KoG•6–2002 Kamil Maleˇcek, Dagmar Szarkov´a: A Method for Creating Ruled Surfaces and its Modifications Professional paper KAMIL MALECEKˇ Accepted 20.05.2002 DAGMAR SZARKOVA´ A Method for Creating Ruled Surfaces and its Modifications Metoda stvaranja pravˇcastih ploha i njihovih A Method for Creating Ruled Surfaces and its modifikacija Modifications SAZETAKˇ ABSTRACT Uˇclanku je dana netradicionalna metoda za definiranje The paper presents a non-traditional method for defining pravˇcastih ploha.Ta metoda omogu´cuje jednostavnu kon- ruled surfaces. This method enables a simple construc- strukciju izvodnica pravˇcaste plohe prvenstveno pomo´cu tion of the ruled surfaces generating lines, not only with raˇcunala, a ne samo u klasiˇcnom smislu. Opisana the classical means, but first of all with a computer. The je metoda za definiranje i konstrukciju poznatih ploha, method for defining and constructing known surfaces and ali i za modeliranje novih. Uvedeni matematiˇcki opis also modelling of new surfaces is described here. The in- omogu´cuje stvaranje interaktivnog modeliranja ploha troduced mathematical description enables creation of the pomo´cu raˇcunala i vrlo brzi dizajn plohe te projekcije interactive modelling of surfaces by using a computer and njezinih odabranih dijelova. Slike prikazuju raˇcunalne very quick surface design and projection of its arbitrary grafiˇcke izlaze. segments. The pictures are presenting the graphical out- put from a computer. Kljuˇcne rijeˇci: pravˇcaste plohe, razvojne plohe, vitopere plohe Key words: developable surface, ruled surface, skew sur- face MSC 2000: 65D17, 51N05, 51N20 1 Definition of a ruled surface and construc- tion of generating lines We will work in the Euclidean space E3 and in the vec- tor space V (E3) with the Cartesian coordinates system O,x1,x2,x3. Let these vector functions be set: y1(x1)=(x1,0, f (x1)), x1 ∈ I1, y2(x2)=(0,x2,g(x2)), x2 ∈ I2 . (1) Let the real functions f and g in (1) be continuous and dif- Fig. 1 ferentiable on the intervals I1 and I2. These intervals can contain many points for which derivative of the functions f and g are improper. Vector functions (1) describe curves x(t)=(x(t),y(t),0), t ∈ I, (2) k1 ⊂ x1x3 and k2 ⊂ x2x3 . We assume that these curves k1 and k2 are not intersected (Fig. 1). dx(t) where for any t ∈ I is x(t) ∈ I , y(t) ∈ I and = x(t) Let the curve m be defined by the vector function (2) in the 1 2 dt plane x1x2 is a non-zero vector. 59 KoG•6–2002 Kamil Maleˇcek, Dagmar Szarkov´a: A Method for Creating Ruled Surfaces and its Modifications Now, we will construct the generating line p in this way Curves k1 and k2 are lines, k1x1 and k2 x2 . Let the curve (Fig. 1): m become a circle defined by the vector function a) We choose a point M on the curve m and mark its or- x(t)=(acost,asint,0), t ∈0,2π . (6) thogonal projections to the axes x1 and x2 as K1 and L1 . The surface ϕ defined in this way has the following para- b) Points K and L are points located on curves k1 and metric representation according to equation (4): k2 respectively, while K1 and L1 are their orthogonal acost projections to the plane x1x2 . x = (1 + u), 1 2 c) The line p joins the points K and L. asint x = (1 − u), 2 2 The line p is a generating line of the ruled surface ϕ and q + q q − q x = 1 2 + u 1 2 , with this method we would construct next generating lines 3 2 2 of the surface ϕ . t ∈0,2π, u ∈ R . (7) 2 Parametric representation of the ruled In Fig. 2a the curves k1, k2 and m are shown. surface ϕ In Fig. 2b a surface segment, which is called the surface of an elliptic movement is shown. We obtain the coordinates of the points K and L with the substitution of (2) to (1). Then K =[x(t),0,F(t)] and L =[0,y(t),G(t)], where F(t)= f (x(t)) and G(t)=g(y(t)). Let the generating line p be defined for example by the centre x(t) y(t) F(t)+G(t) S = , , 2 2 2 of the line segment KL and by the direction vector 1 p(t)= (x(t),−y(t),F(t) − G(t)) . (3) 2 Fig. 2a Then the ruled surface ϕ has the following parametric rep- resentation: x(t) x = (1 + u), 1 2 y(t) x = (1 − u), 2 2 F(t)+G(t) F(t) − G(t) x = + u , 3 2 2 