# Circles on XY, YZ and XZ planes from a Sphere

## Main Question or Discussion Point

I want to find the equations for the circles (formed on the planes) when a sphere cuts the XY, YZ and XZ planes. What I am trying to achieve is a software application that will have a 3D cuboid and inside this cuboid there will be many spheres. Now I want to find the circles created by these spheres when they intersect with the planes. Thank you.

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Any sphere? Then you'll have some equation in the unknowns x,y,z that defines which points (x,y,z) belong to the sphere. The points in the XY plane also must fulfil z=0, so you'll get a new equation in x and y only. That describes the first circle you're looking for.

Suppose there is a sphere with a certain radius in this cuboid and it only intersects with the YZ plane. There is another sphere that has some radius and it intersects with any two planes and so on...

I'm not quite sure what you're starting off with. When you have a sphere of radius r and centre (cx,cy,cz), this sphere is formed of all points (x,y,z) with (x-cx)2+(y-cy)2+(z-cz)2=r2. The points on the XY plane also fulfil the equation z=0. Put those two together, and you get (x-cx)2+(y-cy)2=r2, i.e. the equation of the circle in which the sphere intersects the plane.

Is that what you meant?

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Thanks a lot Michael for your replies and help. If I have a cuboid (front, back, top, bottom, left and right planes) will there be two XY (front OR back), two YZ(left OR right) and two XZ (top OR bottom) planes ? Or am I thinking wrong.

Now keeping my confusion in mind, can you tell me if the sphere intersects the XY plane (which I am thinking of as the front OR back), does your previous answer hold. I mean the equation you had provided. I think you will need to explain to me like a child :-)