lens - Do lenses have focal planes or focal spheres/ellipsoids?

The title says it pretty much all. I'm asking this because I find myself always using the focus point in the center, and after I focused, I rotate my camera to make the framing right. However, if lenses do create a plane in which everything is sharp, then the focus will be slightly off after rotating the body.

Doing some maths: if I focus at distance x meters, and rotate my camera for alpha radians, then the focus plane would be off for:

∆f = x*tan(alpha/2)*tan(alpha)

So, let's plug in: x = 2 meters, alpha = 20 degrees = 0.349 radians, we get: ∆f = 0.128 meters. So this means, my optimal focus point is now almost 13 centimeters away from the original focus point.

If they are not spherical, I will experience my focus being off. So is it safe to use other focus points except for the one at the center and immediately put my framing right? This will slow down the process of taking pictures?

Answer

Yes on normal lenses the area in focus is nearly perfectly described as a plane and the small deviations are rarely taken into account in regular photography. There are two important factors that cause these slight imperfections: the Petzval curvature and Astigmatism. The uncorrected astigmatism is usually more severe but can be over-corrected in order to also somewhat get rid of the petzval field curvature (at the cost of sharpness). The Petzval curvature (sum) is caused only by the curvatures of the surfaces and the refractive index of the lenses involved and not the thickness of the optical system. Some optical systems such as big telescopes actually has curved image-sensor arrays to compensate for the Petzval curvature. In DSLR:s this is not an option since different lenses has different Petzval curvatures and the plane is therefore slightly curved.

When focusing and recomposing one will move the plane of focus from whatever the camera locked it's focus on. This distance will increase with the distance to the subject and increased difference in angle from where focus was acquired.

There are however special lenses where the focused area is not a plane at all. If you for example have a lens-sensor system where the lens can be tilted in one plane and the sensor in another the area of focus can be limited to a line instead.