When is a normal vector to a surface a constant vector?

Question

I have a question about when this property occurs. In my example sheets, there is a question stating: a plane Pi is tangent to S(surface) along a unit speed curve and thus the normal to the surface at that curve is constant. I didn't quite understand this statement and hence I wanted to know, in general, the is a normal vector to a surface constant so that I can apply it to this particular question.

Answer 1

The normal vector is constant because the normal vector to the plane is always in the same direction, and since the plane is tangent to the surface at those points the surface has the same normal vector. Thus the normal vector is always in the same direction along the curve of tangency.

Answer 2

Roughly speaking, the normal vector $N$ to a surface $S$ is a vector which is perpendicular to the surface at a given point (it is actually perpendicular to the tangent plane at the point).

Since you have a condition which says that a plane $\Pi$ is tangent to the surface $S$ along a curve, this means that the tangent plane to the surface is the same along the curve (it is obviously $\Pi$ all the way along the curve). Therefore, the perpendicular vector to this tangent plane (the normal to the surface) points the same direction along this curve, and thus is constant.

Clearly this condition is not general. You can not always find curves along a given surface such that the tangent plane to the surface is the same along the curve.