PARABOLA (Neo-Lat., from Gk. rapagOvi, par/11)W, comparison. juxtaposition, parable, pa rabola). A conic section cut by a plane parallel to the element of the cone. When the plane coincides with its parallel element, the parabola as sumes the limiting form of a straight line. The parabola may also be de fined as the locus of a point whose distance from a fixed point (the focus) is equal to its distance from a fixed straight line (the direetrix), i.e.
its eccentricity (q.v.) is I. From this defini tion its construction readily follows. Let DD' (Fig. 1 ) he the fixed straight line or direc trix, and F the fixed point or focus. Draw a perpendicular to the directrix, passing through the focus, and this will be the axis of the curve. The point 0 on the axis, half way between F and DIY, will evidently be a point on the curve. To find other points on the curve draw a series of lines parallel to the directrix and cutting the axis XX' in M,, M,, F, , With F as a centre and ZNI, as a radius, describe an arc cutting the perpendicular through Al, in 1', and Q,; with F as a centre and ZM, as a radius, describe a circle cutting the perpendicular through in P, and and so on. The points 1•,, Q„ Q,..., are then points on a parabola. PQ is called the lotus rectum or parameter, and, as is evident from the construc tion, equals twice the distance between the focus and directrix. The curve may be described mechanically in the following manner: move a right-angled triangle with one perpendicular side coinciding with the directrix; a string equal in length to the other perpendicular side has one end fastened to the outer vertex of the triangle and the other to the focus; a pencil resting against the lower side of the triangle and holding the string taut will trace a parabola.
The Cartesian equation of the parabola, its axis being taken as the X-axis and its vertex as the origin, is = 4px, where p is the distance between the focus and the vertex. Its polar equation, the focus being the pole, is 2 p 4p coset r— • or r — .
1—cos 0' when the vertex is taken as pole. The following are some of the most important properties of the parabola: (1) Any line RH (Fig. 2) parallel to the X-axis is a diameter, i.e. bisects a sys FIG. 2.
tern of parallel chords, as those parallel to PQ; (2) the subnormal MN is constant and equal to the semi-latus rectum FK; (3) if the tangent through I' cuts the X-axis in T, and if the normal at P cuts the X-axis in N, the focus F is equidistant from P, T, and N; (4) the angle DPT = angle XTP is equal to the angle Tl'F, or the tangent at any point makes equal angles with the axis of the curve and the focal radius to the point of contact (these last two properties furnish simple methods for drawing tangents to the parabola) ; (5) through any point in the plane, three normals can be drawn to the parabola ; (0) the tangents at the ex tremities of any focal chord PL intersect at a point G on the directrix, and at right angles; (7) a perpendicular from the focus F upon a tangent SQ meets this tangent at a point I on the tangent through the vertex; (8) a tangent RE at the end of a diameter RH is parallel to the chords bisected by that diameter, of which PQ is one and the tangents at the extremities of PQ intersect upon the corresponding diameter 511; (0) the area of a parabolic segment ORPQ is of the triangle RPQ on the same base and having the same height ; (10) the parabola has no real finite asymptotes.
Concave reflecting mirrors are often formed so that all axial sections are equal parabolas. In such a mirror, all parallel rays of light are re flected to the focus; and, conversely, if a light be placed at the focus of such a mirror, its rays will be reflected in a parallel pencil. If a body were projected upward and obliquely to the di rection of gravity, it would, if undisturbed by any other force except gravity, accurately de scribe a parabola whose axis is vertical and whose vertex is the highest point reached by the body. The term parabola is used in analysis, more generally, to denote that class of curves in which some power of the ordinate is proportional to a lower power of the corresponding abscissa. Thus the common parabola above given has the equation y' = kx, the cubical parabola has the equation y = a + bx ex' ± (Ix', the simplest form being p = ke and the semi-cubical parabola, = kal or = Le. The last mentioned curve is also known as the Neilean parabola, because it was rectified by William Neil (1657).
For the various curves the name para bola, consult: Brocard, Notes de bibliographic des eourbes geomarigues (Bar-le-Duc, 1897; partie complementaire, 1899).