Traceable Mathematical Curves

Date: 10/27/97 at 09:15:32
From: Anonymous
Subject: Traceable mathematical curves.. help!

Dear Dr. Math,

I am an eighth grader at Gilman School in Baltimore, Maryland. We are
studying Traceable Mathematical Curves. I was wondering - is there any 
way to tell just by looking if a curve is traceable or not? Is there 
some property of a curve that will tell you this? 

Zachary Heaps

P.S. Do curves have formulas?

Date: 10/27/97 at 10:37:32
From: Doctor Rob
Subject: Re: Traceable mathematical curves.. help!

I am not quite sure what you mean by a traceable curve. Do you mean 
that the curve can be drawn without lifting your pencil?

Most curves have formulas. Once you learn the Cartesian coordinate 
system of plane geometry, you will learn that points are given by two 
real numbers called the x-coordinate and the y-coordinate.  Most 
curves are defined by an equation relating x and y to each other, 
such as

   x^2 + y^2 = 100

which represents a circle. Some curves are given by telling what 
x and y are in terms of another variable, say t, such as

   x = 10*(1-t^2)/(1+t^2)
   y = 10*(2*t)/(1+t^2)

which represents the same circle (this is not obvious, but true).

When you have and equation in x and y, as above, or one relating 
x and y to another variable t, as above, the graph of the equation(s) 
is the set of all pairs (x,y) which satisfy them. To be able to draw 
the graph of the equation(s) without lifting your pencil, the graph 
must not be able to be separated into two parts that are not connected 
to each other. 

An example of a curve for which this fails is the hyperbola x*y = 1.  
There are solutions with both x and y positive, which form one 
connected part, and there are solutions with both x and y negative, 
which form a different connected part. These two parts are 
disconnected from each other, however.

You may be asking whether one can tell from the equations whether the 
graph of the equations is connected or not. If that is a correct 
interpretation, the answer is that it is not easy to tell. For 
example, the equation

   y^2 = x^3 - 3*x^2 + 2*x

has a disconnected graph, while the equation

   y^2 = x^3 - 3*x^2 + 3*x

(changing the last 2 to a 3) has a connected graph.

For polynomial equations of degree 2 in x and y, one can tell what 
kind of curve they represent, and this gives you the answer. There are 
several possibilities.  If you want information on this, write again 
and ask.

I hope this helped.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 10/28/97 at 09:08:08
From: Anonymous
Subject: Re: Traceable mathematical curves.. help!

Dr. Math,

You are the man(or woman)! Thank you for taking the time to respond to
my question. The formulas are great and I even understand some of 
them! 

I am sorry that I was unclear about the traceable part of the 
question. Our teacher said that he would give us 20 geometric figures 
and we would have a short time to decide if they were traceable or 
not. He said that you could just look at them and tell right away if 
they could be traced. I have been working really hard and I can't 
figure out how to tell this without tracing it with my eyes and this 
takes a long time. Can you help with this part of the question or 
direct me to some other place?

Thank you so much,
Zachary Heaps

Date: 10/28/97 at 10:23:22
From: Doctor Rob
Subject: Re: Traceable mathematical curves.. help!

Aha!  Probably the geometrical figures are going to be points 
connected by lines. The lines can be straight or curved. The question 
will be to trace the figure with your pencil without lifting the point 
and without retracing any line in the figure. This is a common 
problem, and you can tell by inspection whether or not this is 
possible.

The first thing is to make sure the figure is connected, that is, you 
can get from any part of the figure to any other part without lifting 
the pencil.  That is easy to do by inspection.

The second thing is to look at each point and count how many lines end 
at that point.  The reason this matters is as follows.

Think about a path tracing the figure. If the point is not the 
beginning or end of the tracing path, whenever that path gets to a 
point, it must leave it by a different line. This accounts for two 
lines ending at that point. If the tracing path revisits the point, 
there will be two more lines accounted for for each visit. In order to 
be able to trace without retracing any line, all the points which are 
not beginning or end points of the tracing path must have an even 
number of lines ending there. If there are points with an odd number 
of lines ending there, they must be the beginning and ending points of 
the tracing path.

Thus there must be either zero or two points having an odd number of 
lines ending there. If there are two, one is the beginning and one is 
the end of the tracing path.  If there are zero, any point will do for 
the beginning, and the path must end there.

If there are more than two points with an odd number of lines ending 
there, no tracing path can exist. An example of that is the following 
figure:

         O
         |
         |
   O-----O-----O

The numbers of lines for the four points are 1, 1, 1, and 3, of which 
four are odd, so no tracing path is possible. (I bet you knew that 
anyway!)

Another example is:

        O
       / \
      /   \
     /     \
    O-------O
    |\     /|
    | \   / |
    |  \ /  |
    |   X   |
    |  / \  |
    | /   \ |
    |/     \|
    O-------O

(X is not a point, just a crossing of two lines. The O's are the 
points.) The numbers of lines for the five points are 2, 4, 4, 3, 
and 3, of which two are odd, so a tracing path is possible. Just be 
sure to start at one of the points where three lines meet, the bottom 
corners.

One odd but helpful thing is that once you start at a good point, it 
almost doesn't matter which way you go! Just don't retrace a line you 
have already traced, and don't go to the end point until you have to, 
and eventually you will have a tracing path. Try it - it's fun and 
easy!

I hope that this is what you were talking about.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/