The term conic sections is used when discussing the derivation of a line that is a locus of points equal distance from either a line, a point, both a line and a point, two lines, etc. The term conic sections also can be used when discussing certain planes that are formed when they are intersected with a right circular cone. The planes, or lines as we know them, consist of the circle, the ellipse, the parabola, and the hyperbola. (West, 112) There are different ways to derive each separate curve, and many uses for them to be applied to as well. All of which are an important aspect to conic sections.
The cone is a shape that is formed when you have a straight line and a circle, and the straight line is moved around the circumference of the circle while also always passing through a fixed point at a distance away from the circle. The parts formed are labeled the upper nappe, the lower nappe, and the vertex, (Prime, 1) as described in the diagram below in diagram 1:
The cone is then used with the help of a right plane to form the different circles, parabolas, ellipses, and hyperbolas, as shown below in diagram 2 on the next page. Taking a flat plane that would be parallel to the base of the cone, and intercepting it with a single nappe of the cone produces the circle. The ellipse is formed by the intersection of the cone with a flat plane that intercepts one nappe of the cone, but is not parallel to the base, and is not parallel to any other side of the cone. The parabola is a curve produced by the intersection of the cone with a flat plane of a single nappe, parallel to the remaining slanted side. Finally, the hyperbola is produced when you take a flat plane and intercept it with the cone so that it passes through both nappes. It is nearly perpendicular to a base of the cone, but is more often than not, illustrated with a greater angle of the lower nappe being intercepted. (West, 114)
The easiest way to visually produce these shapes is to use a flashlight and have an empty wall in your house. By holding the flashlight parallel to the floor, you will produce a circle on the wall. In holding the flashlight slightly tilted, an ellipse is formed. When you hold the flashlight so that the side farthest away from the wall is parallel to the wall, a parabola is formed. Finally, when the flashlight is tilted so that it is almost parallel to the floor, you will get half of a hyperbola because the flashlight can only simulate a single cone, as the wall represents the plane. (West, 114-115)
To get into the details of conic sections it is necessary to break them up into each separate conic section. The circle can be considered the simplest conic section because it is the easiest to understand. A Greek Philosopher named Apollonius was studying the first forms of the circle back around 225 B. C., while he was writing Conic Sections, which was a series of eight different books analyzing all the primate forms of these shapes. (West, 113) This was around when someone, not necessarily him, noticed that when the circumference of any circle was divided by the diameter, a similar number, or a constant, somewhere around 3 was always achieved. There are even references to this found in the bible describing some sort of circular container that held a large amount of water. Well, over 2,000 years later, we use the formula C=ƒàD. That is how we would normally come up with the circumference of a circle with either the diameter or the radius (doubled equals the diameter) given. To apply the circle in conic sections, it should be used as a locus of points equal distance from a given point to show how the formulas are derived. Knowing that, the radius is the given distance, and the center of the circle is the point from which the circumference is equal distance of. On a graph, with the center at the origin, one would take any point on the circumference of the circle, (x,y), to get the radius. This is known as the central equation, X2 + Y2 = r2. The way this formula came about was by expressing the radius in terms of the distance formula:
r = „©(X-0)2 + (Y-0)2 , with the right half of the equation under the square root. In taking both sides and squaring them, you result in r2= (X-0) 2+ (Y-0) 2. To simplify this equation, when the center is at the origin it is not necessary to subtract the x and y coordinate of the center of the circle. (Gullberg, 559) For example, if the center of the circle was at (3, -5), then the equation would be r = (X-3)2 + (Y+5)2. That is how you would use the equation is the center of the circle was not located at the origin.
