Which is the intersection?

Question

I am looking at the last question of the following exercise:

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Which exactly is the intersection of any surface from one family of the triply orthogonal system with any surface from another family?

Answer 1

The intersection of two surfaces is a curve $\Sigma(u_0,v_0,w)$ resp. ($\Sigma(u_0,v,w_0)$, $\Sigma(u,v_0,w_0)$) which has two of three parameters fixed. These curves are coordinate lines on a surfaces patches defined by $\Sigma$.

From the second part of the exercise you know that matrices of both first and second fundamental form are diagonal and therefore lines of curvature are coordinate lines.

Answer 2

Let's call the three families the constant-$u$ family ($C_u$), the constant-$v$ family ($C_v$), and the constant-$w$ family ($C_w$).

A surface in the family $C_u$ is of the form $$ G(v,w) = \Sigma(u_0,v,w) $$ for some fixed $u=u_0$. Similarly, a surface in the family $C_v$ is of the form $$ H(u,w) = \Sigma(u,v_0,w) $$ for some fixed $v=v_0$. The intersection of the surfaces $G$ and $H$ is the curve $$ F(w) = \Sigma(u_0, v_0, w) $$ This answers the question you asked, but doesn't tell you how to do the homework problem.