Notes to Instructor
This module is intended to serve as an introduction to parameterizations of curves. At Kenyon we typically assign the project to the students in class, spending fifty minutes to get them started. Beyond an initial set of questions, we have found that students need very little guidance from us. They go away and come back two weeks later with their impressive works of art.
Software and Technology Needs: In order to use this module, you will need MAPLE version 8.0 or higher. This is because the module relies on Maplets, a feature new to version 8.0. A Maplet is a graphical interface that allows the user to access the power of MAPLE without interacting with the complicated underlying code. If you have older versions of MAPLE, then we refer you to an older version of the lab which exists entirely within this MAPLE worksheet: ParametricPlotsProject.mws.
Prerequisite Material: The module assumes no prior knowledge of parametric equations. It begins with an introduction to parameterizations of circles and ellipses, proceeds to parameterizations of lines and functions, and ends with a few "famous curves" (Lissajous curves, cycloids, epicycloids, and hypercycloids). Note that the section covering parameterizations of lines requires that the reader has some familiarity with vectors. At Kenyon, we assign this module in the third semester of Calculus, well after 2- and 3-dimensional vectors are covered. Nonetheless, with some additional discussion about vectors, this module could be covered in the first or second semester of calculus as well.
The prerequisite knowledge of MAPLE is also kept at a minimum in this module. While many of our students (in Multivariable Calculus) have used MAPLE as a visual and computational tool prior to completing this module, most of the syntax necessary for completing this module is presented to the user as the need arises. Of course, the MAPLETS that accompany this module require MAPLE syntax as input. Hence students will need to know the basic MAPLE syntax for functions [link is obsolete, 2014]. For example, the MAPLE syntax for x(t) = t cos(t) is t*cos(t). (The asterisk is necessary to denote multiplication, and the "cos" must be written with a lowercase "c".)
Word of Warning: The graphs produced by the MAPLETS are unconstrained by default, and thus the x- and y- axes are not shown on a one-to-one scale. This is noted in the Using the MAPLET instructions that appear in the first section of the module. We recommend that instructors reiterate this point to students so that they pay special attention to scaling.
Resizing the MAPLETS: MAPLE automatically sizes the MAPLETS that appear in this module so as to best fit your computer screen. However, this automatic sizing is by no means fool-proof, and at times it is not optimal for some operating systems and screen resolutions. Should your MAPLET Window not open in a manner that best fits your screen (for instance, if all features are not visible,) then you can resize the screen manually or you can reprogram the MAPLET to open with a specified window size. To reprogram a MAPLET, simply search for the following lines of code:
use Maplets:-Elements in
maplet := Maplet(Window( 'title' = "-------",
and insert the following height and width specifications:
use Maplets:-Elements in
maplet := Maplet(Window( 'title' = "-------",
height = 500, width = 400,
Awarding a Prize: Our instructions to our students in Part 6 contain the additional statements:
To make life even more interesting, the best picture in the class will win a prize. Judging will be based on both creative and technical merit.
You should decide whether you want to include this or not.