Chapter 3 Curves
Applications in engineering, physics, navigation, orienteering, seismology. How do they design roller coasters? How high, how fast, how long and how do they know? How does the captain of a submarine know where he is? How do rescue teams find lost people in the wilderness?
Roller Coasters

From The Honolulu Advertiser, Sunday July 11, 1999

"Santa Clara, CA - The newest invention being put together by Silicon valley engineers is going to make plenty of people sick. The first ever "flying" roller coaster, unveiled recently at Paramount's Great America, lifts riders off their seats, rolls them onto their knees and inverts them - over and over again - for more than two miles. What makes the as-yet unnamed coaster different from all others is the egg-like seat, which cups riders into a flying position, holding their knees and upper bodies in place. As a result, rather than sitting up for the wild journey, riders spend virtually all of the ride on their backs and fronts sailing down the inversions."

How do they design roller coasters? How do they figure out the shapes, how do they know how steep to make the angles, and how do they know how fast it will go? The answer is, of course, mathematics! Visit the web pages below, to learn more about roller coasters, and even to try your hand at building a roller coaster yourself!

http://www.learner.org/exhibits/parkphysics/ Rollercoasters; "What are the forces behind the fun?"

http://www.learner.org/exhibits/parkphysics/coaster/ Annenberg: Design a rollercoaster

http://www.discovery.com/exp/roller coasters/build.html Discovery Channel: Build a Coaster

There are many different curves in a rollercoaster, and the engineers who design roller coasters need to understand the mathematical properties of these curves, and the physics involved, as well as the strengths of the different materials used. Some of the curves that are found in roller coasters are a group of mathematical curves called The Conics.

The Conics

There are an infinite number of mathematical curves, some yet to be discovered. Historically, perhaps the first curve to be "discovered" was the circle, one of four curves called The Conic Sections, or just The Conics. The Conics are four curves: circle, ellipse, parabola and hyperbola, shown below. Learn more about the conics at the website linked below:

Mathematical Curves: The Conics

Test Question #1: How would you describe a parabola as the result of slicing a cone?

http://xahlee.org/SpecialPlaneCurves_dir/specialPlaneCurves.html

Test Question #2: On the above website, to whom is credit given for discovering the Conic Sections?

Circle

The simplest, and most common curve around us is the circle. The circle, as a wheel, is one of the greatest inventions of all time and the basis of much of our transportation system. Circular gears are important elements in many of the machines we use every day, from CD players to electric saws.

The distance around a circle is called the circumference. The formula for finding the circumference of a circle is "circumference of a circle equals pi times the diameter of the circle":

"Pi" is a very interesting number. It has a long and interesting history. Go to the website Earth Mysteries: Notes on Pi, linked below, to find out more about pi, circles and their applications:

http://www-groups.dcs.st-and.ac.uk:80/~history/HistTopics/Pi_through_the_ages.html

TheGeometryPages: Circle

Test Question #3: When, where, and in what document was the earlies known value of pi written?

You can find applications of the circle in many different fields, from art to science. A rainbow is an arc of a circle; have you ever seen a rainbow that is a full circle? Occasionally, one is visible, in certain types of weather. The following website will explain the physics and geometry of rainbows:

http://www.unidata .ucar.edu/staff/blynds/rnbw.html

Test Question #4: Do two people ever see the same rainbow? Explain why or why not.

Circles have many applications. Scientists use intersecting circles when they are trying to locate the center of an earthquake, Circles are used in navigation through radar and sonar, Geometry, and circles in particular, are also used in the sport of orienteering. Visit the Orienteering Association of South Australia to find out what orienteering is:

http://www.oasa.asn.au/default.htm

The following web pages will show you how to use circles to create geometric graphics:

http://forum.swarthmore.edu/alejandre/circles.html

Ellipse

The ellipse looks like a stretched, or elongated circle. But unlike the circle, which is always the same shape, ellipses come in many shapes, from long and narrow to shorter and "fatter":

The orbits of the planets are ellipses, and the orbits of moons, satellites and comets

A circle is easy to draw, with a compass or with a piece of string and a pencil. Visit the following website to learn how to draw an ellipse:

http://www.maa.org/math land/mathland92.html

Test Question #5: What simple tools do you need to draw an accurate ellipse, as described on the website above?

Visit the following website to learn other methods of constructing an ellipse, and how to draw some beautiful geometric graphics.

The Geometry Pages: Ellipse

The ellipse has many applications: elliptical gears are often used for machine tools, and ellipses are used in optics, in telescopes and microscopes.

Many science museums have a "whispering gallery", an elliptical room: sound waves emanating from one focus bounce off the ceiling and are concentrated at the other focus of the ellipse. A whisper spoken at one focus can be heard clearly across the room at the other focus.

