The Shape of the Locus - April 10-14, 1995 (repeated April 17-21), 1995
Kelly walked into the dining room, where Kim was working on some geometry problems. "What's up?""I'm trying to figure out a couple of problems that we looked at in class," said Kim. "The first one we built in class, and the second one was supposed to be homework. But I'm having problems figuring both of them out. Seems like the teacher used some sort of divine hammer and just knew the answer!"
"I know what you mean. Sometimes these things just fall out of the sky like a cannonball. So show me what you're doing," said Kelly.
"Okay. On the first one, we built this tool--you put two pins on a board, with a loop of string that's longer than the distance between the two, and put a pencil inside the loop and pull it around the pins, and you draw some shape when you loop around the pencils. Like this:
__________________
|O <--- pins ---> O|
\ /
\ /
\ / <-- string
\ /
\ /
\O/
^
|__ pencil
We're supposed to figure out what shape the pencil traces out. This is
an example of a 'locus' or something. We're also supposed to figure out
what a locus is.""Isn't it an insect or something?" asked Kelly.
"No, that's a locust, silly! This has something to do with geometry or math."
"I've never heard of it. You said you had two problems?" asked Kelly.
"On this one, they tell us what the constraints are on the path, and we're supposed to figure out what shape it is, and how we could build a machine like the one above to draw it. This one is supposed to be the path of a point where the lines of sight to two different places from the point are at right angles to each other. Like this:
O O
* *
* *
* *
* *
* *
* *
O
So how does the point move if those two lines are to stay perpendicular to
each other? And we get extra credit if we can build a machine that will
draw it, too."
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