-
Skip These Ads
-
Skip These Ads|
|
 |
Click here and add this page to your favorites!
|
Encyclopedia Britannica - Main :: PAI-PAS |
|
|
PARABOLA , a plane curve of the second degree. It may be defined as a section of a right circular cone
cone
The form of the curve is shown in fig. 1, where P is a point on the curve equidistant from the fixed line AB, known as the directrix, and the fixed point F known as P the focus . The line CD passing through the focus and perpendicular to the directrix is the axis
diameter , and meets the curve in the vertex G. The line FL perpendicular to the axis
diameter , and the parameter of any diameter is measured by the focal chord drawn
parallel to the tangent at the vertex of the diameter and is equal B 748 to four times the focal distance of the vertex. To construct the parabola when the focus and directrix are given, draw the axis CD and bisect CF at G, which gives the vertex. Any number of points on the parabola are obtained by taking any point E on the directrix, joining EG and EF and drawing FP so that the angles PFE and DFE are equal. Then EG produced meets FP in a point on the curve. By joining the points so obtained the parabola may be described. A mechanical construction, when the same conditions are given, consists in taking a rigid bar ABC bent at right angles at B (fig. 2), and fastening a string
and F. Then if a pencil be placed along /p BC so as to keep the string
limb AB be slid along the directrix, the n pencil will trace out the parabola. F Properties which may be readily de- A fundamental property of the curve is that the line at infinity is a tangent (see GEOMETRY, PROJECTIVE), and it follows that the centre and the second real focus and directrix are at infinity. It also follows that a line half-way between a point and its polar and parallel to the latter touches the parabola, and therefore the lines joining the middle points of the sides of a self-conjugate triangle form a circumscribing triangle, and also that the nine-point circle of a self-conjugate triangle passes through the focus. The orthocentre of a triangle ,circumscribing a parabola is on the directrix; a deduction from this theorem is that the centre of the circumcircle of a self-conjugate triangle is on the directrix (" Steiner's Theorem "). In the article GEOMETRY, ANALYTICAL, it is shown that the general equation of the second degree represents a parabola when the highest terms form a perfect square. Analytic This is the analytical expression of the projective Geometry. property that the line at infinity is a tangent. The simplest equation to the parabola is that which is referred to its axis and the tangent at the vertex as the axes of co-ordinates, when it assumes the form y2=4ax where 2a=semilatus rectum; this may be deduced directly from the definition. An equation of similar form is obtained when the axes of co-ordinates are any diameter and the tangent at the vertex. The equations to the tangent and normal at the point x'y' are yy'= 2a(x+x') and 2a(yy')+y'(xx')=o, and may be obtained by general methods (see GEOMETRY, ANALYTICAL, and INFINITESIMAL CALCULUS). More convenient forms in terms of a single parameter are deduced by substituting xi = amt, y' =tam (for on eliminating m between these relations the equation to the parabola is obtained). The tangent then becomes my=x+amt and the normal y=mx+2amam3. The envelope of this last equation is 27ay2=4(x2a)3, which shows that the evolute of a parabola is a semi-cubical parabola (see below Higher Orders). The cartesian equation to a parabola which touches the co-ordinate axes is V ax+1/ by= 1, and the polar equation when the focus is the pole and the axis the initial line is r cos2O/2=a. The equation to a parabola in triangular co-ordinates is generally derived by expressing the condition that the line at infinity is a tangent in the equation to the general conic. For example, in trilinear co-ordinates, the equation to the general conic circumscribing the triangle of reference is l0y+mya+naf=o; for this to be a parabola the line as + b/3 + cy=o must be a tangent. Expressing this condition we obtainV la* 1/ml Vnc=o as the relation which must hold between the co-efficients of the above equation and the sides of the triangle of reference for the equation to represent a parabola. Similarly, the conditions for the inscribed conic S/la+V m#+V ny = o to be a parabola is lbc+mca+nab=o, and the conic for which the triangle of reference is self-conjugate lag+ml2+ny2=c is a2mn+b2nl+c2lm=o. The various forms in areal co-ordinates may be derived from the above by substituting Xa for 1, ub for m and vc for n, or directly by expressing the condition for tangency of the line x+y+z=o to the conic expressed in areal co-ordinates. In tangential (p, q, r) co-ordinates the