Next, the ellipse and the derivation of its central and general formula¡¦s will be discussed. The ellipse is considered the locus of points for which the sum of their distances from two fixed points remain the same, or constant. In the diagram below, F1 and F2 are Foci. For purposes of space, the diagram below will also include the material on the central equation for an ellipse. Let¡¦s say that point P, or (X, Y), is on the perimeter of the ellipse. Then the distance from F-1 to P + the distance from F2 to P added together equals the constant, which also equals the length of the major axis (The farthest end of the ellipse to the other end). The major axis will be called 2A in terms of there being two parts extending from the center of the ellipse, which would also be the same as saying that 2A is equal to the sum of the distances from the foci to point P. From there, one could say that the length of the minor axis, or the line that is labeled Y on the diagram, is B2, and the distance from the center from the center of the ellipse to the focus is C2. Now the A is equal to the focus to the interception of the minor axis on the circle because if that was to be doubled, it would equal the same as the sum of the distances of any point on the perimeter since that number is a constant. If you used the Pythagorean Theorem as A2 = B2 + C2, then you could get any of the parts of an ellipse that you needed.
The next formula that can be derived would be the central equation and the general equation for the ellipse. Let¡¦s say that the major axis, the commonly places horizontal one, was on the X-axis, and the center was at the origin. Then you can deride a formula that would give the ellipse a central equation for any point on the perimeter of the ellipse allowing for it then to be drawn. The way this was brought about was by letting the point on the perimeter named P be labeled (X, Y), and the points of focus equal (-C, 0), and
(C, 0). Now, by using the distance formula, one could add these two together to get the ¡§A,¡¨ which would be doubled in the equation so that the square root of a square root does not have to be used. Well for example in this case, the distance formula would give you:
With PF1 and PF2 being the two distances to the foci, this formula can be rewritten several times. First one would set the sum of the two distances equal to 2a like above. Then you would subtract PF1 from one side to the other and then square both sides. One would result in the following equation:
X2 ¡V 2CX + C2 + Y2 = 4A2 ¡V 4A „©((X + C)2 + Y2) + X2 + 2CX + C2 + Y2
After being simplified, the following equation would result:
A „©((X + C)2 + Y2) = A2 + CX
Squaring both sides again results in the equation as follows (After simplifying):
Eq. #1 (A2 - C2)X2 + A2Y2 = A4 + 2A2CX + C2X2
Since after playing around a bit with the Pythagorean Theorem you can come up with equations for each the A2, the B2, and the C2, you can substitute them for their equivalent as follows:
As for C: C2 = A2 - B2, or for B: B2 = A2 - C2, or for A: A2 = B2 + C2
In substituting B2 for (A2 ¡V C2) in Eq. #1, you will come up with the following formula with its simplified form located below it.
B2X2 + A2Y2 = A2B2, which would then simplify itself to (After dividing the A2B2):
That is the final equation for the central formula of the ellipse. The general equation would be derived by substituting in the original distance formulas the distance the ellipse is moved at either the domain or the range from the original ¡§C,¡¨ in which the final general formula is located below (h being the domain moved, k being the range difference:
Next, the derivation of the central and general formula¡¦s for the parabola will be discussed. The parabola is considered to be the locus of points equal distance from fixed point (the focus), and a line (the directrix). The distance from the focus to any point on the parabola will always be equal to the distance from the directrix to that parabola. The vertex as labeled below is considered to be on the axis of symmetry, or the midpoint of the line extending from the focus to the midpoint. See the diagram below for visual aid:
The central equation is used to get a constant for which the certain points on the line can be determined and eventually leading to the capability of drawing a parabola. There is a quite simple formula that is commonly used, and it is: X2 = 4PY. Taking the distance formula and setting the distance of the central point equal to the distance to the directrix derives this formula. The directrix is Y = -P so when using the formula, the distance of the point on the parabola to the directrix is equal to Y + P. With this known, the formula below is derived when the two distances are set equal to each other:
„©((X ¡V 0)2 + (Y ¡V P)2 = Y - (-P) Which can be simplified to:
X2 = 4PY
The general equation would then be equal to the same formula, but in the initial distance formula one would subtract the new point, (U, V) from the X and Y, as illustrated below (after being simplified): (X ¡V U)2 = 4P(Y ¡V V)
The final specific conic section that will be discussed is the hyperbola. The hyperbola is a locus of points for which the difference of the distances from the foci to the line always gives a constant. The diagram below shows the hyperbola and the method for which a constant is achieved.
