CURVES AND SURFACES, S.L. Rueda SUPERFACES. 2.3 Ruled Surfaces

Transcription

1 SUPERFACES. 2.3 Ruled Surfaces

2 Definition A ruled surface S, is a surface that contains at least one uniparametric family of lines. That is, it admits a parametrization of the next kind α : D R 2 R 3 α(u, v) = γ(u) + vω(u), where γ(u) and ω(u) are curves in R 3. A parametrization which is linear in one of the parameters (in this case v) it is called a ruled parametrization. The curve γ(u) is called directrix or base curve. The surface contains an infinite family of lines moving along the directrix. For each value of the parameter u = u 0, we have a line that will be called generatrix. γ(u 0 ) + vω(u 0 )

3 Let us suppose that γ (u) 0 and ω(u) 0 for every u. Examples ELLIPTIC CYLINDER CONE x z2 9 = 1, α(u, v) = x2 + z 2 = y 2, α(u, v) = (2cos(u), 0, 3 sin(u)) + v(0, 1, 0) (cos(u), 1, sin(u)) + v( cos(u), 1, sin(u)) (u, v) [0, 2π] [0, 1] (u, v) [0, 2π] [0, 2]

4 Hiperboloid of one sheet x2 + y2 a 2 b 2 z2 c 2 admits two ruled parametrizations, = 1. This a doubly ruled surface. It α(u, v) = γ(u) + v(±γ (u) + (0, 0, c)), (u, v) [0, 2π) R, γ(u) = (acos(u), b sen(u), 0) is a parametrization of the ellipse x2 a 2 + y2 b 2 = 1. If a = 2, b = 3, c = 5 and v [ 1, 1] we have:

5 Plücker Conoid Given by the ruled parametrization α(u, v) = (0, 0, sen(2u)) + v(cos(u), sen(u), 0), (u, v) [0, 2π) R. If v [0, 2] we have:

6 Möbius Strip Given by the ruled parametrization ( u ( u α(u, v) = (cos(u), sen(u), 0) + v(cos cos(u), cos sen(u), sen 2) 2) (u, v) [0, 2π) R. If v [0, 2] we have: ( u 2) ), u [0, π/4] u [0, π] u [0, 2π] u [0, 3π] u [0, 4π]

7 2.3.1 Curvature of a ruled surface The Gauss or total curvature of a ruled surface is always less than or equal to zero. Let us check that K(u, v) = f 2 EG F 2 = [γ (u), ω(u), ω (u)] 2 α u (u, v) α v (u, v) 4 0. Thus, the points of a ruled surface are all hyperbolic or parabolic (in particular planar points). Definition We call distribution parameter to the value of the triple product (also called mixed or box product) p(u) = [γ (u), ω(u), ω (u)].

8 If p(u) = 0 then K(u, v) = 0 and the point P = α(u, v) is a parabolic point ( or planar point), for every v. At such point, one of the principal curvatures is zero, therefore the asymptotic directions are directions of maximal or minimal curvature. If p(u) 0 then K(u, v) < 0 and the point P = α(u, v) is a hyperbolic point, for every v. At such point one of the principal curvatures is positive and the other is negative.

9 2.3.2 Classification if ruled surfaces Definition A surface S (not necessarily ruled) is a planar surface if its Gauss curvature is zero at every point. Such surfaces are called developable surfaces and they can be constructed bending a sheet of paper. Therefore, a ruled surface S is Otherwise S is non-developable. developable f = 0 p(u) = 0. Ruled developable surfaces: cylinders, cones...

10 Classification of ruled developable surfaces Being p(u) = [γ (u), ω(u), ω (u)] = 0 in this case, we have several possibilities: If ω (u) = 0 then ω(u) = ω is constant and the surface is called generalized cylinder. If γ (u) ω 0 (γ (u) and ω are not parallel), then N(u, v) α u α v α u α v = γ (u) ω γ (u) ω, which is a vector depending on v. Thus, the direction of the tangent plane is constante along a generatrix (fixing u = u 0 ).

