See also:

• ELLIPSE (adapted from Gr. EXkei 'tc, a deficiency, iXXeirely, to fall behind)

ELLIPSE (adapted from Gr. EXkei 'tc, a deficiency, iXXeirely, to fall behind)  , in See also:

• MATHEMATICS (Gr. /saBnµar1Kil, Sc. vOcvn or E7rlQTt'µn; from p iN.La, "learning" or "science ")

mathematics, a conic See also:

• SECTION
• SECTION (Lat. sectio, cutting, secare, to cut)

section, having the See also:

• FORM (Lat. forma)

form of a closed See also:

• OVAL (Lat. ovum, egg)

oval . It admits of several See also:

• DEFINITIONS

definitions framed according to the aspect from which the See also:

• CURVE (Lat. curvus, bent)

curve is considered . In solido, i.e. as a section of a See also:

• CONE (Gr. Kwvos)

cone or See also:

• CYLINDER (Gr. KvAwSpos, from KvXivaety, to roll)

cylinder, it may be defined, after Menaechmus, as the perpendicular section of an "acute-angled" cone; or, after See also:

• APOLLONIUS

Apollonius of See also:

• PERGA (mod. Murtana)

Perga, as the section of any cone by a See also:

• PLANE

plane at a less inclination to the See also:

• BASE

base than a generator; or as an oblique section of a right cylinder . Definitions in piano are generally more useful; of these the most important are: (I) the See also:

• ELLIPSE (adapted from Gr. EXkei 'tc, a deficiency, iXXeirely, to fall behind)

ellipse is the conic section which has its eccentricity less than unity: this involves the notion of one directrix and one See also:

• FOCUS (Latin for " hearth " or " fireplace ")

focus; (2) the ellipse is the See also:

• LOCUS (Lat. for " place "; in Gr. rinros)

locus of a point the sum of whose distances from two fixed points is See also:

• CONSTANT, BENJAMIN (1845-1902)

constant: this involves the notion of two foci . Other geometrical definitions are: it is the oblique See also:

• PROJECTION

projection of a circle; the polar reciprocal of a circle for a point within it; and the conic which intersects the See also:

• LINE

line at infinity in two imaginary points . Analytically it is defined by an See also:

• EQUATION (from Lat. aequatio, aequare, to equalize)

equation of the second degree of which the highest terms represent two imaginary lines . The curve has important See also:

• MECHANICAL

mechanical relations, in particular it is the See also:

• ORBIT (from Lat. orbita, a track, orbis, a wheel)

orbit of a particle moving under the See also:

• INFLUENCE (Late Lat. influentia, from influere, to flow in)

influence of a central force which varies inversely as the square of the distance of the particle; this is the gravitational See also:

• LAW
• LAW (0. Eng. lagu, M. Eng. lawe; from an old Teutonic root lag, " lie," what lies fixed or evenly; cf. Lat. lex, Fr. loi)
• LAW, JOHN (1671—1729)
• LAW, WILLIAM (1686-1761)

law of force, and the curve consequently represents the orbits of the See also:

• PLANETS, MINOR

planets if only an individual See also:

• PLANET (Gr. ssXavirrns, a wanderer)

planet and the See also:

• SUN
• SUN (0. Eng. sonne, Ger. sonne. Fr. soleil, Lat. sol, Gr. ijXior, from which comes helio- in various English compounds)

sun be considered; the other planets, however, disturb this orbit (see See also:

• MECHANICS

MECHANICS) . The relation of the ellipse to the other conic sections is treated in the articles CONIC SECTION and See also:

• GEOMETRY

GEOMETRY; in this See also:

• ARTICLE (from Lat. articulus, a joint)

article a See also:

• SUMMARY

summary of the properties of the curve will be given . To investigate the form of the curve use may be made of the See also:

• DEFINITION (Lat. definitio, from de-finire, to set limits to, describe)

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 . I) be the directrix, S the focus, and X the See also:

• FOOT

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 See also:

• MAJOR
• MAJOR (Lat. for " greater ")
• MAJOR (or MAIR), JOHN (1470-1550)

major See also:

• AXIS (Lat. for " axle ")

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 See also:

• DRAWN

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 See also:

• MINOR
• MINOR (Lat. for smaller, lesser)
• MINOR, ROBERT CRANNELL (1839-1904)

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 See also:

• FOCAL

focal distances are readily obtained by the methods of geometrical conics or See also:

• ANALYTIC (the adjective of " analysis," q.v.)

analytic-ally . The semi-major axis is generally denoted by a, and the semi-minor axis by b, and we have the relation b2=See also:

