An ellipse obtained as the intersection of a cone
with an inclined plane.
In mathematics, an ellipse is a curve on 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 that has 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.
Ellipses are the closed type of conic section: a plane curve that results from the intersection of a cone by a plane. (See figure to the right.) Ellipses have many similarities with the other two forms of conic sections: the parabolas and the hyperbolas, both of which are open and unbounded. The cross section of a cylinder is an ellipse, unless the section is parallel to the axis of the cylinder.
Analytically, an ellipse can 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, called the eccentricity of the ellipse.
Ellipses are common in physics, astronomy and engineering. For example, the orbit of each planet in the solar system is an ellipse with the barycenter of the planet-Sun pair at one of the focal points. The same is true for moons orbiting planets and all other systems having two astronomical bodies. The shape of planets and stars are often well described by ellipsoids. Ellipses also arise as images of a circle under parallel projection and the bounded cases of perspective projection, which are simply intersections of the projective cone with the plane of projection. It 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, emphasizing the connection of the curve with "application of areas".
Elements of an ellipse
The ellipse and some of its mathematical properties.
The distance traveled from one focus to another, via some point on the ellipse, is the same regardless of the point selected.
Ellipses have two perpendicular axes about which the ellipse is symmetric. These axes intersect at the center of the ellipse due to this symmetry. The larger of these two axes, which corresponds to the largest distance between antipodal points on the ellipse, is called the major axis. (On the figure to the right it is represented by the line segment between the point labeled −a and the point labeled a.) The smaller of these two axes, and the smallest distance across the ellipse, is called the minor axis. (On the figure to the right it is represented by the line segment between the point labeled −b to the point labeled b.)
The semi-major axis (denoted by a in the figure) and the semi-minor axis (denoted by b in the figure) are one half of the major and minor axes, respectively. These are sometimes called (especially in technical fields) the major and minor semi-axes, the major and minor semiaxes, or major radius and minor radius.
The four points where these axes cross the ellipse are the vertices and are marked as a, −a, b, and −b. In addition to being at the largest and smallest distance from the center, these points are where the curvature of the ellipse is maximum and minimum.
The two foci (the term focal points is also used) of an ellipse are two special points F1 and F2 on the ellipse's major axis that are equidistant from the center point. The sum of the distances from any point P on the ellipse to those two foci is constant and equal to the major axis (PF1 + PF2 = 2a). (On the figure to the right this corresponds to the sum of the two green lines equaling the length of the major axis that goes from −a to a.)
The distance to the focal point from the center of the ellipse is sometimes called the linear eccentricity, f, of the ellipse. Here it is denoted by f, but it is often denoted by c. Due to the Pythagorean theorem and the definition of the ellipse explained in the previous paragraph: f2 = a2 −b2.
A second equivalent method of constructing an ellipse using a directrix is shown on the plot as the three blue lines. (See the Directrix section of this article for more information about this method). The dashed blue line is the directrix of the ellipse shown.
The eccentricity of an ellipse, usually denoted by ε or e, is the ratio of the distance between the two foci, to the length of the major axis or e = 2f/2a = f/a. For an ellipse the eccentricity is between 0 and 1 (0 < e < 1). When the eccentricity is 0 the foci coincide with the center point and the figure is a circle. As the eccentricity tends toward 1, the ellipse gets a more elongated shape. It tends towards a line segment (see below) if the two foci remain a finite distance apart and a parabola if one focus is kept fixed as the other is allowed to move arbitrarily far away. The eccentricity is also equal to the ratio of the distance (such as the (blue) line PF2) from any particular point on an ellipse to one of the foci to the perpendicular distance to the directrix from the same point (line PD), e = PF2/PD.
Drawing an ellipse with two pins, a loop, and a pen
Ellipse construction applying the parallelogram 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 will become the ellipse's foci. A string tied at each end to the two pins and the tip of a pen is used to pull the loop taut so as to form a triangle. The tip of the pen will then trace an ellipse if it is moved while keeping the string taut. Using two pegs and a rope, this procedure is traditionally used by gardeners to outline an elliptical flower bed; thus it is called the gardener's ellipse.
