I am interested in finding out how to represent the lower half of a sphere in the form of z=f(x,y) with r=2 and a centre point (3,1).

I am an engineer that has not done this for some time!!!

Many thanks


Hello Yvette.

A sphere's equation centered at the position (a,b,c) with radius r is

(x-a)2 + (y-b)2 + (z-c)2 = r2

So a full sphere of radius 2 and centre point (3,1,0) is

(x-3)2 + (y-1)2 + z2 = 22

When you solve this for z, you get

z = sqrt(4 - ((x-3)2 + (y-1)2) ) or z = - sqrt(4 - ((x-3)2 + (y-1)2) )

If z is the up/down axis, then you only want z < 0 if you want just the "lower half" of the sphere, that is

z = - sqrt(4 - ((x-3)2 + (y-1)2) ).

If you intended a solid sphere, rather than the shell, then change the first equation to

- sqrt(4 - ((x-3)2 + (y-1)2) ) ≤ z ≤ 0

Hope this helps,
Stephen and Penny.