Ellipse
ELLIPSE (adapted from Gr., a deficiency, to fall behind), in mathematics, a conic section, having the form of a closed oval. It admits of several definitions framed according to the aspect from which the curve is considered. In solido, i.e. as a section of a cone or cylinder, it may be defined, after Menaechmus, as the perpendicular section of an "acute-angled" cone; or, after Apollonius of Perga, as the section of any cone by a plane at a less inclination to the base than a generator; or as an oblique section of a right cylinder. Definitions in plano are generally more useful; of these the most important are: (1) the ellipse is the conic section which has its eccentricity less than unity: this involves the notion of one directrix and one focus; (2) the ellipse is the locus of a point the sum of whose distances from two fixed points is constant: this involves the notion of two foci. Other geometrical definitions are: it is the oblique projection of a circle; the polar reciprocal of a circle for a point within it; and the conic which intersects the line at infinity in two imaginary points. Analytically it is defined by an equation of the second degree of which the highest terms represent two imaginary lines. The curve has important mechanical relations, in particular it is the orbit of a particle moving under the influence of a central force which varies inversely as the square of the distance of the particle; this is the gravitational law of force, and the curve consequently represents the orbits of the planets if only an individual planet and the Sun be considered; the other planets, however, disturb this orbit (see Mechanics).
The relation of the ellipse to the other conic sections is treated in the articles Conic Section and Geometry; in this article a summary of the properties of the curve will be given.
To investigate the form of the curve use may be made of the definition: the ellipse is the locus of a point which moves so that the ratio of its distance from a fixed point (the focus) to its distance from a straight line (the directrix) is constant and is less than unity. This ratio is termed the eccentricity, and will be denoted by e. Let KX (fig. 1) be the directrix, S the focus, and X the foot of the perpendicular from S to KX. If SX be divided at A so that SA/AX = e, then A is a point on the curve. SX may be also divided externally at A', so that SA'/A'X = e, since e is less than unity; the points A and A' are the vertices, and the line AA' the major axis of the curve. It is obvious that the curve is symmetrical about AA'. If AA' be bisected at C, and the line BCB' be drawn perpendicular to AA', then it is readily seen that the curve is symmetrical about this line also; since if we take S' on AA' so that S'A' = SA, and a line K'X' parallel to KX such that AX = A'X', then the same curve will be described if we regard K'X' and S' as the given directrix and focus, the eccentricity remaining the same. If B and B' be points on the curve, BB' is the minor axis and C the centre of the curve.
Metrical relations between the axes, eccentricity, distance between the foci, and between these quantities and the co-ordinates of points on the curve (referred to the axes and the centre), and focal distances are readily obtained by the methods of geometrical conics or analytically. The semi-major axis is generally denoted by a, and the semi-minor axis by b, and we have the relation b = a (1 − e). Also a = CS·CX, i.e. the square on the semi-major axis equals the rectangle contained by the distances of the focus and directrix from the centre; and 2a = SP + S'P, where P is any point on the curve, i.e. the sum of the focal distances of any point on the curve equals the major axis. The most important relation between the co-ordinates of a point on an ellipse is: if N be the foot of the perpendicular from a point P, then the square on PN bears a constant ratio to the product of the segments AN, NA' of the major axis, this ratio being the square of the ratio of the minor to the major axis; symbolically PN = AN·NA' (CB/CA). From this or otherwise it is readily deduced that the ordinates of an ellipse and of the circle described on the major axis are in the ratio of the minor to the major axis. This circle is termed the auxiliary circle.
Of the properties of a tangent it may be noticed that the tangent at any point is equally inclined to the focal distances of that point; that the feet of the perpendiculars from the foci on any tangent always lie on the auxiliary circle, and the product of these perpendiculars is constant, and equal to the product of the distances of a focus from the two vertices. From any point without the curve two, and only two, tangents can be drawn; if OP, OP' be two tangents from O, and S, S' the foci, then the angles OSP, OSP' are equal and also SOP, S'OP'. If the tangents be at right angles, then the locus of the point is a circle having the same centre as the ellipse; this is named the director circle.
The middle points of a system of parallel chords is a straight line, and the tangent at the point where this line meets the curve is parallel to the chords. The straight line and the line through the centre parallel to the chords are named conjugate diameters; each bisects the chords parallel to the other. An important metrical property of conjugate diameters is the sum of their squares equals the sum of the squares of the major and minor axis.
