Finding a Locus of Points from a Description

Date: 03/30/2004 at 12:35:54
From: Rene
Subject: locus in geometry

I am trying to help my son figure out these three questions.  We have 
asked several people and no one can help us.  Is there a formula to 
use?  I know nothing when it comes to geometry, so hope you can help.

They all have to do with the locus in geometry.

1. What is the locus of points in the plane of an angle that are 
   equidistant from the sides of angle?

2. What is the locus of points in space that are equidistant from 
   two parallel planes?

3. What is the locus of points in a plane that are equidistant from 
   points A and B in the plane?

Date: 03/30/2004 at 13:03:47
From: Doctor Peterson
Subject: Re: locus in geometry

Hi, Rene.

This isn't a matter of formulas; I'm not sure exactly what is the key 
to recognizing a locus.  Each problem is a little different.  In the 
hardest cases, you have to use analytic geometry, and only when you 
get an equation will you see what the locus actually is.  In many 
cases, you just "see" it; the answer is obvious when you look at it 
the right way.  These problems are in between; they're obvious to me 
because I know enough geometry, but you can figure them out visually 
or analytically if you don't immediately see it.

First I'd like to point you to some discussions of what locus is, and 
some examples:

  Locus
    http://mathforum.org/library/drmath/view/55121.html 

  The Meaning of Locus
    http://mathforum.org/library/drmath/view/62350.html 

  Locus and Equations of Lines
    http://mathforum.org/library/drmath/view/52853.html 

Let's start your first problem:

  1. What is the locus of points in the plane of an angle that are 
     equidistant from the sides of angle?

I would start by drawing a picture:

       /
      +   P
     /   o
    /    |
   o-----+--
  O

I've drawn an angle, and a point P that is equidistant from the sides 
of the angle (that is, the same distance from each ray). I drew in (or 
would if I could draw!) perpendicular segments from P to each ray, to 
remind myself how to find the distance from a point to a line (or 
ray).  The locus is not yet visible; it will be the set of all places 
where P might be ("locus" is just the Latin word for "place"), or 
equivalently the set of all points P that fit the definition.

Now what you do next will depend on your thinking style.  If you can 
visualize easily, imagine moving P around while keeping the two 
distances the same; you'll find that it slides along in a particular 
direction, tracing out a certain line.  If you prefer experimenting 
with reality rather than picturing things in your mind, try finding 
several points that fit the description, and look for what they have 
in common.  Then you can fill in between, to locate all possible 
points that work.  On the other hand, if you like theory and have an 
analytical mind, you can just look for a theorem or two that will 
determine where P can be; in this case, try drawing in segment OP and 
you'll notice a pair of congruent triangles.  What do they tell you 
about the location of P?

Whatever method you use to discover the locus, or just guess at it, 
you need to convince yourself that your answer is correct, either with 
a real proof, or just enough thinking (along the lines of the 
analytical approach I listed last) to feel confident that you've got 
it right.

So it takes a certain amount of "playing" and imagining to find the 
answer, and a certain amount of familiarity with theorems or facts of 
geometry to see that you have it right; locus problems test your 
knowledge and a bit more.  But usually if you just think about the 
meaning of the terms in the problem (what is the distance from a 
point in space to a plane, for example?) and think concretely, you 
should be able to get it.

If you need more help, please write back and show me how far you got 
on your problems.

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

Date: 04/05/2004 at 11:33:22
From: Rene
Subject: Thank you (locus in geometry)

Thank you for your prompt and excellent help!  We thought about the 
questions and hopefully got them right.