I got that, but what about the converse? If , a fixed point that lies in every normal plane of a curve , then the curve must be spherical.
Here's my thinking so far: A curve is spherical if and only if . I would like to use the assumption that there is a fixed point in every normal plane to reach this equality, but I'm pretty lost.
I'm not clear on what it means for p to be in every normal plane of . Does this mean that such that ? But if this is so, I don't technically get a point I get a vector from the origin to the point p (I guess this probably isn't critical, but I still feel uneasy somewhat).
If I'm on the right track, I still don't know how to tie these things together. If I'm not on the right track, could someone help me find where to tap my cane?