are ellipses. The concept extends to an arbitrary number of elements of the random vector, in which case in general the iso-density contours are ellipsoids. A special case is the multivariate normal distribution. The elliptical distributions are important in finance because if rates of return on assets are jointly elliptically distributed then all portfolios can be characterized completely by their mean and variance--that is, any two portfolios with identical mean and variance of portfolio return have identical distributions of portfolio return.
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. Another efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken.
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. These algorithms need only a few multiplications and additions to calculate each vector.
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.
Drawing with Bézier paths
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.
It is sometimes useful to find the minimum bounding ellipse on a set of points. The ellipsoid method is quite useful for attacking this problem.