The central equation for the hyperbola is derived from using several distance formulas and many substitutions of these formulas to derive a final and simple equation. This equation is based upon the center of the hyperbola being located at the origin. We will let 2A equal the length of the major axis, and let 2C equal the distance from one focus to the other focus. After doing so, you can rotate the C, after taking the distance from the center of the hyperbola to a focus, and make a right triangle when creating ¡§B.¡¨ Also, you would take just half of the ¡§2A¡¨ and make it just a single ¡§A.¡¨ (This is shown in the diagram on the next page, just imagine a circle with a perimeter on the foci, then the point where C and B meet would also be on that perimeter.) Now, by assuming that the hyperbola is centered at the origin with the foci on the X-axis, you can then once again apply distance formulas with multiple simplifications and substitutions derive the final equation of the hyperbola.
Since the distance from point F2, or (-C, 0), to the point
(X, Y) on the graph would be equal to „©((X-C)2 + (Y-0)2) which in turn would be equal to „©((X-C)2 + Y2), and the distance from point F1, or (C, 0), is equal to the equation,
„©((X+C)2 + (Y-0)2), which is reduced to „©((X+C)2 + Y2), one then could subtract them and set it equal to 2A, since we know the constant is the same length as the major axis. This would result in the early stage of a formula, and look like this:
„©((X-C)2 + Y2) - „©((X+C)2 + Y2) = 2A
After moving the distance formula for the larger distance over, squaring both sides, and reducing two times, one will come up with the following:
(C2 ¡V A2)X2 ¡V A2Y2 = A2(C2 ¡V A2)
With the previously assessed knowledge of B2 being equivalent to C2 ¡V A2, we can substitute the B2 in both cases that its equal occurs. This is shown after occurring below:
B2X2 ¡V A2Y2 = A2B2
Then, after dividing both of the sides by ¡§A2B2,¡¨ you receive the final formula:
By modifying the formula so that the center of the equation can be moved to elsewhere on the graph, and not restricted to just the origin, you can name the new central point
¡§(H, K)¡¨ and come up with a new formula that would yield the general equation. This is done by placing the H and the V in places in the original distance formula like the central equation, then working the problem out again by squaring the sides and simplifying many times. An example of what one would start with when deriving the general equation would be:
„©((X ¡V(H ¡V C))2 + (Y ¡V K)2) + „©((X ¡V (H + C)2 + (Y ¡V K)2) = 2A
After some of the reducing we have known and grown to love, you get this:
That is the final general equation for the hyperbola not centered at the origin, or the point on an ordinary graph labeled (0, 0).
There is a formula that is used to come up with the eccentricity of any type of conic section, with just a single formula. This equation is known as the vertex equation. The vertex equation is:
Y2 = 2PX ¡V (1 ¡V E2) X2
The letters all are the same as mentioned earlier, but the ¡§E¡¨ stands for eccentricity. The eccentricity is derived from the ratio of the distance of the focus to the point on the curve, to the distance of the directrix to the same point on the curve. Examples below:
The formula for eccentricity in this graph is:
The eccentricities for each of the pairs of ratios remains constant as noted because the ellipse is still the same. There is also a formula for the eccentricity of a hyperbola.
