A Method for Creating Ruled Surfaces and Its Modifications

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 omogućuje jednostavnu kon- ruled surfaces. This method enables a simple construc- strukciju izvodnica pravˇcaste plohe prvenstveno pomoću 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- omogućuje stvaranje interaktivnog modeliranja ploha troduced mathematical description enables creation of the pomoću 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 surface MSC 2000: 65D17, 51N05, 51N20

1 Deﬁnition of a ruled surface and construction of generating lines

We will work in the Euclidean space E3 and in the vector 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 representation: 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 parallel 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 according 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 function

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 representation 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 . (16) ( ) ( ) y1 t , y2 t and (3) must be linearly dependent. From this condition we get equation: The equation (16) is the equation of a one-parametric line d f (x ) system and its envelope can be found by differentiating of x(t)y(t) F(t) − x(t) 1 dx the equation (16) according to parameter t: 1 x1=x(t) g(x2) − G(t) − y(t) = 0. (15) y(t)x + x(t)x − x(t)y(t) − x(t)y(t)=0. (17) dx 1 2 2 x2=y(t)

If the equation (15) is identity on the interval I, the surface From the equations (16) and (17) we get: ϕ is created by torsal lines only and it is a developable surface. If the equation (15) is not identity, the surface ϕ is a x2(t)y(t) skew surface on which torsal generating lines can exist. x = , 1 x(t)y(t) − x(t)y(t) The equation (15) of the surface of an elliptic movement has the form: − 2( ) ( ) = y t x t , x2 2 x(t)y (t) − x (t)y(t) a (q1 − q2)sint cost = 0, t ∈0,2π.

Then the lines for parameters t = 0, π/2, π, 3π/2 are tor- x3 = 0, t ∈ I . (18) sal lines located in the planes x1x3 and x2x3 . In the case when the cylindrical surface has the parametric If an envelope exists and it is a curve marked as m, then presentation (13) we can simply verify that the equation the equations (18) are its parametric representation. The (15) is an identity and the cylindrical surface will be a de- points for which x(t)y(t)− x(t)y(t)=0 do not have to be velopable surface. The intersection points K1 and L1 of the necessarily troublesome points. This problem will be not line l with the semicircles k and k from the example 2 1 1 2 investigated here. (Fig. 3a) are examples of points in which derivative of the functions f and g is improper. The tangent lines of the The envelope m depends only on the curve m, what is evi- curves k1 and k2 in the points K1 and L1 are parallel with dent from the equations (18) and the geometric view, too. the axis x3 . Analogously for the line l2 . If the curve m is a circle parametrized by the function (6), then according to (18) the envelope m has the following 5 Continuity between the surfaces ϕ and parametric representation: skew surfaces x = acos3 t, x = asin3 t, x = 0, t ∈0,2π, (19) Continuity between the mentioned ruled surfaces ϕ and 1 2 3 skew surfaces, which are deﬁned by three basic curves, is clearly seen on the surface of an elliptic movement. If the the curve m is an asteroid (Fig. 4a).

62 KoG•6–2002 Kamil Maleˇcek, Dagmar Szarkov´a: A Method for Creating Ruled Surfaces and its Modiﬁcations

Example 3:

Let the surface ϕ be deﬁned by curves k1 and k2 , which are parametrized by the vector functions (5) and the curve m ⊂ x1x2 is a parabola expressed by parametric representation ’ 1 x = − t2, x = t, x = 0, t ∈ R . 1 2p 2 3 The vector function 1 1 y(v)= − t2 − tv,t + v,0 , v ∈ R (20) 2p p

of the parameter v describes the system of tangent lines to the parabola m for any value of parameter t ∈ R (Fig. 4b).

The intersections of the tangent lines with the axes x1 and Fig. 4a x2 are the points K1 and L1 with the following coordinates 1 t K = t2,0,0 and L = 0, ,0 . (21) 1 2p 1 2

The coordinates of the points K and L are ’ 1 t K = t2,0,q and L = 0, ,q . 2p 1 2 2

The surface parametric representation according to (4) has the following form:

t2 x = (1 + u), 1 4p Fig. 4b t x = (1 − u), 2 4 Orthographic views of the cylindrical surface generating q + q q − q lines (example 2) in the plane x1x2 are examples for the x = 1 2 + u 1 2 , one-parametric system of lines which has not any enve- 3 2 2 lope. t ∈ R, u ∈ R . (22)

The set of points M which are projected orthogonally to 7 Modification of the method for the cre- the points K1 and L1 on the axes x1 and x2 can be parame- ation of ruled surfaces terized according to (21) by the function t2 t The idea described above allows us to define and create x(t)= , ,0 , t ∈ R . ruled surfaces by a method which we could call dual for 2p 2 defining and creating the surfaces ϕ. The surface is defined by the curves k1 and k2 which are parametrized by the The points M are therefore located on the parabola m which functions (1). Let the curve m be without singular points is the generatrix of the ruled surface ϕ constructed by the in the plane x1x2 . Tangent lines of the curve m create a method described in the first part. one-parametric system of lines. Let K and L be the inter- 1 1 The both modifications of the presented method are illus- sections of one tangent line (which is the intersecting line trated in Fig. 4b showing the construction of the surface ϕ with the axes x1 and x2 too) of the one-parametric system projected orthogonally to the plane x 1x2. A similar con- with the axis x1 and x2 . We can construct the surface gen- struction can be seen in Fig. 4a, where the curve m is a erating line p by means of points K1 and L1 with the same circle and the curve m is an asteroid. method as in the first part (see Fig. 1).