An ellipse can also be drawn using a ruler, a set square, and a pencil:
- Draw two perpendicular lines M,N on the paper; these will be the major (M) and minor (N) axes of the ellipse. Mark three points A, B, C on the ruler. A->C being the length of the semi-major axis and B->C the length of the semi-minor axis. With one hand, move the ruler on the paper, turning and sliding it so as to keep point A always on line N, and B on line M. With the other hand, keep the pencil's tip on the paper, following point C of the ruler. The tip will trace out an ellipse.
The trammel of Archimedes, or ellipsograph, is a mechanical device that implements this principle. The ruler is replaced by a rod with a pencil holder (point C) at one end, and two adjustable side pins (points A and B) that slide into two perpendicular slots cut into a metal plate. The mechanism can be used with a router to cut ellipses from board material. The mechanism is also used in a toy called the "nothing grinder".
In the parallelogram method, an ellipse is constructed point by point using equally spaced points on two horizontal lines and equally spaced points on two vertical lines. It is based on Steiner's theorem on the generation of conic sections. Similar methods exist for the parabola and hyperbola.
Mathematical definitions and properties
In Euclidean geometry
In Euclidean geometry, the ellipse is usually defined as the bounded case of a conic section, or as the set of points such that the sum of the distances to two fixed points (the foci) is constant. The ellipse can also be defined as the set of points such that the distance from any point in that set to a given point in the plane (a focus) is a constant positive fraction less than 1 (the eccentricity) of the perpendicular distance of the point in the set to a given line (called the directrix). Yet another equivalent definition of the ellipse is that it is the set of points that are equidistant from one point in the plane (a focus) and a particular circle, the directrix circle (whose center is the other focus).
The equivalence of these definitions can be proved using the Dandelin spheres.
The equation of an ellipse whose major and minor axes coincide with the Cartesian axes is . This can be explained as follows:
If we let
Then plotting x and y values for all angles of θ between 0 and 2π results in an ellipse (e.g. at θ = 0, x = a, y = 0 and at θ = π/2, y = b, x = 0).
Squaring both equations gives:
Dividing these two equations by a2 and b2 respectively gives:
Adding these two equations together gives:
Applying the Pythagorean identity to the right hand side gives:
This means any noncircular ellipse is a compressed (or stretched) circle. If a circle is treated like an ellipse, then the area of the ellipse would be proportional to the length of either axis (i.e. doubling the length of an axis in a circular ellipse would create an ellipse with double the area of the original circle).
The distance from the center C to either focus is f = ae, which can be expressed in terms of the major and minor radii:
The sum of the distances from any point P = P(x,y) on the ellipse to those two foci is constant and equal to the major axis (proof):
The eccentricity of the ellipse (commonly denoted as either e or ) is
(where again a and b are one-half of the ellipse's major and minor axes respectively, and f is the focal distance) or, as expressed in terms using the flattening factor
Other formulas for the eccentricity of an ellipse are listed in the article on eccentricity of conic sections. Formulas for the eccentricity of an ellipse that is expressed in the more general quadratic form are described in the article dedicated to conic sections.
Each focus F of the ellipse is associated with a line parallel to the minor axis called a directrix. Refer to the illustration on the right, in which the ellipse is centered at the origin. The distance from any point P on the ellipse to the focus F is a constant fraction of that point's perpendicular distance to the directrix, resulting in the equality e = PF/PD. The ratio of these two distances is the eccentricity of the ellipse. This property (which can be proved using the Dandelin spheres) can be taken as another definition of the ellipse.
Besides the well-known ratio e = f/a, where f is the distance from the center to the focus and a is the distance from the center to the farthest vertices (most sharply curved points of the ellipse), it is also true that e = a/d, where d is the distance from the center to the directrix.
The ellipse can also be defined as the set of points that are equidistant from one focus and a circle, the directrix circle, that is centered on the other focus. The radius of the directrix circle equals the ellipse's major axis, so the focus and the entire ellipse are inside the directrix circle.
Ellipse as hypotrochoid
An ellipse (in red) as a special case of the hypotrochoid
The ellipse is a special case of the hypotrochoid when R = 2r.
