Expert: Steve Holleran - 5/7/2007

Question
QUESTION: I have a project where I need to be able to find the intercepts of two circles.  I am given the latitude and longitude (Y and X respectively) of the center points of the circles, and then need to get an equation for the two intercepts.  In the end, I need an X = ? and Y = ? (except there should be two X and Ys as there should be two intercepts - and I cannot have any Y's in the X equation or X's in the Y equation).  Luckily, the radius of the circles will always be the same.

So, I have:
(X - Long1)^2 + (Y + Lat1)^2 = r^2
(X - Long2)^2 + (Y + Lat2)^2 = r^2

Since the r are equal I can get:

(X - Long1)^2 - (X - Long2)^2 = (Y + Lat2)^2 - (Y+ Lat1)^2

I can solve this equation for X or Y and then I know I will need to plug that back into this, but it gets so messy that I have no idea how to get it past that.  I am not sure if there is an easier way to do it (even a solver online that could simplify it would be fine with me) but any help you could give would be greatly appreciated.  Thanks,

Mark


ANSWER: Hi Mark,

I'm not real sure why your latitude parentheses are in the form (y + lat1)^2.  The basic form for the equation of a circle with center at (h, k) and radius r is:

       (x - h)^2 + (y - k)^2 = r^2

So I would think yours would be :

     (x - long1)^2 + (y - lat1)^2 = r^2, no?

In any case, lets stay with the equations the way you have them--its only a minor difference anyway, and what I'm going to describe here would basically be done the same.

If you're going after the x-intercepts of the circle, I assume you're looking for the points where the circle intersects the x-axis.  All you have to do is set y = 0 in each equation:

For the first one, it would go:

 (x - long1)^2 + (0 + lat1)^2 = r^2

 (x - long1)^2 + (lat1)^2 = r^2

 (x - long1)^2 = r^2 - lat1^2

 (x - long1) = +/- sqrt(r^2 - lat1^2)

        x = long 1 +/- sqrt(r^2 - lat1^2)

As you can see, because of the +/-, you get two intercepts.

The algebra for the y-intercepts is exactly similar.

Just a note here, Mark.  When you say you are trying to find the intercepts, I hope you are not trying to find the INTERSECTION points of the two circles.  If that is the case, then we are in a world of mess.

I even looked online on how to do this, and its really bad.
The only advice I can give is to use a graphing calculator in parametric mode to graph the circles, and then find the intersection points graphically.  Not much help there, I know.

Well, I hope you were looking for the intercepts and not the intersection points.  In any case, its not an easy problem.

Steve

---------- FOLLOW-UP ----------

QUESTION: Thanks for the fast response.  Sadly, yes I am trying to find the intersection points of the two circles.  When I was working it out I saw that it was really bad and I do not have the math skills to solve it out.

I do not need to know all of the steps to get it, is there any equations solver that I could plug this into and have it return what X and Y are equal to (in terms of Lat and Long, with no X and Y in the equatutions):

(X - Long1)^2 - (X - Long2)^2 = (Y + Lat2)^2 - (Y+ Lat1)^2


Answer
Hi Mark,

Unfortunately, I know of no such solver, but that doesn't mean that there isn't one out there.  Perhaps some college math site that uses Mathematica or Maple software could steer you in the right direction.  

In my experience at the high school level, I was unaware of any such program.  Sorry I can't be of more help.

Steve