Practical Conic Sections

The Geometric Properties of Ellipses, Parabolas and Hyperbolas

J. W. Downs

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486148885
• Topic: Mathematics
• Subtopic: Analytic Geometry
• Index: Mathematics

Illustrated with interesting examples from everyday life, this text shows how to create ellipses, parabolas, and hyperbolas and presents fascinating historical background on their ancient origins. The text starts with a discussion of techniques for generating the conic curves, showing how to create accurate depictions of large or small conic curves and describing their reflective properties, from light in telescopes to sound in microphones and amplifiers. It further defines the role of curves in the construction of auditoriums, antennas, lamps, and numerous other design applications. Only a basic knowledge of plane geometry needed; suitable for undergraduate courses. 1993 edition. 98 figures.

CHAPTER ONE

DERIVING ELLIPSES

METHOD 1

At the risk of being obvious, ellipses (and the other conic sections) may be obtained by cutting up (sectioning) a cone. Although this may not be the most convenient way of obtaining an ellipse, it must be listed as a legitimate means of deriving one. The intersection of a cone and a plane that passes completely through the cone is an ellipse. Ellipses are also generated at the intersection of a cylinder and a plane, but a cylinder must be considered to be a part of a special kind of cone having an apex angle of 0°. Figure 1-1 shows the shadow of a ball illuminated by a point source of light. The shadow cast on the table is an ellipse, with the ball touching the surface at one focus. (The shadow of a sphere is always conical, regardless of the angle from which the sphere is illuminated.)

FIGURE 1-1

An elliptical shadow cast by a ball. The ball rests on one focus of the ellipse (proof given in Chapter 9).

METHOD 2

Ellipses occur naturally in free orbital motion. Such motion ranges from planets having nearly circular orbits to the extremely eccentric orbits of recurrent comets.

METHOD 3

For those who enjoy working algebra problems and putting dots on graph paper, the equation

describes an ellipse in the xy plane with major and minor axes of length 2a and 2b. The standard nomenclature for an ellipse described in analytical geometry is shown in Figure 1-2.

FIGURE 1-2

METHOD 4

Ellipses may be defined as the locus of a point, the sum of whose distances to two fixed points is a constant. Put into practice, this method resolves itself to the two-pins-and-a-string method of constructing ellipses. Two pins are placed at the foci and a loop of string is adjusted to a length that allows the pencil point to touch a point on the ellipse. This point is usually at the major or minor axis intercept, but it may be any point known to be on the ellipse. See Figure 1-3.

FIGURE 1-3

This is a very practical way of drawing ellipses, and it is often the most convenient approach to be used for laying out large ellipses, such as elliptical flower beds or large outdoor signs.

It is possible to accomplish the same thing without the use of pins and string. Going back to the definition of an ellipse as the locus of a point whose distance to two fixed points is a constant, we should establish the fact that this constant is always equal to the length of the major axis of the ellipse. If we establish the major axis on a line and mark off arbitrary points along this line, we may take the distances (with a compass) from point A to one of these points and swing an arc from F₁, as shown in Figure 1-4. From point B we adjust the compass to measure the length from B to the same point and swing another arc from F₂. The intersection of the two arcs will be a point on the ellipse. By repeating this operation several times and connecting these points of intersection, we may draw the ellipse. Although this appears to be a practical way to draw an ellipse, in practice it becomes difficult to draw through the points as we approach the ends of the ellipse.

FIGURE 1-4

It is important to remember that the constant is always equal to the length of the major axis of the ellipse. In Chapter 2 we shall see that the constant involved in generating hyperbolas is also equal to the major axis (the distance between vertices) of a pair of hyperbolas, the only difference being that we subtract the two distances instead of adding them when determining points.

METHOD 5

The trammel method is an easy way to draw ellipses; it requires no pins or construction lines except the major and minor axes. For this reason it is frequently preferred by drafters. Two approaches may be used. In Figure 1-5(a) one-half the lengths of the major and minor axes are marked off on a piece of cardboard or plastic and placed over two lines drawn perpendicular to each other. The point P will be on the ellipse as long as the points M and N are on the x and y axes. Similarly, Figure 1-5(b) shows a trammel marked with one-half the minor axis inside one-half the major axis. Again, if the points M and N are positioned over the axis lines, point P will fall on the ellipse.

FIGURE 1-5

A mechanical device known as an ellipsograph, or the trammel of Archimedes, is used for drawing ellipses and is shown in Figure 1-6. The pen (P) is shown at the end of the movable arm, but any point on the arm will describe an ellipse. Note that this method is no different from the method shown in Figure 1-5(b) but is presented in a more practical mechanical form. The point P will cross the major axis when M is centered and will cross the minor axis when N is centered.

FIGURE 1-6

Ellipsograph.

METHOD 6

The parallelogram method starts with a pair of intersecting axes centered on a parallelogram that is to circumscribe the ellipse. Divide AO and AE into the same number of equal parts, as shown in Figure 1-7. From D, draw lines through points 1, 2, and 3 on AO; and from C, draw lines through points 1, 2, and 3 on AE. The intersections of these lines will be points on the ellipse. Although any parallelogram will work, it is more convenient if the parallelogram is a rectangle; otherwise the axes will not correspond to the major and minor axes of the ellipse.

FIGURE 1-7

METHOD 7

The Directing Circle method has several advantages over the...