Parabola

The Parabola is an interesting mathematical curve, one of the conics, and a curve with many applications from space travel to the sport of baseball. Visit the web page below to learn more about the parabola:

The Geometry Pages: Parabola

The parabola is the form taken by the path of any object thrown in the air, and is the mathematical curve used by engineers in designing some suspension bridges. The properties of the parabola make it the ideal shape for the reflector of an automobile headlight. In an automobile headlight, the light from a bulb at the focus point of the metal back of the headlight reflects and is sent outward to light our way in the dark.

If a ray of light leaves the focus and strikes the parabola, it will be reflected in a path parallel to the symmetry axis. This property is an essential feature of satellite dishes, radio telescopes, and reflecting telescopes, including liquid mirror telescopes.

Whenever we throw an object in the air, the path it follows is a parabola. Therefore, the mathematics of this curve is important in many sports, including baseball. Visit the site below to find out more:

http://www.maa.org/ mathland/mathtrek4_13_98.html

Test Question #6: C.H. Jackson invented something that helped baseball? What is it called, and what is it?

Hyperbola

The Hyperbola, the last of the curves called The Conics, is shown below on the x-y coordinate system. The Hyperbola has two branches.

When a plane intersects a pair of cones, the resulting slice is in the shape of a curve called the Hyperbola. The Hyperbola has two branches:

http://www.sisweb.com/math/algebra/conics.htm

You can find out more about the hyperbola by visiting the following website:

The Geometry Pages: Hyperbola

Historically, Euclid and Aristaeus wrote about the general hyperbola but only studied one branch of it while the hyperbola was given its present name by Apollonius who was the first to study the two branches of the hyperbola. There are other curves associated with the hyperbola, and you can see them by following the links on this web page:

http://history.math.csusb.edu/Curves/Hyperbola.html

The Hyperbola has many applications: the path of a comet often takes the shape of a hyperbola, and many telescopes use hyperbolic (hyperbola-shaped) lenses. Hyperbolic gears are used in many machines, and in industry. Sound waves travel in hyperbolic paths, and so there are applications of the hyperbola in navigation.

You will find some very interesting activities with the hyperbola at the web pages below:

http://www.geocities.com/CapeCanaveral/Lab/3550/hyperbol.htm

Test Question #7: How many straight lines are there on a hyperboloid?

More Curves

The Cardioid is a mathematical curve that is heart-shaped. The name comes from the Greek cardia: heart, as in cardiovascular and cardiology. The Cardioid shown below came from a fascinating website which has information and images of dozens of other mathematical curves. Go to this website and click on the link for Cardioid to answer the question below:

http://www.best.com/~xah/SpecialPlaneCurves_dir/specialPlaneCurves.html

The name cardioid (heart-shaped; from the Greek word cardi, meaning heart). You will find much more information about the cardioid at the following web page:

The Geometry Pages: More Curves

The following website will tell you more about the cardioid, including some interesting historical information:

http://www.maa.org/editorial/knot/Cardi.html

Test Question #8: Who (and when) gave the cardioid its name?

The cardioid can be defined as the path of a point on a circle that rolls around a fixed circle of the same size without slipping. Go to the following website to learn how to construct a cardioid:

The following website will give you the names of dozens of interesting mathematical curves. You can click on any name and go to a page that shows you the curve and gives you interesting information about it:

http:/ /www-history.mcs.st-and.ac.uk/history/Curves/Curves.html

Test Question #9: What is a cycloid?

The Spiral is a mathematical curve that has a long and interesting history. There are a number of different kinds of spirals. One of these is the Fibonacci spiral, which is also called the Golden Spiral. Learn more about this spiral by visiting the website below:

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

Test Question #10: What are the first ten numbers in the Fibonacci sequence?

Projects (Choose 2 of the 3)

1) List and describe (with drawings of your own) 4 applications of each of the following curves: circle, ellipse, parabola, spiral. The applications should be from astronomy, navigation, nature, architecture, music, etc, and not the exact same ones you read about in the MathApplications pages. This report must be at least 250 words, and include a drawing for each application. You might want to begin with the last link above, on Fibonacci spirals, which has many very interesting links.You might also want to use the Math Forum search, at the following website: http://forum.swarthmore.edu/library/

2) Graph the circle and the parabola that have the equations below:

Use graph paper, and make a table of values using the following values for x: -3,-2,-1,0,1,2,3 and find the corresponding values for y by plugging these x-values into the equation. Turn in two sheets of graph paper, one with the table of x and y values for the circle and a carefully drawn graph of the circle, and the other with the table of x and y values for the parabola and a carefully drawn graph of the parabola. Show all your work.

3) Construct a colorful geometric graphic using one of the following geometric curves: cardioid, spiral,ellipse. Use the construction methods described in the Geometry Pages links. (Construct an ellipse with tacks and string.) Use compass to construct arcs and circles, protractor to measure angles, and a ruler to draw all lines. Color your design with colored pens or pencils.

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