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.
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Practical Conic Sections
The Geometric Properties of Ellipses, Parabolas and Hyperbolas
By J. W. Downs
Dover Publications, Inc.Copyright © 1993 J. W Downs
All rights reserved.
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.)
Ellipses occur naturally in free orbital motion. Such motion ranges from planets having nearly circular orbits to the extremely eccentric orbits of recurrent comets.
For those who enjoy working algebra problems and putting dots on graph paper, the equation x2/a2 + y2/b2] = 1 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.
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-astring 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.
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.
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.
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.
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.
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.
The Directing Circle method has several advantages over the other methods described. It gives tangent lines instead of points to connect and has the further advantage of being part of a system for drawing all conic curves. This method of deriving conic curves is so important that an entire chapter (Chapter 4) is devoted to it.
Draw a circle with a diameter equal to the major axis of the desired ellipse. Draw the diameter line that will also be the major axis of the ellipse; then establish the foci at two points equidistant from the center. Place a drafting triangle over the circle so that one edge passes through one focus and the right angle falls on some point of the circle. The other arm of the right angle will be tangent to an ellipse at some point. By drawing several tangent lines, an ellipse will be derived inside the directing circle (Figure 1-8). Although either focus may be used, it is best to use the one more distant from the tangent, since the near focus is so close to the tangent that it is difficult to maintain accuracy.
Another advantage of this method is that it allows us to derive unstated parameters quickly and easily. For example, if the foci and major axis are known and we wish to find the length of the minor axis, we simply construct a perpendicular at either focus. The length of the line F1P1(Figure 1-9) is identical to the minor axis. This follows logically, since a perpendicular to this line constructed at P1 will be the tangent to the ellipse that is parallel to the major axis, which can only be at the intercept of the minor axis.
If only the major and minor axes of an ellipse are given, the foci may be found by swinging an arc whose radius is equal to one-half the major axis (the distance a) from either minor axis intercept. The two intersections of the arc and the major axis will be the foci. Many people do not realize that this is exactly what they have been doing mathematically by establishing foci with the formula c2 = a2 - b2. The resulting right triangle has a hypotenuse equal in length to one-half the major axis (a), the distance b is given, and the distance c must satisfy the Pythagorean theorem (Figure 1-10). The advantage of establishing foci with a compass is that numerical values need not be assigned to any of these dimensions.
The two-circle method may also be useful to drafters although it requires the use of construction lines. Draw two concentric circles, the smaller having a diameter equal to the minor axis and the larger having a diameter equal to the major axis. Draw several radials extending through both circles. Next, draw a horizontal line at the point of intersection of each radial and the smaller circle and a vertical line at its intersection with the larger circle. The intersection of the two perpendicular lines drawn from each radial is a point on the ellipse. See Figure 1-11.
Ellipses show up in another way that is not really practical for construction but merits some discussion. Two sources of harmonic motion (sine waves) that are of the same frequency but are not in phase with each other may be described as having elliptical properties. An oscilloscope fed by two such wave sources (one to the vertical axis and the other to the horizontal axis) will display an ellipse. If the two signals have exactly the same amplitude and have a phase difference of 90° or 270°, the figure, called a Lisajous figure, will be a circle—but remember that circles are special ellipses having equal major and minor axes. If the phase angles or the amplitudes are changed, an ellipse will be traced on the oscilloscope.
In optics and microwave transmission, when two components of transmitted energy are displaced in phase and in space, the signal is described as having elliptical or circular polarization.CHAPTER 2
A plane cutting a cone that does not pass completely through the cone no matter how far the cone and the plane are extended will produce a curve that increases in width to infinity. Since the curve will never close, there is no formula for its area. When such a curve is formed by a plane that does not cut the cone parallel to its side, the curve is a hyperbola. Although the plane never leaves the cone in one direction, it will intersect the other nappe of the cone to form a pair of hyperbolas. Whether the other nappe of the cone is real or imagined, this second curve must be considered. Since construction of a pair of solid cones joined at the apex is difficult, a lamp can be used to illustrate hyperbolas.
Figure 2-1 shows an ideal situation in which the lampshade is cylindrical and the light bulb is located precisely in the center of the shade. In this case, a pair of identical hyperbolas is cast on the wall. Usually, however, the shade is conical and the bulb is located arbitrarily. The resulting hyperbolas will be accurate but will not be geometrically or mathematically related. Should the lamp be tilted toward the wall in such a manner that the far (from the wall) edge of the cone is parallel to the wall, the curve cast on the wall will be a parabola. If the lamp is tilted farther, an ellipse will be shown.