11 The ruled surface S parametrized by α(u, v) = (2cos(u), 0, 3 sen(u)) + v(2, 1, 5) is a generalized cylinder. The tangent plane to S at points of one generatrix α(p i, v) is 3+x 3y = 0.

12 If γ (u) = 0 then γ(u) = γ is constant (it is a point), and the surface is called generalized cone. If ω (u) ω 0 (ω (u) and ω are not parallel), then N(u, v) vω (u) ω(u) vω (u) ω(u) = ω (u) ω(u) ω (u) ω(u), which does not depend on v. Thus, the direction of the tangent plane is constant along a generatrix (fixing u = u 0 ). The ruled surface parametrized by α(u, v) = (2, 0, 3) + v(u 3, u, cos(u)), (u, v) [0, 4π) [0, 1] is a generalized cone.

13 If ω (u) 0 and γ (u) 0 then the condition p(u) = 0 implies that γ (u), ω(u) and ω (u) are in the same plane. Hence, the vectors γ (u) ω(u) and ω (u) ω(u) are parallel. This means that the vector α u α v = (γ (u) ω(u)) + (vω (u) ω(u)) is proportional to the vector ω (u) ω(u), which does not depend on v. That is N(u, v) = ω (u) ω(u) ω (u) ω(u), does not depend on v. Therefore, the direction of the tangent plane is once more constant along one generatrix (fixing u = u 0 ). We call this surface tangent developable. Observation In all the cases, if p(u) = 0 the tangent plane is the same for all points of a given generatrix.

14 2.3.3 Striction curve. Let S be the ruled surface parametrized by α(u, v) = γ(u) + vω(u), (u, v) D. Definition We call striction curve of S to the curve ( (γ (u) ω(u)) (ω ) (u) ω(u)) β(u) = γ(u) ω(u). ω (u) ω(u) 2 Assuming ω (u) ω(u) 0, the values of v for which (α u (u, v) α v (u, v)) (ω (u) ω(u)) = 0, verify ( (γ (u) ω(u)) (ω ) (u) ω(u)) v =, ω (u) ω(u) they belong to the striction curve.

15 Hence, the striction curve contains: The singular points, α u (u, v) α v (u, v) = 0. The points for which ω (u) ω(u) is orthogonal to the vector α u (u, v) α v (u, v). Definition We call central points of S to the regular points that belong to the striction curve. The Gauss curvature K(u, v) = [γ (u), ω(u), ω (u)] 2 α u (u, v) α v (u, v) 4 0. reaches its minimum value in the values of v for which α u (u, v) α v (u, v) 2 is minimal.

16 Using that we obtain α u (u, v) α v (u, v) = γ (u) ω(u) + v(ω (u) ω(u)). 0 = α u(u, v) α v (u, v) 2 v = 2(ω (u) ω(u)) (α u (u, v) α v (u, v)) and therefore the minimum is obtained at the points of the striction line. Observation At the central points, the absolute value of the Gauss curvature K(u, v) is maximal.

17 If the surface is tangent developable, the vectors γ (u), ω(u) and ω (u) are coplanar. In this case, the vector ω (u) ω(u) is not orthogonal to α u (u, v) α v (u, v). Thus, all the points in the striction curve are singular points. As they are coplanar, γ (u) = λ(u)ω(u) + µ(u)ω (u) and it turns out that ( (γ (u) ω(u)) (ω ) (u) ω(u)) µ(u) =. ω (u) ω(u) 2 Definition In this case, we call edge of regression to the striction curve. β(u) = γ(u) µ(u)ω(u).

18 If λ(u) = µ (u) then β (u) = 0 and the surface is generalized conic. If λ(u) µ (u). The vector ω(u) is proportional to β (u) and S is a tangent developable surface. We can parametrize the surface using the edge of regression: v α(u, v) = β(u) + λ(u) µ (u) β (u).