The formula for eccentricity of this conic section is ¡§C¡¨ over ¡§A.¡¨ It is proven through distance formulas and substitutions. Also shown, is that the directrixes have the equations of positive and negative ¡§A¡¨ over ¡§E.¡¨ With the parabola, the distance from the point on a curve to the focus and to the directrix are the same, the eccentricity is therefore equal to one. In the ellipse, it is less than one because the ¡§A¡¨ is always greater than the ¡§C.¡¨ In the hyperbola the eccentricity is always greater than one because the opposite of the ellipse occurs. In the circle, the eccentricity is always equal to zero. The name¡¦s for each of the conic sections came directly from their eccentricities. For example, the circle gets its name from the Greek word kirkos, meaning a ring. The ellipse gets its name from the Greek word elleipein, meaning to deduct, the parabola from the Greek word paraballein, and the hyperbola from the Greek word hyperballein, meaning to exceed, or be in excess. In using these noted eccentricities, doing some substitutions, rotating of the axes, and a lot of simplifying, you come up with a shorter version of the vertex equation. But to make it work for every conic section, you must add the ¡§ ¡V (1 ¡V E2) X2¡¨ to the Y2 = 2PX. Therefore you will result in the final vertex equation of:
Y2 = 2PX ¡V (1 ¡V E2) X2
There are many ways in which conic sections apply to our environment and the rest of our surroundings and the things in them. Foremost, the ellipse has been studied as a part of the universal geometric code since in 1609 when a German astronomer by the name of Johannes Kepler proved with astronomical data that the patterns of travel by an object, a planet, in space, could be described by using a geometric curve. This curve was then known as an ellipse. The studies that were then culminated led to the creation of three separate laws that relate conic sections to planetary motion. These laws are today known as Kepler¡¦s Laws. The first law of planetary motion pretty much stated that planets move in an elliptical pattern around their focal object, in this case, the sun. The second law of planetary motion explains that there is a steady rate of time and distance at which the distance of the object from the focus changes. The third and final law, added ten years later, said that the time period for one revolution squared, was proportional to the measure of half the distance across the ellipse at its maximum cubed. In other words, the time it took to complete a revolution was based upon the size of the elliptical orbit. (West, 116) There are a few examples of which conic sections have been used in architecture such as ceilings. Examples of this are in the dome-shaped elliptical ceilings of the Mormon Tabernacle in Salt Lake City, Utah, and at the Whispering Gallery in Washington, D. C. The interesting part of these structures is that when a sound wave is created near one of the focal points, the decibel level is exactly the same at the other focal point, which was quite some distance away. This was due to the ray-like wave hitting the focal point at an angle, which was projected in the other direction at the exact same angle, and while doing so, keeping the sound waves concentrated. The second conic section that has applications to common things around us is the parabola. Some examples of its usage are in suspension bridge cables, and in the arches of common walkways and/or bridges. The reason for it being used is that the weight is best distributed evenly when a parabola curved cable or archway is used, and therefore it will hold the strongest. The more common recognition¡¦s of the parabola in everyday events would be in the kicking or throwing of a ball, the path of water coming out of an ordinary garden hose when pointed upward, and the pathway of a bullet from a fired gun, although not as easily visible. One more way a parabola is used is in headlights. The mirror reflection of the light and being reflected multiple times causes a quite strong beam of light to be developed and creates a better viewing sight. The same concept goes into the creation of telescopes and its reflection of an image that is quite some distance away. Finally, the last conic section that has significant relation to the world we live in is the hyperbola. The easiest way to imagine this event would be to picture a strong light with a perfectly cylindrical lampshade covering it. The light that infiltrates through the lampshade in the openings creates a hyperbolic curve. A jet plane that flies at a speed of over 770 miles per hour will create a sound wave that will hit the ground and create sound in a hyperbolic shape. The sonic boom will be heard quite some distance from the plane itself as an almost half cone image is given. The interception of the ground, taking place as the plane, forms the curve of the hyperbola. It is not a parabola because it would intercept the cone at the top had there been one. Along the line where the hyperbola hits the ground, all different people in many places hear the same noise just as loud. That is the sonic boom. Another way that the hyperbola is applied to everyday science is in a popular radar system known as LORAN. This system uses foci and hyperbolic curves to determine the location of ships and other such objects.
As one can see, there are many formulas that can be derived and applied when using conic sections. Their understanding and comprehension is important for laws pertaining to astronomy, and many other applications as previously mentioned.
Mr. Finta, I would just like to let you know that in the original version of this report there was 14 and a half pages with the vertex equations thoroughly covered and with a great proofreading job. The essay was lost on my computer and I had to start from what I last saved it at, 4 pages. I re-built it as best I could but it is nowhere near the quality of its original counterpart as this one was rushed to be completed.