ELLIPSE
, or Ellipsis, is one of the conic sections, popularly called an oval; being called an Ellipse or Ellipsis by Apollonius, the first and principal author on the conic sections, because in this figure the fquares of the ordinates are less than, or defective of, the rectangles under the parameters and abscisses.
This figure is differently defined by different authors; either from some of its properties, or from mechanical construction, or from the section of a cone, which is the best and most natural way. Thus; Fig. 1. Fig. 2.
1. An Ellipse is a plane figure made by cutting a cone by a plane obliquely through the opposite sides of it; or so as that the plane makes a less angle with the base than the side of the cone makes with it; as ABD fig. 1.
The line AB connecting the uppermost and lowest points of the section, is the transverse axis; the middle of it C, is the centre; and the perpendicular to it DCE, through the centre, is the conjugate axis. The parameter, or latns rectum, is a 3d proportional to the transverse and conjugate axes; and the foci are two points in the transverse axis, at such equal distances from the centre, that the double ordinates passing through those points, and perpendicular to the transverse, are equal to the parameter.
2. The Ellipse is also variously described from some of its properties. As first, That it is a figure of such a nature that if two lines be drawn from two certain points C and D in the axis, fig. 2, to any point E in the circumference, the sum of those two lines CE and DE will be every where equal to the same constant quantity, viz, the axis AB. Or, secondly, that it is Fig. 3. Fig. 4. a figure of such a nature, that the rectangle AGXGB (fig. 3) of the abscisses, tending contrary ways, is to GH2 the square of the ordinate, as AB2 to IK2, the square of the transverse axis to the square of the conjugate, or, which is the same thing, as the transverse axis is to the parameter. And so of other properties.
3. Or the Ellipse is also variously described from its mechanical constructions, which also depend on some of its chief properties. Thus; 1st, If in the axis AB, there be taken any point I (fig. 4); and if with the radii AI, BI, and centres F and f, the two foci, arcs be described, these arcs will intersect in certain points E, E, e, e, which will be in the curve or circumference of the figure: and thus several points I being taken in the axis AB, as many more points E, e, &c, will be found; then the curve line drawn through all these points E, e, will be an Ellipse. Or, thus; if there be taken a thread of the exact length of the transverse axis AB, and the ends of the thread be fixed by pins in the two foci F and f; (fig. 5) then moving a pen or pencil within the thread, so as to keep it always stretched out, it will describe the curve called an Ellipse. Fig. 5. Fig. 6.
To Construct an Ellipse. There are many other ways of describing or constructing an Ellipse, besides those just now given: as
1st. If upon the given transverse axis there be described a circle AGB (fig. 6), to which draw any ordinate DG, and DE a 4th proportional to the transverse, the conjugate, and the ordinate DG; then E is a point in the curve. Or if the circle agb be described on the conjugate axis ab, to which any ordinate dg is drawn, in which taking dE in like manner a 4th proportional to the conjugate, the transverse, and ordinate dg, then | shall E be in the curve. Or, having described the two circles, and drawn the common radius CgG cutting them in G and g; then dgE drawn parallel to the transverse, and DGE parallel to the conjugate, the intersection E of these two lines will be in the curve of the ellipse. And thus several points E being found, the curve may be drawn through them all with a steady hand. Fig. 7. Fig. 8.
2. If there be provided three rulers, of which the two GH and FI (fig. 7) are of the length of the transverse axis LK; and the third FG equal to HI the distance between the foci; then connecting these rulers so as to be moveable about the foci H and I, and about the points F and G, their intersection E will always be in the curve of the ellipse; so that by moving the rulers about the joints, with a pencil passed through the slits made in them, it will trace out the Ellipse. Fig. 9. Fig. 10.
