Intersection of Two Spheres

Date: 10/13/2003 at 10:48:31
From: D.S.
Subject: Intersection of 2 spheres

I was able to figure out from your previous answers how to find the
intersecting point when we have 3 spheres:

  (x-x1)^2 + (y-y1)^2 + (z-z1)^2 = R1^2
  (x-x2)^2 + (y-y2)^2 + (z-z2)^2 = R2^2
  (x-x3)^2 + (y-y3)^2 + (z-z3)^2 = R3^2

Now I'm trying to figure out how to find the equation of the 
intersecting circle when I have 2 spheres:

  (x-x1)^2 + (y-y1)^2 + (z-z1)^2 = R1^2
  (x-x2)^2 + (y-y2)^2 + (z-z2)^2 = R2^2

I would like to display this circle as the illustration to the 
intermediate step before displaying the final intersection point that 
is created by 3 spheres.  Hence, I would also need to find the 
equation for the radius of the circle.

The equation for the intersecting circle should look something like

  (x-a)^2 + (y-b)^2 = r^2

although it won't, in general, be parallel to the x-y plane.  

I would greatly appreciate your help!

D.S.

Date: 10/13/2003 at 13:05:38
From: Doctor Rob
Subject: Re: Intercection of 2 spheres

Thanks for writing to Ask Dr. Math, D.S.!

Curves in 3-space are generally described by a pair of simultaneous
equations, rather than a single equation.  A single equation
generally describes a surface, not a curve.  That means that an
equation of the form 

  (x-a)^2 + (y-b)^2 = r^2 

cannot be expected to appear for your circle, unless the other
equation is something like z = c.  This works for circles whose plane
is parallel to the xy-plane, but not otherwise.

To find the intersection point of three spheres whose equations are
those you gave above, pick one of the equations and subtract it
from the other two.  That will make those other two equations into
linear equations in the three unknowns.  

Use them to find two of the variables as linear expressions in the
third.  These two equations are those of a line in 3-space, which
passes through the two points of intersection of the three spheres.  

Then substitute these into the equation of any of the original
spheres.  This will give you a quadratic equation in one variable,
which you can solve to find the two roots.  

These values will allow you to determine the corresponding values of
the other two variables, giving you the coordinates of the two
intersection points.

For the intersection of two spheres, you can subtract one equation
from the other, to get a linear equation in the three variables.
This is the equation of the plane in which the intersecting circle
lies.  This you can put in the standard form

   a*x + b*y + c*z = d.

Next, you can find the line through the center of the circle by
finding the line through the center of the first sphere which is
perpendicular to the above plane, in the form of the parametric
equations

   x = x1 + a*t,
   y = y1 + b*t,
   z = z1 + c*t.

Substituting this into the equation of the plane will give you one
equation in the parameter t, which you can solve for t = t0, and
find the coordinates

   (x1+a*t0, y1+b*t0, z1+c*t0)

of the center of the circle.  To find its radius r,

   r^2 = R1^2 - (a^2+b^2+c^2)*t0^2.

Feel free to write again if I can help further.

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

Date: 10/13/2003 at 14:33:34
From: D.S.
Subject: Thank you (Intercection of 2 spheres)

Thanks so much for your prompt answer. It helped me a 
great deal.

Best regards,
D.S.