Question

What is the equation of a sphere in cylindrical coordinates?

Answer 1

Hint: Extending our cartesian coordinate system from two dimensions to three, then we need to add the new axis to the model to get the third dimension. Beginning with polar coordinates we can do the same for cylindrical coordinates also.

Complete step-by-step answer:
Defining surfaces with rectangular coordinates sometimes becomes complicated so for that we need to get the simplified form to solve. Cylindrical coordinates can simplify the plot which is symmetric with the z-axis such as paraboloids and cylinders. The paraboloid z=x2+y2 and it will become z=r2and the cylinder x2+y2=1 would become r=1.
For conversion to cylindrical coordinates from rectangular coordinates we need to use the following conversions:
X=rcos(p)
Y=rsin(p)
Z=z
Here, in these conversions r is the radius in the x-y plane and p is the angle in the x-y plane.
Now, we need to find the equation of a sphere in spherical coordinates as per the required condition in question.
So, for that we know that the equation of a sphere in rectangular coordinates system is ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}={{r}^{2}}$ where r is the radius of the sphere. Since ${{x}^{2}}+{{y}^{2}}={{r}^{2}}$ in cylindrical coordinates, the equation of a sphere in cylindrical coordinates can be written as ${{r}^{2}}+{{z}^{2}}={{R}^{2}}$.


Note: We just need to make sure that we are using the terms wisely and not getting confused with the conversions of polar, cylindrical and rectangular coordinate systems. Simply we can use simple conic section equations.