SUPERFACES. 2.3 Ruled Surfaces

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CURVES AND SURFACES, S.L. Rueda

SUPERFACES.

2.3 Ruled Surfaces CURVES AND SURFACES, S.L. Rueda DeﬁnitionA ruled surface S, is a surface that contains at least one unipara- metric family of lines. That is, it admits a parametrization of the next kind

2 3 α : D ⊆ R −→ R α(u, v) = γ(u) + vω(u), where γ(u) and ω(u) are curves in R3. 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 = u0, we have a line

γ(u0) + vω(u0) that will be called generatrix. CURVES AND SURFACES, S.L. Rueda Let us suppose that γ0(u) =6 0 and ω(u) =6 0 for every u. Examples

ELLIPTIC CYLINDER CONE x2 z2 2 2 2 4 + 9 = 1, α(u, v) = x + z = y , 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] CURVES AND SURFACES, S.L. Rueda

x2 y2 z2 Hiperboloid of one sheet a2 + b2 − c2 = 1. This a doubly ruled surface. It admits two ruled parametrizations, 0 α(u, v) = γ(u) + v(±γ (u) + (0, 0, c)), (u, v) ∈ [0, 2π) × R, x2 y2 γ(u) = (acos(u), b sen(u), 0) is a parametrization of the ellipse a2 + b2 = 1. If a = 2, b = 3, c = 5 and v ∈ [−1, 1] we have: CURVES AND SURFACES, S.L. Rueda Plucker¨ 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: CURVES AND SURFACES, S.L. Rueda Mobius¨ Strip Given by the ruled parametrization u u u α(u, v) = (cos(u), sen(u), 0) + v(cos cos(u), cos sen(u), sen ), 2 2 2 (u, v) ∈ [0, 2π) × R.

If v ∈ [0, 2] we have:

u ∈ [0, π/4] u ∈ [0, π] u ∈ [0, 2π] u ∈ [0, 3π] u ∈ [0, 4π] CURVES AND SURFACES, S.L. Rueda 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 −f 2 [γ0(u), ω(u), ω0(u)]2 K(u, v) = 2 = − 4 ≤ 0. EG − F ||αu(u, v) ∧ αv(u, v)|| Thus, the points of a ruled surface are all hyperbolic or parabolic (in parti- cular planar points). Deﬁnition We call distribution parameter to the value of the triple product (also called mixed or box product) p(u) = [γ0(u), ω(u), ω0(u)]. CURVES AND SURFACES, S.L. Rueda

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) =6 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. CURVES AND SURFACES, S.L. Rueda 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 developable ⇔ f = 0 ⇔ p(u) = 0. Otherwise S is non-developable. Ruled developable surfaces: cylinders, cones... CURVES AND SURFACES, S.L. Rueda Classification of ruled developable surfaces Being p(u) = [γ0(u), ω(u), ω0(u)] = 0 in this case, we have several possibili- ties:

If ω0(u) = 0 then ω(u) = ω is constant and the surface is called generalized cylinder. If γ0(u) ∧ ω =6 0 (γ0(u) and ω are not parallel), then 0 αu ∧ αv γ (u) ∧ ω N(u, v) = 0 , ||αu ∧ αv|| ||γ (u) ∧ ω|| which is a vector depending on v. Thus, the direction of the tangent plane is constante along a generatrix (ﬁxing u = u0). CURVES AND SURFACES, S.L. Rueda

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. CURVES AND SURFACES, S.L. Rueda

If γ0(u) = 0 then γ(u) = γ is constant (it is a point), and the surface is called generalized cone. If ω0(u) ∧ ω =6 0 (ω0(u) and ω are not parallel), then vω0(u) ∧ ω(u) ω0(u) ∧ ω(u) N(u, v) = , ||vω0(u) ∧ ω(u)|| ||ω0(u) ∧ ω(u)|| which does not depend on v. Thus, the direction of the tangent plane is constant along a generatrix (ﬁxing u = u0).

The ruled surface parametrized by α(u, v) = (2, 0, 3) + v(u3, u, cos(u)), (u, v) ∈ [0, 4π) × [0, 1] is a generalized cone. CURVES AND SURFACES, S.L. Rueda

If ω0(u) =6 0 and γ0(u) =6 0 then the condition p(u) = 0 implies that γ0(u), ω(u) and ω0(u) are in the same plane. Hence, the vectors γ0(u) ∧ ω(u) and ω0(u) ∧ ω(u) are parallel. This means that the vector

0 0 αu ∧ αv = (γ (u) ∧ ω(u)) + (vω (u) ∧ ω(u)) is proportional to the vector ω0(u) ∧ ω(u), which does not depend on v. That is ω0(u) ∧ ω(u) N(u, v) = , ||ω0(u) ∧ ω(u)|| does not depend on v. Therefore, the direction of the tangent plane is once more constant along one generatrix (ﬁxing u = u0). 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. CURVES AND SURFACES, S.L. Rueda 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 (γ0(u) ∧ ω(u)) · (ω0(u) ∧ ω(u)) β(u) = γ(u)− ω(u). ||ω0(u) ∧ ω(u)||2

Assuming ω0(u) ∧ ω(u) =6 0, the values of v for which

0 (αu(u, v) ∧ αv(u, v)) · (ω (u) ∧ ω(u)) = 0, verify (γ0(u) ∧ ω(u)) · (ω0(u) ∧ ω(u)) v = , ||ω0(u) ∧ ω(u)|| they belong to the striction curve. CURVES AND SURFACES, S.L. Rueda Hence, the striction curve contains:

The singular points, αu(u, v) ∧ αv(u, v) = 0. 0 The points for which ω (u) ∧ ω(u) is orthogonal to the vector αu(u, v) ∧ αv(u, v).

Deﬁnition We call central points of S to the regular points that belong to the striction curve. The Gauss curvature [γ0(u), ω(u), ω0(u)]2 K(u, v) = − 4 ≤ 0. ||αu(u, v) ∧ αv(u, v)|| 2 reaches its minimum value in the values of v for which ||αu(u, v) ∧ αv(u, v)|| is minimal. CURVES AND SURFACES, S.L. Rueda Using that

0 0 αu(u, v) ∧ αv(u, v) = γ (u) ∧ ω(u) + v(ω (u) ∧ ω(u)). we obtain ∂||α (u, v) ∧ α (u, v)||2 0 = u v = 2(ω0(u) ∧ ω(u)) · (α (u, v) ∧ α (u, v)) ∂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. CURVES AND SURFACES, S.L. Rueda If the surface is tangent developable, the vectors γ0(u), ω(u) and ω0(u) are 0 coplanar. In this case, the vector ω (u) ∧ ω(u) is not orthogonal to αu(u, v) ∧ αv(u, v). Thus, all the points in the striction curve are singular points. As they are coplanar, γ0(u) = λ(u)ω(u) + µ(u)ω0(u) and it turns out that (γ0(u) ∧ ω(u)) · (ω0(u) ∧ ω(u)) µ(u) = . ||ω0(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). CURVES AND SURFACES, S.L. Rueda

If λ(u) = µ0(u) then β0(u) = 0 and the surface is generalized conic. If λ(u) =6 µ0(u). The vector ω(u) is proportional to β0(u) and S is a tangent developable surface. We can parametrize the surface using the edge of regression: v α(u, v) = β(u) + β0(u). λ(u) − µ0(u)