LOVEWRIT
Spelling out LOVE with functions
BRIEF ABSTRACT
Here's a quick challenge for students learning how to plot parametrically: write a word with parametric curves.
GENERAL INFORMATION
FileName:   LOVEWRIT
Full title:  Spelling out LOVE with functions
Last Update: 6/5/96
Developer: 
Lynn Kiaer, Department of Mathematics,  Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA
Contact:  
Aaron Klebanoff, Department of Mathematics, 
Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. 
Phone:  812-877-8151.  Email:  Klebanoff@rose-hulman.edu.  
FAX:  812-877-3198.
Support:  
The production of this material is supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849:  Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN. 
STATEMENT OF PROBLEM 
Write a set of parametric equations whose graph is the word LOVE in block capitals, with the bottom left corner of L at (1, 1).
KEYWORDS  
Functions, parametric plotting, geometry.
TEACHER NOTES
ISSUES RELATED TO THE PROBLEM 
It's not necessary to specify the location of the first letter.  It was provided here to make the problem more concrete for the students.
Prerequisites
Understanding of lines and circles and their parametric plots.
Time allotment - time management
This can be done in 30 minutes with groups of students - each taking parts of the word L O V E.
Expectations 
Future payoffs
Extensions
Students can parameterize their names, other words, and they can even solve the ultimate problem of writing in cursive!
References and Sources
POSSIBLE SOLUTION(S)
Suppose we decide that the O will be a circle with radius 1.  Then the heights of the letters should be 2, and the other letters should probably be roughly of width 1.5.  (Other decisions are possible, of course.  Making the O an ellipse with small eccentricity is somewhat more visually pleasing.)
Here is the vertical bar of the L
In[2]:=
x[t_] := 1 /; t<=1 && t >= 0
y[t_] := 3 - 2t /; t<=1 && t >= 0
And here is the horizontal
In[4]:=
x[t_] := t /; t>1 && t<2.5
y[t_] := 1 /; t>1 && t<2.5
In[6]:=
L = ParametricPlot[{x[t],y[t]},{t,0,2.5}, Axes->False]
ParametricPlot::pptr: 
   {CompiledFunction[{t}, <<2>>][t], <<1>>}
     does not evaluate to a pair of real numbers at t
     = 2.5.

Out[6]=
-Graphics-
Now for the O.  It will have its center at (3.5, 2).
In[7]:=
x[t_] := 3.5 + Cos[t Pi] /; t>2.5 && t<4.5
y[t_] := 2 + Sin[t Pi] /; t>2.5 && t<4.5
In[9]:=
Oh = ParametricPlot[{x[t],y[t]},{t,2.50001,4.5}, 
	Axes->False,AspectRatio->Automatic]
ParametricPlot::pptr: 
   {CompiledFunction[{t}, <<2>>][t], <<1>>}
     does not evaluate to a pair of real numbers at t
     = 4.5.

Out[9]=
-Graphics-
In[10]:=
Show[L, Oh]

Out[10]=
-Graphics-
Now for the V.  We'll put the upper left end at (4.5,3), the vertex at (5.5,1) and the upper right at (6.5,3).
In[11]:=


In[11]:=
x[t_] := t /; t>4.5 && t<=6.5
y[t_] := 3 - 2(t-4.5) /; t>4.5 && t<=5.5
y[t_] := 1 + 2(t-5.5) /; t>5.5 && t<=6.5
V = ParametricPlot[{x[t],y[t]},{t,4.50001,6.5}, 
	Axes->False,AspectRatio->Automatic]

Out[14]=
-Graphics-
In[15]:=
Show[L,Oh,V]

Out[15]=
-Graphics-
All that's left is the E.  But what a letter!  The upper left corner is at about (6.6,3), and it starts out a lot like the L, only translated.
In[16]:=
x[t_] := 6.6 /; t<=6.6 && t>6.5
y[t_] := 3 - 20(t - 6.5) /; t<=6.6 && t>6.5
In[18]:=
x[t_] := t /; t>6.6 && t<=8.1
y[t_] := 1 /; t>6.6 && t<=8.1
In[20]:=
x[t_] := t-1.5 /; t>8.1 && t<=9.6
y[t_] := 3 /; t>8.1 && t<=9.6
In[22]:=
x[t_] := t-3 /; t>9.6 && t<=10.6
y[t_] := 2 /; t>9.6 && t<=10.6
In[24]:=
Ebottom = ParametricPlot[{x[t],y[t]},{t,6.50001,8.1}, 
	Axes->False]

Out[24]=
-Graphics-
In[25]:=
Etop = ParametricPlot[{x[t],y[t]},{t,8.10001,9.6}, 
	Axes->False]

Out[25]=
-Graphics-
In[26]:=
Emiddle = ParametricPlot[{x[t],y[t]},{t,9.60001,10.6},
	Axes->False]

Out[26]=
-Graphics-
In[27]:=
EE = Show[Ebottom,Emiddle,Etop]

Out[27]=
-Graphics-
In[28]:=
Show[L,Oh,V,EE]

Out[28]=
-Graphics-
In[29]:=
Show[{L,Oh,V,EE},AspectRatio->Automatic]

Out[29]=
-Graphics-
ISSUES IN SOLUTION
One of the frustrations in solving this problem is the fact that Mathematica wants to connect everything.  If you ParametricPlot the whole thing, you get
In[30]:=
ParametricPlot[{x[t],y[t]},{t,0,10.6},Axes->False]

Out[30]=
-Graphics-
We can make the E look okay by defining it from the upper right to the lower right, and traversing the center crossbar twice, but I haven't figured out how to get around the other problems.  If anyone knows how to make a Mathematica function discontinuous...