Mathematically, Cartesian coordinates for hyperbolas may be plotted using the formula x2/a2 - y2/b2 = 1. (A hyperbola whose asymptotes are the xy axes may be drawn by plotting xy = k, where k is any constant.) The standard nomenclature used in analytical geometry for a hyperbola is shown in Figure 2-2.
Hyperbolas are infrequently observed in nature. Comets having orbital eccentricities greater than unity follow a hyperbolic path. Since hyperbolas are open-ended figures, these comets will not return.
The interference of two systems of radiating concentric circles forms a family of hyperbolas, as shown in Figure 2-3. This is best illustrated by dropping two stones into a quiet pond and watching the ripples as they cross each other. The interference of two sources of high-frequency energy will generate a hyperbolic pattern of reinforcement and cancellation. In Optics of the Electromagnetic Spectrum, Dr. Charles Andrews describes tracing a family of hyperbolas on a school lawn using a meter to detect the lines of minimum intensity of two interfering radio signals (of the same frequency) and marking these with a tennis-court line marker.
A hyperbola may be defined as the locus of a point, the difference of whose distances from two fixed points (the foci) is a constant. As might be expected, a device may be constructed somewhat similar to that described in the two-pins-and-a-string method (Chapter 1) for constructing ellipses. Since this method, illustrated in Figure 2-4, is considerably more complicated and requires drilling holes in the drawing table, it is not likely to be the most popular way to draw hyperbolas. The crank mounted below the table extends or retracts the same length of string through each hole. The difference between m and n remains constant, thus complying with the definition of a hyperbola.
There is another method of constructing a hyperbola that is similar to Method 5 but does not require drilling holes in the drawing board. It also utilizes the principle that a hyperbola is the locus of a point, the difference of whose distances from two fixed points is a constant.
Once we have decided on the distance between foci and the distance ab, which serves both as a distance between vertices and as the constant, we may construct the hyperbola. Draw a line slightly longer than the distance between foci. Mark a and b (the vertices) and F1 and F2 on this line. The center point is not used except for laying out these points symmetrically. As an aid to construction, draw another line (apart from the drawing), mark the distance AB, and add the arbitrary points L, M, N, O, and P. (At this point it would be helpful to have two compasses, since the adjustment from F1 to F2 is considerable.) Place the compass at the point marked B and measure the distance BL Transfer this to the drawing, placing the compass point at F1 and drawing an arc. Again place the compass on the auxiliary line at the point marked A and measure the distance AL. Place the compass point at F2 on the drawing and draw an arc that intersects the arc drawn from F1. This will be a point on the hyperbola. In a similar manner, repeat the process from points M, N, O, and P. Connecting these points will give a hyperbola. See Figure 2-5.
If we set the distance between vertices of a pair of hyperbolas at zero, the resulting "curve" will be a straight line perpendicular to the axis, as shown in Figure 2-6. This follows all of the geometric rules for a hyperbola and has been used in compound reflective antennas. We will discuss this further in Chapters 5-6.
The parallelogram method of constructing hyperbolas is similar to that shown for ellipses in the last chapter. If we know the distance AB and one point D on the hyperbola, we may construct the rectangle DEGF. Divide lines DC and DF into the same number of equal parts and number them from point D. From point A, draw lines to points 1, 2, and 3 on DC; and from point B, draw lines to points 1, 2, and 3 on DF. The points of intersection of similarly numbered lines may be connected to form a hyperbola (Figure 2-7). This method will work when the construction lines are not perpendicular, but the lines will not be axes of the hyperbola, so it is more practical to use a rectangle instead of another parallelogram.
The Directing Circle method is practical and has advantages over most of the other methods of drawing hyperbolas. As discussed in Chapter 1, this method uses tangent lines instead of points.
Draw a circle that will touch the hyperbola at the vertices. Draw a line through the circle, extending beyond the circle in both directions. Using the center of the circle, select the focal length and mark the foci with a compass. Place a right angle over the circle positioned so that one arm of the right angle passes through one focus and the right angle occurs on the directing circle. The other arm of the right angle will be tangent to the hyperbola. Again, as with the ellipse, it is more convenient to use the focus that is farther from the hyperbola being drawn, but the other focus may be used. When a representative number of tangents have been drawn, the curve envelope takes shape, as shown in Figure 2-8.
The asymptotes of a pair of hyperbolas may be found by placing the right angle over the circle so that one arm passes through a focus and the other arm passes through the center of the circle. This is discussed in greater detail in Chapter 4.
Excerpted from Practical Conic Sections by J. W. Downs. Copyright © 1993 J. W Downs. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
1. Deriving Ellipses.
2. Deriving Hyperbolas.
3. Deriving Parabolas.
4. The Directing Circle.
5. Reflective Properties of Solid Conic Curves.
6. Compound Reflectors.
8. For Practical Purposes.
9. Unusual Properties of Cones and Conic Curves.