• A2(z)

a2(i—e2) . Also a2= 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 PN2 = AN.NA'(CB/CA)2 . 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 See also:

• AUXILIARY (from Lat. auxilium, help)

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 See also:

• LIE, JONAS LAURITZ EDEMIL (1833—1908)
• LIE, MARIUS SOPHUS (1842–1899)

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 OF, OP' be two tangents from 0, 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 See also:

• MIDDLE

middle points of a See also:

• SYSTEM

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 See also:

• PROPERTY

property of conjugate diameters is the sum of their squares equals the sum of the squares of the major and minor axis . In ;See also:

• ANALYTICAL

analytical geometry, the equation ax2+2hxy-{-by2+2gx+2fy+ c=o represents an ellipse when ab>h2; if the centre of the curve be the origin, the equation is aix2+2hixy-I-biy2 =C1, and if in addition a pair of conjugate diameters are the axes, the equation is further simplified to Ax2+By2=C . The simplest form is x2/a2+y2/b2 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 See also:

• ABSCISSA (from the Lat. abscissus, cut off)

abscissa being a See also:

• COS
• COS, or STANKO (Ital. Stanchio, Turk. Islan-keui, by corruption from Etc rav KW)

cos 4i, and the See also:

• ORDINATE

ordinate b See also:

• SIN
• SIN (O. Eng. syn: a common Teutonic word, cf. Dutch zonde, Ger. Siinde)

sin 4), since on eliminating(/' between x =a cos and y =b sin ¢ we obtain the equation to the ellipse . The See also:

• ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

angle 41 is termed the See also:

• ECCENTRIC (from Gr. F,c, out of, and Kivrpov, centre)

eccentric angle, and is geometrically represented as the angle between the axis of x (the major axis of the ellipse) and the See also:

• RADIUS

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 B is x cos 8/a+y sin B/b =1, and to the normal ax/cos 6—by/sin 0 =a2—b2 . The See also:

• AREA

area of the ellipse is crab, 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 See also:

• SERIES (a Latin word from serere, to join)

series, the analytical evaluation leading to an integral termed elliptic (see See also:

• FUNCTION

FUNCTION, ii .

Complex) . There are several approximation formulae:—S=vr(a+b) makes the perimeter about 1/2ooth too small; s=vry (a2+b2) about 1/200th too See also:

• GREAT

great; 2s=1r(a+b)+sr/ (a2+b2) is within 1/30,000 of the truth . An ellipse can generally be described to satisfy any five conditions . If five points be given, See also:

• PASCAL, BLAISE (1623-1662)
• PASCAL, JACQUELINE (1625-T661)

Pascal's theorem affords a See also:

• SOLUTION (from Lat. solvere, to loosen, dissolve)

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, &c . If a tangent and its point of contact be given, it is only necessary to remember that a See also:

• DOUBLE
• DOUBLE (from the Mid. Eng. duble, the form which gives the present pronunciation, through the Old Er. duble, from Lat. duplus, twice as much)

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 See also:

• PRACTICAL

practical importance are the following constructions:—(I) 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 . (i) 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 See also:

• STRIP

strip of See also:

• PAPER
• PAPER (Fr. papier, from Lat. papyrus)

paper or See also:

• RULE

rule and See also:

• MARK
• MARK, CARL (1858– )
• MARK, GOSPEL OF ST
• MARK, ST

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 See also:

• DIAMETER (from the Cr. &a, through, µErpov, measure)

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 See also:

• THREAD (0. Eng. praed, literally, that which is twisted, prawan, to twist, to throw, cf. " throwster," a silk-winder, Ger. drehen, to twist, turn, Du. draad, Ger. Draht, thread, wire)

thread of length AA', and See also:

• FIX, THEODORE (180o-1846)

fix it at its extremities by means of pins at the foci . The thread is now stretched taut by a See also:

• PENCIL (Lat. penicillus, brush, literally little tail)

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 See also:

• GENERAL
• GENERAL (Lat. generalis, of or relating to a genus, kind or class)

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 . See also:

• DIVIDE

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 See also:

• RAM
• RAM, PIERRE FRANCOIS XAVIER DE (18o4-1865)

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 (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 0 and radius OB forms See also:

• PART

part of a curve . Let this arc on the See also:

• REVERSE

reverse See also:

• SIDE
• SIDE (mod. Eski Adalia)

side to P intersect a line through 0 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 O, .

Then with centre O, and radius OJ, =0A', describe an arc . By reflecting the two arcs thus described over the centre the ellipse is approximately described .