Geometry
Chapter 10, Transformations and Patterns
The motion of a sailboat can be tracked by using a series of isometries.
Below, you will describe the isometries found in diagrams of sailboat movement.
In these diagrams, you will see roundabout paths to get from one place to
another. These are necessary in sailing to optimize the power of the wind.
Part A, Isometries
1. Go to the Match Racing Web site and scroll down to the first diagram after the title "Pre-start circling manoeuvres." Have you ever thought about how sailboats start a race? The sailors cannot just sail their boats to the starting line and stop until the race begins. One of the tactics for gaining optimal starting position is called "circling." One type of circling is described in this diagram.
a. Use transformations (reflections, rotations, and translations) to describe the series of isometries which occur from position #1 to position #5 for sailboat A. Be careful! Each of the shifts in position requires at least two different transformations. If the transformation is a reflection, describe the line of reflection. If it is a rotation, describe the center, angle and direction of rotation. If it is a translation, give the distance of the translation in terms of boat length, and describe the direction of the translation.
b. Which of the three basic transformations which are isometries will probably not be used to describe the motion of a sailboat. Why?
Part B, Compositions of Transformations
2. Now, you will look at some instructions about steering a sailboat into the wind. Go to the Practical Aspects of Sailing Web site for basic instructions on sailing.
a. Scroll down to figure 23 titled "Beating Upwind." This diagram shows how to make forward motion when you are sailing into the wind. Describe the sailboat's path using the language of transformations as you did in #1a.
b. Describe one transformation which is mathematically equivalent to the composition of transformations described above. Is this a viable path for the sailboat?