Parametric Equations, Maple and TubePlots

The Maple tubeplot function graphs in a three-dimensional world with parametric equations, although in a sense we will be working only in two dimensions, because the value of x(t) in the functions we use will always be zero. The tubeplot function has the form:

Multiple functions (tubes) can be graphed using one tubeplot command by including more square brackets:

tubeplot({[x₁(t),y₁(t),z₁(t),t=t-min..t-max], [x₂(t),y₂(t),z₂(t),t=t-min..t-max]}).

We begin by describing a block-style letter in the y-z plane with a set of parametric equations (See Figure 1). The letter "T" is described by two sets of equations. The vertical segment of the "T" is given by

x(t)=0

y(t)=10

z(t)=t, where 0  t  10

and the horizontal segment by

x(t)=0

y(t)=t, where 6  t  14

z(t)=10

Producing these parametric plots is an engaging way to get students involved in such concepts as variables, constants and parameters. To draw each segment in the letter, they must consider which variable (y(t) or z(t)) stays constant and which one changes, and find the proper boundaries for the parameter. The first project students do after an introduction and demonstration on the computer, is to write their first names with tubeplots, using only block-style letters formed from horizontal and vertical segments (See Figure 2). The "S" is composed of five separate segments, three vertical and two horizontal. Thus the first five sets of square brackets in the tubeplot command contain the information for the "S":

[0,t,10,t=10..14],[0,10,t,t=5..10],[0,t,5,t=10..14], [0,14,t,t=0..5],[0,t,0,t=10..14].

I believe it is better to have students work in pairs until every student has a firm grasp of what he or she is doing. This a wonderful activity for generating student discourse about mathematics. Students will make conjectures and can immediately test their hypotheses. If the letters do not come out the way they intended, they can adjust their hypotheses until the letters attain the desired shape. Students will begin to ask questions about previously studied topics, such as roots, transformations, and the fundamental theorem of algebra, as they attempt to create the letters in their names and actually fit different functions to a given set of points. Students are quick to grasp the concepts required to create the tubeplots, and this activity has been used very successfully with Algebra II, Precalculus and Calculus students.

Students are given the block-letter name assignment first. The next assignment is to write their names using only non-constant functions such as polynomials, absolute value, conic sections and trigonometric functions, whenever possible (See Figure 3). The letter"S", which consisted of five separate segments in Figure 2, is now created from a single set of parametric equations using a sine function. The corresponding parametric equations for the "S" are given by

x(t)=0

y(t)=

z(t)=t, where 0 t 10

The "A" is written using an absolute value function combined with a horizontal segment. The "M" is created by fitting a fourth degree polynomial to the given roots, the middle part of the "M" being a double root. The"-.5" coefficient was arrived at by guessing and checking. "SAM" is a relatively simple name, yet it is rich in mathematics. Graphing tubeplots leads very naturally into the concept of inverse relations as students must think about reflecting graphs and "reversing" the axes to get the letter they want. Letters such as"S," "B" and "C" are excellent for applying inverse relations. Some students may want to get more cursive letters. They can use trigonometric and exponential functions, or conic sections to accomplish it. This can lead to frequency and amplitude changes of periodic functions, rotated conics or parametric representations (such as x(t)=cos(t) and y(t)=sin(t)) of conics in order to get their letters exactly the way they want them. Some students may want to use all three dimensions. Students generated many of the offshoots of the original assignment through their own curiosity and exploration. There is plenty of opportunity for extension and investigation.

Maple "welds" the segments of the tubes together very nicely, when they are close enough (compare the "T" in Figure 1 with the "S" in Figure 2), and has many different coloring schemes. A few of the optional parameters in Maple's tubeplot function are the number of points it evaluates to create the graph, the tube radius, the style of axes, the coloring scheme and the orientation. The more points used to generate the tubeplot, the longer the plot will take to create and the more memory it will require, so a compromise must be reached between output quality, computer processing power and memory. Students will explore with various functions when trying to create a specific letter, for instance-a sine function versus a third degree polynomial, to make the letter "S".  You may want students to use only certain functions, such as lines and absolute value functions, or open it up to any type of relation that will work. As an introductory activity, you may want to have all students do the same word, such as "MATH," before graphing their names. Give students plenty of time for investigation and discovery, especially when just beginning.
 

Sample Maple Equations for the Lesson