Lines and Curves

Date: 04/21/99 at 09:42:41
From: Laura Bordoni
Subject: How to make curves from straight lines

My friend and I are doing a project at school and our teacher has 
shown us one way to do it, but we can't do the whole project on that 
one piece of information: when you make an L shape and draw a line 
from the top to the lefthand bottom and keep repeating that, you have 
made some curves from straight lines. 

Please help us.

Date: 04/21/99 at 13:03:22
From: Doctor Peterson
Subject: Re: How to make curves from straight lines

Hi, Laura.

Thanks for stating your problem clearly.

This way of constructing curves from straight lines is called an 
"envelope," because the curve just fits around the lines. When you 
draw all the lines, you imagine you see a curve touching all of them, 
even though it's not really there. What you are really doing is 
approximating the curve by lots of straight lines. It is often done 
using strings stretched between nails; here's a page I found that 
illustrates several examples:

  Mathematical String Art - Richard and Donna Goldstein
  http://www.providence.edu/mcs/rbg/strngart.htm   

Here's a page from a computer programming class showing how to draw 
something like what you asked about; just look at the pictures!

  Math 150 - Introduction to BASIC Programming - Larry Riddle
  http://www.mathforum.org/dr.math/gifs/string_art_lab.doc   

For your first try, just draw two lines in an "L", as you said, maybe 
six inches long; from the point where they meet, mark every inch along 
each line. On the vertical line, label these points 1, 2, 3, 4, 5, 6, 
7, with "1" at the intersection of the lines; on the other line, label 
them 7, 6, 5, 4, 3, 2, 1 going to the right. Now draw a line between 
the two 1's, another between the two 2's, and so on. You should see 
something like the pictures on these pages.

In general, you can use any two lines, not necessarily perpendicular; 
just mark the same number of points along them, equally spaced, and 
then connect corresponding pairs of points. The shape you get is 
called a parabola. Try experimenting with different lines going in 
different directions to see what works best. You can stretch out the 
points on one line and squeeze them together on the other; make the 
lines close together or far apart; have them intersecting or parallel; 
and so on.

Here's a page on the parabola from Xah Lee's Visual Dictionary of 
Special Plane Curves that tells a little about this; look 
about halfway down at the section on "tangents of parabola":
  
  http://xahlee.org/SpecialPlaneCurves_dir/Parabola_dir/parabola.html   

There are other ways to make similar constructions that will make 
different curves. The first page I mentioned shows several, though it 
doesn't tell how to actually figure out where the points should go. 
The easiest is the astroid (which they also call a "sliding ladder"), 
which is made by marking pairs of points on the two lines that are the 
same distance apart. You can even invent your own rule for selecting 
pairs of points and see what happens!

There's another neat way to make a curve using straight lines. Make a 
circle out of paper (it has to be a perfect circle) and mark a dot 
somewhere about halfway from the center to the outside. Now pick any 
point on the outside (circumference) of the circle and fold it over so 
that point is on the dot you made. Crease the paper to make a line, 
then unfold it. Do this with many different points on the outside, and 
the folds will form the envelope of a curve called an ellipse.

Have fun playing with this.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/