Let and let S be the isotimic surface: . Find all the points on S that have tangent planes with normals .
I'm not even sure where to begin...
We may use the fact that the normal to any point on the level curve of is parallel to at that point. We know this because
for a differentiable curve on the surface , which means that is perpendicular to the curve and thus to the plane itself.
In our case, we are to find all those points of such that
for some constant .