t ∈ I, u ∈ R . (4) Example 1: The surface of an elliptic movement The vector functions (1) are y1(x1)=(x1,0,q1), x1 ∈ R, y2(x2)=(0,x2,q2), x2 ∈ R , (5) where q1 and q2 are non-zero constants from R , q1 = q2 . Fig. 2b 60 KoG•6–2002 Kamil Maleˇcek, Dagmar Szarkov´a: A Method for Creating Ruled Surfaces and its Modifications 3 The section of the surface ϕ by the plane and it is a cylindrical surface. Its generating lines are par- allel to the plane x1x2 . The curves k1 and k2 are congruent. x1x2 o Revolving the curve k1 about the axis x3 by the angle 90 we would get the curve k2 . If the curve k3 is the section of the surface ϕ by the plane x1x2, (x3 = 0), then we get its parametric representation The section k3 of the cylindrical surface (13) by the plane from (4) x1x2 is composed from the surface generating lines. Their number is equal to the number of common points of the curve k1 and the axis x1 . G(t)x(t) −F(t)y(t) Example 2: Circular cylindrical surface x = , x = , x = 0, t ∈ I . (8) 1 G(t) − F(t) 2 G(t) − F(t) 3 Curves k1 and k2 are semicircles with centres S1 ∈ x1 , ∈ | | = | | = If for any t ∈ I is S2 x2, with the same radius r and OS1 OS2 p. The semicircles are parametrized by the vector functions G(t)=F(t), (9) 2 2 y1(x1)= x1,0, r − (x1 − p) , then the corresponding generating line px 1x2 and its in- x ∈p − r, p + r, tersection point with the plane x1x2 is a point at infinity. 1 The section of the surface ϕ of elliptic movement by the 2 2 plane x1x2 (from the example 1 according to (8)) has the y2(x2)= 0,x2, r − (x1 − p) , following parametric representation x2 ∈p − r, p + r, 0 < p − r . aq −aq x = 2 cost, x = 1 sint, ϕ 1 q − q 2 q − q The surface is a half of the circular cylindrical surface 2 1 2 1 which has the following parametric representation accord- ing to (13) x3 = 0, t ∈0,2π. (10) t x1 = (1 + u), This section is the ellipse k3 with the centre in the origin of 2 the coordinate system, values of the semiaxes are t x = (1 − u), 2 2 aq2 aq1 . 2 2 and x3 = r − (t − p) , q2 − q1 q2 − q1 t ∈p − r, p + r, u ∈ R . (14) In the case when the equation (9) expresses identity for the interval I all generating lines of the surface ϕ are parallel Fig. 3a illustrates curves k1 , k2 , m, and the section of the surface by the plane x1x2 which is created by lines l1 to the plane x1x2 and the section of the surface by the plane and l2 . x1x2 cannot be described by equations (8). In Fig. 3b the segment of the circular cylindrical surface is If we choose the vector functions (1) as shown. y1(x1)=(x1,0, f (x1)), x1 ∈ I1, y2(x2)=(0,x2, f (x2)), x2 ∈ I1, I1 = I2 (11) and the curve m is a line parametrized by the vector func- tion x(t)=(t,t,0), t ∈ R, (12) then F(t)=G(t)= f (t). Fig. 3a The ruled surface ϕ has the following parametric represen- tation according to (4): t x = (1 + u), 1 2 t x = (1 − u), 2 2 x3 = f (t), t ∈ I1, u ∈ R (13) Fig. 3b 61 KoG•6–2002 Kamil Maleˇcek, Dagmar Szarkov´a: A Method for Creating Ruled Surfaces and its Modifications 4 Developable and skew surfaces ϕ section of the surface ϕ by the plane x1x2 is the curve k3 , then it is possible to define the surface ϕ by basic curves ϕ The generating line of the surface is defined by choice k1 , k2 and k3 . The generating lines of the surface ϕ are ∈ of the parameter t I. When we substitute the parametric lines intersecting the basic curves. representation (2) of the curve m to the vector functions (1) and differentiate the vector functions (1) according to argument t, then we get: 6 Envelope of orthographic views of the ruled surface generating lines in the plane d f (x1) y (t)=x (t) 1,0, and x1x2 1 dx 1 x1=x(t) ( ) Generating lines of the ruled surface are orthogonally pro- ( )= ( ) , , dg x2 . y2 t y t 0 1 jected to the plane x1x2 and parametric representation of dx2 = ( ) x2 y t these orthographic views can be given by the first two These vector functions for chosen t ∈ I are defining the di- equations in (4) without the parameter u: rection vectors of the tangent lines of curves k 1 and k2 .To ϕ make the generating line of the surface torsal, vectors y(t)x1 + x(t)x2 − x(t)y(t)=0, t ∈ I .