inscribed and circumscribed conics take the forms Xqr+rp+vpq=o and V Ap+ A/,uqV ye = o; these are parabolas when X ++v = o and V X= V,u* V v= o respectively. The length of a parabolic arc can be obtained by the methods of the infinitesimal calculus; the curve is directly quadrable, the area of any portion between two ordinates being two thirds of the circumscribing parallelogram. The pedal equation with the focus as origin is p2=ar; the first positive pedal for the vertex is the cissoid (q.v.) and for the focus the directrix. (See INFINITESIMAL CALCULUS.) REFERENcEs.Geometrical constructions of the parabola are to be found in T. H. Eagles' Plane Curves (1885). See the bibliography to the articles CONIC SECTIONS; GEOMETRY, ANALYTICAL; and GEOMETRY, PROJECTIVE. In the geometry of plane curves, the term parabola is often used to denote the curves given by the general equation a'nxn= ye.+n, thus ax=y2 is the quadratic or Apollonian parabola; a2x=y3 is the cubic parabola, a3x=y4 is Higher Orders. the biquadratic parabola; semi parabolas have the general equation axn-1=yn, thus ax2=y3 is the semicubical parabola and ax3=y4 the semibiquadratic parabola. These curves were investigated by Rene Descartes, Sir Isaac Newton, Colin Maclaurin and others. Here we shall treat only the more important forms. The cartesian parabola is a cubic curve which is also known as the trident of Newton on account of its three-pronged form. Its equation is xy=ax3-+-bx2+cx+d, and it consists of two legs asymptotic to the- axis of y and two parabolic legs (fig. 3). The simplest form is axy=x3a3, in this case the serpentine position shown in the figure degenerates into a point of inflexion. Descartes used the curve to solve sextic equations by determining its inter-sections with a circle; mechanical constructions were given by Descartes (Geometry, lib. 3) and Maclaurin (Organica geometrica). The cubic parabola (fig. 4) is a cubic curve having the equation y=ax3+bx2+cx+d. It consists of two parabolic branches tending in opposite directions. John Wallis utilized the intersections of this curve with a right line to solve cubic equations, and Edmund Halley solved sextic equations with the aid of a circle. Diverging parabolas are cubic curves given by the equation y2=ax3+bx2+cx+d. Newton discussed the five forms which arise from the relations of the roots of the cubic equation. When all the roots are real and unequal the curve consists of a closed oval
oval
curve to be rectified. This was accomplished in 1657 by Neil in England, and in 1659 by Heinrich van Haureat in Holland. Newton showed that all the five varieties of the diverging parabolas may be exhibited as plane sections of the solid of revolution of the semi-cubical parabola. A plane oblique to the axis and passing below the vertex gives the first variety; if it passes through the vertex, the second form; if above the vertex and oblique or parallel to the axis, the third form; if below the vertex and touching the surface, the fourth form, and if the plane contains the axis, the fifth form results (see CURVE). The biquadratic parabola has, in its most general form, the equa tion y=ax++bx8+cx2+dx+e, and consists of a serpentinous and two parabolic branches (fig. 8). If all the roots of the quartic in x are equal the curve assumes the form shown in fig. 9, the axis of x being a double
End of Article: PARABOLA If you wish, you can link directly to this article.
<a href="http://jcsm.org/StudyCenter/Encyclopedia/PAI_PAS/PARABOLA.html"> PARABOLA </a> |
|
|
(Previous) PARABLE (Gr. 7apa(3oXi7, a comparison or simili... |
(Next) PARACELSUS (c. 1490-1541) |
Jesus Christ Saves Ministries, P.O. Box 70696, Pasadena, CA 91117JCSM is a 501(c)(3), non-profit organization. Copyright © 1997-present. |
Free & Cheap Cell
Phones |
Cheap Long Distance
Phone Service Carriers |
Talk America Local Phone Service
|
Ztel & MCI - Unlimited Long Distance
Compare
Cell Phone Plans & Companies |
International Calling Cards & Prepaid Phone Cards |
Voice Over IP Broadband Internet Phone
Service | Wireless
Phone Plans & Cheap Cell Phones
|
_____________________________________________________________________________
Online First Aid and CPR Certification . The Online Christ Centered Ministries . The Skeptic's Annotated Bible: Corrected and Explained . The Inerrancy Discussion Board . Free Email Accounts . Home Equity Loans . JasonGastrich.com . The Missions, Apologetics, and Creation Bible Conference . Young Earth Creation Science . San Diego Music Lessons . 10,000 Wise Quotes and Spiritual Sayings . Gastrich.net . Maximizing the Internet: 12 Keys to Success . Louisiana Baptist University . NKJV Web Hosting and Services . Michael Newdow . San Diego Soccer Training . Christian Guitar Lessons . Jesus Christ Saves Ministries . Eternal Security
