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(so) the intersection curve of the normal plane and the surface of tangents is
approximately a cubic parabola with turning point x(so).
We can see this as follows:
The intersection curve is z(s) = x(s) + u(s) x .(s) = x(so+h) + u(so+h) x .(so+h), where u is chosen in such a way that the component of z(s) - x(so) on x .(so) is equal to 0.
In approximation we have z(s) = x(so) + h x .(so) + h2 x ..(so)/2 + h3 x ...(so)/6 + u(so+h) (x .(so) + h x ..(so) + h2 x ...(so)/2) = x(so) + t (h + u(so+h) - κ2(h3/6 + h2u(so+h)/2)) + n (κh2/2 + κuh + h3κ ./6 + h2uκ ./2) + b (h3κτ/6 + h2uκτ/2).
With our choice of u, the t-component is 0, so u(so+h) = -h + ...
So, in first approximation, the n- and b-coordinates are β = -h2κ/2 and γ = -h3κτ/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(uo), 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 x2/a2 + y2/b2 - z2/c2 = +1.
EP: Prove that the surface with parameter representation x(u,v) = (2u, 3u2+2uv-v2, 4u3+3u2-2v3-v2+2uv+6u2v) is a surface of tangents.