In analytical geometry, the equation ax + 2hxy + by + 2gx + 2fy + c = 0 represents an ellipse when ab > h; if the centre of the curve be the origin, the equation is ax + 2hxy + by = C, and if in addition a pair of conjugate diameters are the axes, the equation is further simplified to Ax + By = C. The simplest form is x/a + y/b = 1, in which the centre is the origin and the major and minor axes the axes of co-ordinates. It is obvious that the co-ordinates of any point on an ellipse may be expressed in terms of a single parameter, the abscissa being a cos φ, and the ordinate b sin φ, since on eliminating φ between x = a cos φ and y = b sin φ we obtain the equation to the ellipse. The angle φ is termed the eccentric angle, and is geometrically represented as the angle between the axis of x (the major axis of the ellipse) and the radius of a point on the auxiliary circle which has the same abscissa as the point on the ellipse.
The equation to the tangent at θ is x cos θ/a + y sin θ/b = 1, and to the normal ax/cos θ − by/sin θ = a − b.
The area of the ellipse is πab, where a, b are the semi-axes; this result may be deduced by regarding the ellipse as the orthogonal projection of a circle, or by means of the calculus. The perimeter can only be expressed as a series, the analytical evaluation leading to an integral termed elliptic (see Function, ii. Complex). There are several approximation formulae: - S = π(a + b) makes the perimeter about 1/200th too small; s =π√(a + b) about 1/200th too great; 2s = π(a + b) + π√(a + b) is within 1/30,000 of the truth.
An ellipse can generally be described to satisfy any five conditions. If five points be given, Pascal's theorem affords a solution; if five tangents, Brianchon's theorem is employed. The principle of involution solves such constructions as: given four tangents and one point, three tangents and two points, etc. If a tangent and its point of contact be given, it is only necessary to remember that a double point on the curve is given. A focus or directrix is equal to two conditions; hence such problems as: given a focus and three points; a focus, two points and one tangent; and a focus, one point and two tangents are soluble (very conveniently by employing the principle of reciprocation). Of practical importance are the following constructions: - (1) Given the axes; (2) given the major axis and the foci; (3) given the focus, eccentricity and directrix; (4) to construct an ellipse (approximately) by means of circular arcs.
(1) If the axes be given, we may avail ourselves of several constructions, (a) Let AA', BB' be the axes intersecting at right angles in a point C. Take a strip of paper or rule and mark off from a point P, distances Pa and Pb equal respectively to CA and CB. If now the strip be moved so that the point a is always on the minor axis, and the point b on the major axis, the point P describes the ellipse. This is known as the trammel construction.
(b) Let AA', BB' be the axes as before; describe on each as diameter a circle. Draw any number of radii of the two circles, and from the points of intersection with the major circle draw lines parallel to the minor axis, and from the points of intersection with the minor circle draw lines parallel to the major axis. The intersections of the lines drawn from corresponding points are points on the ellipse.
(2) If the major axis and foci be given, there is a convenient mechanical construction based on the property that the sum of the focal distances of any point is constant and equal to the major axis. Let AA' be the axis and S, S' the foci. Take a piece of thread of length AA', and fix it at its extremities by means of pins at the foci. The thread is now stretched taut by a pencil, and the pencil moved; the curve traced out is the desired ellipse.
(3) If the directrix, focus and eccentricity be given, we may employ the general method for constructing a conic. Let S (fig. 2) be the focus, KX the directrix, X being the foot of the perpendicular from S to the directrix. Divide SX internally at A and externally at A', so that the ratios SA/AX and SA'/A'X are each equal to the eccentricity. Then A, A' are the vertices of the curve. Take any point R on the directrix, and draw the lines RAM, RSN; draw SL so that the angle LSN = angle NSA'. Let P be the intersection of the line SL with the line RAM, then it can be readily shown that P is a point on the ellipse. For, draw through P a line parallel to AA', intersecting the directrix in Q and the line RSN in T. Then since XS and QT are parallel and are intersected by the lines RK, RM, RN, we have SA/AX = TP/PQ = SP/PQ, since the angle PST = angle PTS. By varying the position of R other points can be found, and, since the curve is symmetrical about both the major and minor axes, it is obvious that any point may be reflected in both the axes, thus giving 3 additional points.
(4) If the axes be given, the curve can be approximately constructed by circular arcs in the following manner: - Let AA', BB' be the axes; determine D the intersection of lines through B and A parallel to the major and minor axes respectively. Bisect AD at E and join EB. Then the intersection of EB and DB' determines a point P on the (true) curve. Bisect the chord PB at G, and draw through G a line perpendicular to PB, intersecting BB' in O. An arc with centre O and radius OB forms part of a curve. Let this arc on the reverse side to P intersect a line through O parallel to the major axis in a point H. Then HA will cut the circular arc in J. Let JO intersect the major axis in O1. Then with centre O1 and radius OJ = OA, describe an arc. By reflecting the two arcs thus described over the centre the ellipse is approximately described.
Note - this article incorporates content from Encyclopaedia Britannica, Eleventh Edition, (1910-1911)
