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DIFFERENTIAL GEOMETRY COURSE

19. *DEVELOPMENT*

*Definition 167 :* We call a surface developable if it is a bent of a plane, so if it is isometric to a plane.

*Proposition 168 :* Any ruled surface whose tangent plane is constant along each rule, is developable.

*Proof:* Let __z__(s) be an orthogonal trajectory of the rules, where the parameter s is arc length.

Let __v__(s) be a unit normal in the direction of the rule, so that __y__(s,u) = __z__(s) + u __v__(s) is a parametrization of the ruled surface.

Because the tangent plane is constant along each rule, so for each s the same plane for all u, the three vectors __v__ ' , __z__ ' and __v__ all are lying in the
plane through the origin parallel to the tangent plane.

Since ||__v__|| = 1, __v__ ' is perpendicular to __v__, and so is the direction vector __z__ ' of the orthogonal trajectory.

So __v__ ' (s) = λ(s) __z__ ' (s). We can now check by simple calculation that a_{1 1} = (1+uλ)^{2}, a_{1 2} = 0, and a_{2 2} = 1.

Now draw a planar curve __x__(s) with κ(s) = -λ(s), and let __n__(s) be the principal normal vector of __x__(s). According to Frenet we have
__n__ ' = -κ __x__ ' = λ __x__ ' .

Check by calculation that __y__(s,u) → __x__(s) + u __n__(s) is an isometry from the ruled surface to the plane wherein __x__(s) is lying.
To do this, you only have to calculate the first fundamental form of this plane with this parametrization.

*Explanation 169 :* We call the curve __x__(s) trace curve of __z__(s). We get it as a trace of __z__(s) if we draw __z__(s) with wet ink
on the ruled surface and roll the surface over the plane. Since the tangent plane is constant along the rule, this rolling also gives the isometry (development).

*Remark 170 :* Proposition 168 is convertible. Also see 163 and 164.

*Problem 171 :* Check the following assertions in the context of the proof of 168.

i) For the second fundamental form we find h_{1 1} = (1+uλ) __z__ "**.** __N__ , h_{1 2} = __v__ ' **.** __N__ = 0,
h_{2 2} = __0__ **.** __N__ = 0.

ii) According to 152, __z__(s) and the rules are curvature lines.

iii) According to 141, the rules are asymptotic lines. The total curvature is 0 everywhere (see 151). Each point is parabolic (see 134 2) ).

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