63 KoG•6–2002 Kamil Maleˇcek, Dagmar Szarkov´a: A Method for Creating Ruled Surfaces and its Modiﬁcations

Description of the surface ϕ construction is shown in and therefore the surface has only one torsal basic line cor- Fig. 5a. The segment of the surface is shown in Fig. 5b. respondent to the parameter t = 0. Now we will show some examples of the surfaces ϕ.

Example 4: Conical surface

The vector functions (1) are

( )=(, , ), ∈ , y1 x1 x1 0 q1 x1 R 1 y (x )= 0,x , x2 + q , x ∈ R . 2 2 2 2p 2 2 2

The curve k1 is a line, the curve k2 is a parabola. Let the curve m be a line parallel to the axis x2 deﬁned by the vector function Fig. 5a x(t)=(k,t,0), t ∈ R, (23)

where k is a non-zero constant from R (Fig. 6a). The surface ϕ has the following parametric representation according to (4):

k x = (1 + u), 1 2 t x = (1 − u), 2 2 2p(q + q )+t2 2p(q − q ) −t2 x = 1 2 + u 1 2 , 3 4p 4p t ∈ R, u ∈ R . (24)

Fig. 5b

The section of the surface ϕ by the plane x1x2 has the following parametric representation according to (8): − q2 2 q1 x1 = t , x2 = t, x3 = 0, t ∈ R , 2p(q2 − q1) 2(q2 − q1) the curve k3 is a parabola. The equation (15) has the form

1 t(q − q )=0 2p 2 1 Fig. 6a

64 KoG•6–2002 Kamil Maleˇcek, Dagmar Szarkov´a: A Method for Creating Ruled Surfaces and its Modiﬁcations

This surface is so called Frezier’s cylindroid and its segment is shown in Fig. 7b. The surface is a skew surface which has two torsal generating lines. The equation (15) has the form

t(t − p) r2 − (t − p)2 + r2 − (t − p)2 t(t − p) = r2 − (t − p)2 + q + r2 − (t − p)2

and this is fulﬁlled only for the points t = p ± r, in which derivative of the functions f and g is improper. Ortho- graphic views of torsal lines in the plane x1x2 are the lines l1 and l2 (Fig. 7a). It is possible to construct cylindroid generating lines analogously using the one-parametric system of lines as at a conical surface. It this case the system of lines is parallel to the line l1.

Fig. 6b

It is evident that the surface ϕ is a conical surface with a vertex in the point K, so this surface is developable. In this case the equation (15) is an identity, because in the vector function (23) is x(t)=0 for all t ∈ R. The segment of the surface is illustrated in Fig. 6b. We could construct generating lines by the means of a one- parametric system of lines in the plane x1x2 , too. In this case, the system of lines would be a pencil of lines with the centre in the point K1 (without the line m). The line p1 is one line from the pencil of lines (see Fig. 6a). This is Fig. 7a nothing new for us, it is a classical construction of conical surface generating lines.

Example 5: Frezier’s cylindroid

The vector functions (1) are 2 2 y1(x1)= x1,0, r − (x1 − p) ,

x1 ∈p − r, p + r, 0 < p − r, Fig. 7b 2 2 y2(x2)= 0,x2, r − (x2 − p) + q , At the end of this paper there are illustrated two compli- x2 ∈p − r, p + r, cated surfaces (see Figs 8 and 9). The segment of the surface demonstrated in Fig. 8 is defined by curves k 1 , k2 where q is a non-zero constant from R. and m, where the curve k1 is the Witch of Agnési, k2 is The curves k1 and k2 are semicircles as in the example 2 for a parabola and the curve m is an epicycloid. In Fig. 9 is a the cylindrical surface, but the circle k2 is translated by the segment of the surface for which the curve k 1 is a parabola, translation vector (0,0,q). The curve m is a line defined by the curve k2 is Witch of Agnési and the curve m is a circle the vector function (12), Fig. 7a. with its centre in the origin.

65 KoG•6–2002 Kamil Maleˇcek, Dagmar Szarkov´a: A Method for Creating Ruled Surfaces and its Modiﬁcations

Fig. 8 Fig. 9

References

[1] MEDEK, V., ZAMO´ Zˇ IK,´ J.: Konˇstrukt«õvna geometria pre technikov, Alfa-Bratislava, SNTL-Praha, 1978 [2] BUDINSKY´ B., KEPR. B: Zaklady« diferencialn« «õ geometrie s technickymi« aplikacemi, SNTL, Praha 1970

Kamil Maleˇcek Dagmar Szarková Department of Mathematics Department of Mathematics Faculty of Civil Engineering Faculty of Mechanical Engineering Czech Technical University Slovak University of Technology Thákurova 7 Námestie Slobody 17 166 29 Praha 6, Czech Republic 812 31 Bratislava, Slovakia tel: +420 2 24354384 tel: +421 2 57296 394 e-mail: [email protected] e-mail: [email protected]