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DIFFERENTIAL GEOMETRY COURSE

18. *RULED SURFACES*

*Definition 157 :* A ruled surface is a surface with a parametrisation __x__(u) + v __e__(u), where __e__(u) is unit vector.
We call the v-lines rules.

*Definition 158 :* A surface of tangents is a ruled surface where __e__(u) is tangent vector of __x__(u), so with a parametrisation
__x__(s) + v __x__ ^{.}(s) .

*Definition 159 :* A cylinder is a ruled surface where all rules are parallel, a cone is a ruled surface where all rules go through one point (the top).

*Explanation 160 :* We may consider a cylinder as a special case of a cone, namely a cone whose top is a point at infinity.

*Proposition 161 :* The singular points of a surface of tangents __x__(s) + v __x__ ^{.}(s) are the points on the curve __x__(s) (turning curve).

*Proof:* The tangent vectors __x__ ^{.}(s) and __x__ ^{.}(s) + v __x__ ^{..}(s) have only the same or opposite direction if
v=0 (indeed, __x__ ^{.}(s) is perpendicular to __x__ ^{..}(s)).

*Explanation 162 :* The name turning curve comes from the fact that in each point __x__(s_{o}) the intersection curve of the normal plane and the surface of tangents is
approximately a cubic parabola with turning point __x__(s_{o}).

We can see this as follows:

The intersection curve is __z__(s) = __x__(s) + u(s) __x__ ^{.}(s) =
__x__(s_{o}+h) + u(s_{o}+h) __x__ ^{.}(s_{o}+h), where u is chosen in such a way that the component of __z__(s) - __x__(s_{o})
on __x__ ^{.}(s_{o}) is equal to 0.

In approximation we have __z__(s) = __x__(s_{o}) + h __x__ ^{.}(s_{o}) + h^{2} __x__ ^{..}(s_{o})/2 +
h^{3} __x__ ^{...}(s_{o})/6 + u(s_{o}+h) (__x__ ^{.}(s_{o}) + h __x__ ^{..}(s_{o}) +
h^{2} __x__ ^{...}(s_{o})/2) = __x__(s_{o}) + __t__ (h + u(s_{o}+h) - κ^{2}(h^{3}/6 +
h^{2}u(s_{o}+h)/2)) + __n__ (κh^{2}/2 + κuh + h^{3}κ ^{.}/6 + h^{2}uκ ^{.}/2) +
__b__ (h^{3}κτ/6 + h^{2}uκτ/2).

With our choice of u, the __t__-component is 0, so u(s_{o}+h) = -h + ...

So, in first approximation, the __n__- and __b__-coordinates are β = -h^{2}κ/2 and γ = -h^{3}κτ/3.

*Proposition 163 :* With a surface of tangents, a cylinder and a cone, the tangent plane is constant along each rule.

*Proof:* With a surface of tangents, the tangent plane is spanned by __x__ ^{.}(s) and __x__ ^{.}(s) + v __x__ ^{..}(s),
so also by __x__ ^{.}(s) and __x__ ^{..}(s), independent of v. So, with a fixed s, the tangent plane along the rule __x__(s) +
v __x__ ^{.}(s) is the osculating plane in __x__(s).

With a cylinder with direction __e__, the tangent plane is spanned by __x__ '(u) and __e__, so again independent of v.

With a cone with top __x__(u) = __x__(u_{o}), the tangent plane is spanned by __e__ and v __e__ '(u), so also by __e__ and __e__ '(u),
again independent of v.

*Proposition 164 :* If the tangent plane is constant along each rule of a ruled surface, then, everywhere locally, the ruled surface is a surface of tangents, a cylinder or a cone.

*Proof:* The tangent plane is spanned by __e__(u) and __x__ '(u) + v __e__ '(u); so if it is independent of v, then
__e__, __x__ ' and __e__ ' lie in one plane.

Furthermore, __e__ ' is perpendicular to __e__, because ||__e__|| = 1.

So we find the singular points from (__e__ ').(__x__ ' + v __e__ ' ) = 0, from which it follows that
v = -((__e__ ').(__x__ ' ))/((__e__ ').(__e__ ' )).

Then the candidate turning curve is __x__(u) - __e__(u)((__e__ ').(__x__ ' ))/((__e__ ').(__e__ ' )).

Now there are three cases, namely: __x__ ' = __0__; __e__ ' = __0__; __x__ ' ,__e__ ' ≠ __0__.

*Problem 165: *Complete the proof of the proposition by considering these three cases in more detail.

*Note 166 :* Wellknown examples of ruled surfaces whose tangent planes are *not* constant along each rule are the hyperboloids x^{2}/a^{2} + y^{2}/b^{2} -
z^{2}/c^{2} = __+__1.

*EP: * Prove that the surface with parameter representation __x__(u,v) = (2u, 3u^{2}+2uv-v^{2},
4u^{3}+3u^{2}-2v^{3}-v^{2}+2uv+6u^{2}v) is a surface of tangents.

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