Rulings of a surface

• Sep 2nd 2010, 08:54 PM
ulysses123
Rulings of a surface
The curve γ(t)=[(u(t),v(t)]=[e^{λt},t]
lies on the cone parametrized by:
σ(u,v)=[ucos z,usin v,u]

show that the curve intersects each of the rulings of the cone at the same
angle.

i get that a surface that is the union of straight lines is a ruled surface and that the lines are the rulings, but i cannot figure out how to find the rulings of any surface.
• Sep 2nd 2010, 11:53 PM
Opalg
Quote:

Originally Posted by ulysses123
The curve γ(t)=[(u(t),v(t)]=[e^{λt},t]
lies on the cone parametrized by:
σ(u,v)=[ucos z,usin v,u]

show that the curve intersects each of the rulings of the cone at the same
angle.

i get that a surface that is the union of straight lines is a ruled surface and that the lines are the rulings, but i cannot figure out how to find the rulings of any surface.

I assume that the parametrisation of the cone should be σ(u,v)=[ucos v,usin v,u] (with v in place of z). The lines on the cone are given by fixing v and letting u vary. The set of points $\displaystyle \{(u\cos v, u\sin v, u)\}$ (with v fixed and u varying) is just the set of all scalar multiples of the point $\displaystyle (\cos v, \sin v, 1)$, and therefore represents a straight line.
• Sep 3rd 2010, 01:36 AM
ulysses123
but how can i find the angle between the two tangent curves and thus the angle between the rulings and the curve since the curve has two components and the rulings have three components?
• Sep 3rd 2010, 02:07 AM
Opalg
Quote:

Originally Posted by ulysses123
but how can i find the angle between the two tangent curves and thus the angle between the rulings and the curve since the curve has two components and the rulings have three components?

Both curves lie on the surface of the cone. The straight line is given by $\displaystyle (u\cos v, u\sin v, u)$, with u as the parameter. The other curve is given by $\displaystyle (e^{\lambda t}\cos t, e^{\lambda t}\sin t, e^{\lambda t})$, where the functions u(t) and v(t) are used in place of u and v, and t is the parameter.

To find the angle between the curves, differentiate with respect to the parameter to find the tangent to each curve. The tangent to the straight line is $\displaystyle x = (\cos t, \sin t, 1)$, and the tangent to the other curve is $\displaystyle y = (e^{\lambda t}(\lambda\cos t - \sin t), e^{\lambda t}(\lambda\sin t + \cos t), \lambda e^{\lambda t})$. Use the inner product formula $\displaystyle x\cdot y = \|x\|\|y\|\cos\theta$ to find the angle $\displaystyle \theta$ between those two vectors (which should be independent of t if the curve intersects each of the rulings of the cone at the same angle).
• Sep 3rd 2010, 02:25 AM
ulysses123
ok, this might sound stupid but i'd like to understand this properly, the curve specified as gamma lies on the surface, why is it necassary to subsitute it into the actual equation for the cone, is this because on its own it specifies a parametrised plane curve but this curve is not 'on" the surface of the cone?
• Sep 3rd 2010, 12:10 PM
Opalg
Quote:

Originally Posted by ulysses123
ok, this might sound stupid but i'd like to understand this properly, the curve specified as gamma lies on the surface, why is it necassary to subsitute it into the actual equation for the cone, is this because on its own it specifies a parametrised plane curve but this curve is not 'on" the surface of the cone?

Specifying the parameters u and v tells you where a point lies on the cone, but you need three coordinates to specify the position of a point in 3D-space.

The curve gamma is not a plane curve, it is a spiral which coils round the surface of the cone.

I think that the wording of the question, "γ(t)=[(u(t),v(t)]=[e^{λt},t]", is not very satisfactory. In those equations, u(t) and v(t) are not supposed to represent coordinates of the point γ(t), they are the values to take for u and v in the parametric representation of the cone.
• Sep 9th 2010, 05:32 PM
ulysses123
when you put it like that it makes sense