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Conic Sections
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A conic section, as the name suggests, is a section taken through a cone. At the intersection of the sectional plane and the surface of the cone, curves having many different shapes are produced, depending on the inclination of the plane, and it is these curves which are referred to generally as conic sections. Although the origin of conic sections lies in solid geometry, the properties are readily expressed in terms of plane geometrical curves. In Fig. B.1a, a reference line for conic sections, known as the directrix, is shown as Z-D. The axis for the conic sections is shown as line Z-Z . The axis is perpendicular to the directrix. The point S on the axis is called the focus. For all conic sections, the focus has the particular property that the ratio of the distance SP to distance PQ is a constant. SP is the distance from the focus to any point P on the curve (conic section), and PQ is the distance, parallel to the axis, from point P to the directrix. The constant ratio is called the eccentricity, usually denoted by e. Referring to Fig. B.1a,
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(B.1)
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The conic sections are given particular names according to the value of e, as shown in the following table:
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e<1 e=1 e>1
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Figure B.1
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These curves are encountered in a number of situations. In this book they are used to describe: 1. The path of satellites orbiting the earth 2. The ellipsoidal shape of the earth 3. The outline curves for various antenna reflectors A polar equation for conic sections with the pole at the foci can be obtained in terms of the fixed distance p, called the semilatus rectum. The polar equation relates the point P to the radius r and the angle Fig. B.1b). From Fig. B.1b,
(B.2)
Also,
(B.3)
Combining Eqs. (B.2) and (B.3) and simplifying gives the polar equation as
(B.4)
If the angle is measured from SA, shown as in Fig. B.1c, then, since = 180 + , the polar equation becomes
(B.5)
The Ellipse
For the ellipse, e < 1. Referring to Eq. (B.4), when = 0 , r = p/(1 e). Since e < 1, r is positive. At = 90 , r = p, and at = 180 , r = p/(1 + e). Thus r decreases from a maximum of p/(1 e) to a minimum of r = p/(1 + e), the locus of r describing the closed curve A BMA (Fig. B.2). Also, since cos( ) = cos , the curve is symmetrical about the axis, and the closed figure results (Fig. B.2a). The length AA is known as the major axis of the ellipse. The semimajor axis a = AA /2 and e are the parameters normally specified for an ellipse. The semilatus rectum p can be obtained in terms of these two quantities. As already shown, the maximum value for r is SA = p/(1 e) and the
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Figure B.2 minimum value is SA = p/(1 + e). Adding these two values gives
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Since a = AA /2, it follows that
(B.6)
Substituting this into Eq. (B.5) gives
(B.7)
which is Eq. (2.23) of the text. Equation (B.7) can be written as
When = 360 ,
(B.8)
which is Eq. (2.6) of the text. When = 180
(B.9)
which is Eq. (2.5) of the text. Referring again to Fig. B.2a, and denoting the length SO = c, it is seen that AS + c = a. But, as shown above, AS = p/(1 + e) and p = a(1 e2). Substituting these for AS and simplifying gives
(B.10)
Point O bisects the major axis, and length BO is called the semiminor axis, denoted by b (BB is the minor axis) (Fig. B.2b). The semiminor axis can be found in terms of a and e as follows: Referring to Fig. B.2b, 2a = SA + SA = e(AZ + A Z) = e(2OZ), and therefore,
(B.11)
But OZ = BC = SB/e, and therefore, SB = a. SB is seen to be the radius at B. From the right-angled triangle so formed,
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