# Surface given by an arc-length parametrized curve

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?

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