1
$\begingroup$

Given $\alpha(s)$ a smooth arc-length parametrized curve, how can you write the equation for the surface f(s,t) = \alpha(s) + t \alpha'(s) component-wise?

That is, I want to write $f = (f_x, f_y, f_z)$.

I can write $\alpha(s) = (x(s), y(s), z(s))$, but that doesn't give much. For instance, I don't think f_x(s,t) = x(s) + t x'(s). Is that right?

1 Answers 1

2

f(s,t)=(x(s)+tx'(s),y(s)+ty'(s),z(s)+tz'(s)) just as you're saying.