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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.

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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/
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