The area enclosed by an ellipse is:
where a and b are the semi-major and semi-minor axes (1⁄2 of the ellipse's major and minor axes), respectively.
An ellipse defined implicitly by has area .
The area formula πab is intuitive: start with a circle of radius b (so its area is πb2) and stretch it by a factor a/b to make an ellipse. This intuitively justifies the area by the same factor: πb2(a/b) = πab. However, a more rigorous proof requires integration as follows:
For the ellipse in standard form, , and hence , with horizontal intercepts at ± a, the area can be computed as twice the integral of the positive square root:
The second integral is the area of a circle of radius , i.e., ; thus we have:
The area formula can also be proven in terms of polar coordinates using the coordinate transformation
Any point inside the ellipse with x-intercept a and y-intercept b can be defined in terms of r and , where and .
To define the area differential in such coordinates we use the Jacobian matrix of the coordinate transformation times :
We now integrate over the ellipse to find the area:
The circumference of an ellipse is:
where again a is the length of the semi-major axis and e is the eccentricity and where the function is the complete elliptic integral of the second kind (the arc length of an ellipse, in general, has no closed-form solution in terms of elementary functions and the elliptic integrals were motivated by this problem). This may be evaluated directly using the Carlson symmetric form. This gives a succinct and rapidly converging method for evaluating the circumference.
The exact infinite series is:
where is the double factorial. Unfortunately, this series converges rather slowly; however, by expanding in terms of , Ivory and Bessel derived an expression which converges much more rapidly,
Ramanujan gives two good approximations for the circumference in §16 of; they are
The errors in these approximations, which were "obtained empirically", are of order and , respectively.
More generally, the arc length of a portion of the circumference, as a function of the angle subtended, is given by an incomplete elliptic integral.
The inverse function, the angle subtended as a function of the arc length, is given by the elliptic functions.
Some lower and upper bounds on the circumference of the canonical ellipse with a ≥b are
Here the upper bound is the circumference of a circumscribed concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound is the perimeter of an inscribed rhombus with vertices at the endpoints of the major and minor axes.
The midpoints of a set of parallel chords of an ellipse are collinear.:p.147
The chords of an ellipse which are perpendicular to the major axis and pass through one of its foci are called the latera recta of the ellipse. The length of each latus rectum is 2b2/a.
The curvature is given by . A local normal to the ellipse bisects the angle shown in the figure above. This is evident graphically in the parallelogram method of construction, and can be proven analytically, for example by using the parametric form in canonical position, as given below.
In a projective geometry defined over a field, a conic section can be defined as the set of all points of intersection between corresponding lines of two pencils of lines in a plane which are related by a projective, but not perspective, map (see Steiner's theorem). By projective duality, a conic section can also be defined as the envelope of all lines that connect corresponding points of two lines which are related by a projective, but not perspective, map.
In a pappian projective plane (one defined over a field), all conic sections are equivalent to each other, and the different types of conic sections are determined by how they intersect the line at infinity, denoted by Ω. An ellipse is a conic section which does not intersect this line. A parabola is a conic section that is tangent to Ω, and a hyperbola is one that crosses Ω twice. Since an ellipse does not intersect the line at infinity, it properly belongs to the affine plane determined by removing the line at infinity and all of its points from the projective plane.
An ellipse is also the result of projecting a circle, sphere, or ellipse in a three dimensional affine space onto a plane (flat), by parallel lines. This is a special case of conical (perspective) projection of any of those geometric objects in the affine space from a point O onto a plane P, when the point O lies in the plane at infinity of the affine space. In the setting of pappian projective planes, the image of an ellipse by any affine map (a projective map which leaves the line at infinity invariant) is an ellipse, and, more generally, the image of an ellipse by any projective map M such that the line M−1(Ω) does not touch or cross the ellipse is an ellipse.
In analytic geometry
In analytic geometry, the ellipse is defined as the set of points of the Cartesian plane that, in non-degenerate cases, satisfy the implicit equation
To distinguish the degenerate cases from the non-degenerate case, let ∆ be the determinant
Then thee 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.:p.63
The general equation's coefficients can be obtained from known semi-major axis , semi-minor axis , center coordinates Source: this wikipedia article, under CC-BY-SA.