3. If one end A of any two equal rulers AB, DB, (fig. 9 and 10) which are moveable about the point B, like a carpenter's joint-rule, be fastened to the ruler LK, so as to be moveable about the point A; and if the end D of the ruler DB be drawn along the side of the ruler LK; then any point E, taken in the side of the ruler DB, will describe an ellipse, whose centre is A, conjugate axis = 2DE, and transverse = 2AB+2BE.
Another method of description is by the Elliptical Compass. See that article, below.
Some of the more Remarkable Properties of the Ellipse. —1. The rectangles under the abscisses are proportional to the squares of their ordinates; or as the square of any axis, or any diameter, is to the square of its conjugate, so is the rectangle under two abscisses of the former, to the square of their ordinate parallel to the latter; or again, as any diameter is to its parameter, so is the said rectangle under two abscisses of that diameter, to the square of their ordinate. So that if d be any diameter, c its conjugate, p its parameter = c2/d<*> x the one absciss, d-x the other, and y the ordinate; then, as . From either of which equations, called the equation of the curve, any one of the quantities may be found, when the other three are given.
2. The sum of two lines drawn from the foci to meet in any point of the curve, is always equal to the transverse axis; that is, CE + DE = AB, in the 2d fig. Consequently the line CG drawn from the focus to the end of the conjugate axis, is equal to AI the semitransverse. Fig. 11.
3. If from any point of the curve, there be an ordinate to either axis, and also a tangent meeting the axis produced; then half that axis will be a mean proportional between the distances from the centre to the two points of intersection; viz, CA a mean proportional between CD and CT. And consequently all the tangents TE, TE, meet in the same point of the axis produced, which are drawn from the extremities E, E, of the common ordinates DE, DE, of all Ellipses described on the same axis AB.
4. Two lines drawn from the foci to any point of the curve, make equal angles with the tangent at that point: that is, the [angle] FET = [angle]fEt.
5. All the parallelograms are equal to each other, that are circumscribed about an Ellipsis; and every such parallelogram is equal to the rectangle of the two axes.
6. The sum of the squares of every pair of conjugate diameters, is equal to the same constant quantity, viz, the sum of the squares of the two axes.
7. If a circle be described upon either axis, and from any point in that axis an ordinate be drawn both to the circle and ellipsis; then shall the ordinate of the circle be to the ordinate of the Ellipse, as that axis is to the other axis: viz, AB : ab :: DG : DE, and ab : AB :: dg : dE. (in the 6th fig.) And in the same proportion is the area of the circle to the area of the Ellipse, or any corresponding segments ADG, ADE. Also the area of the Ellipse is a mean proportional between the areas of the inscribed and circumscribed circles. Hence therefore,
8. To find the Area of an Ellipse. Multiply the two axes together, and that product by .7854, for the area. Or
9. To find the Area of any Segment ADE. Find the | area of the corresponding segment ADG of a circle on the same diameter AB; then say, as the axis AB: its conj. ab :: circ. seg. ADG: elliptic seg. ADE.
10. To find the length of the whole circumference of the Ellipse. Multiply the circumference of the circumscribing circle by the sum of the series &c, for the area: where d is the difference between an unit and the square of the less axis divided by the square of the greater.
Or, for a near approximation, take the circumference of the circle whose diameter is an arithmetical mean between the two axes, or half their sum: that is, (t + c)/2 X 3.1416 = the perimeter nearly; being about the 200th part too little; where t denotes the transverse, and c the conjugate axis.
Or, again, take the circumference of the circle, the square of whose diameter is half the sum of, or an arithmetical mean between the squares of the two axes: that is, the perimeter nearly; being about the 200th part too great.
Hence combining these two approximate rules together, the periphery of the Ellipse will be very nearly equal to half their sum, or equal to , within about the 30000th part of the truth.
For the length of any particular arc, and many other parts about the Ellipse, see my Mensuration, pa. 283, &c, 2d edit. See also my Conic Sections, for many other properties of the Ellipse, especially such as are common to the hyperbola also, or to the conic sections in general.
Infinite Ellipses. See Elliptoide.
