Kamil Maleček
Dagmar Szarková

A Method for Creating Ruled Surfaces
and its Modifications

1. Definition of 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:

Let the real functions f and g in (1) be continuous and differentiable 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 k₁x₁x₃ and k₂x₂x₃ . We assume that these curves k₁ and  k₂ are not intersected (Fig.1). 

Fig. 1.

 Let the curve m be defined by the vector function (2) in the plane x₁x₂

(2)          , 

where for any t I is x(t) I₁, y(t) I₂ and is a non-zero vector.

Now, we will construct the generating line p in this way (Fig. 1):

a)   We choose a point M on the curve m and mark its orthogonal projections to the axes x₁ and x₂ as K₁ and L₁.

b)   Points K and L are points located on curves k₁ and k₂ respectively, while K₁ and L₁ are their orthogonal projections to the plane x₁x₂ .

c)   Line p = KL.

Line p is a generating line of the ruled surface  and with this method we would construct next generating lines of the surface .

2. Parametric representation of the ruled surface .

We obtain the coordinates of points K and L with the substitution (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

of the line segment KL and by the direction vector

(3)          

Then the ruled surface has the following parametric representation

(4)          

Example 1: The Surface of an elliptic movement

The vector functions (1) are

(5)          

where q₁ and q₂ are non-zero constants from R , q₁ q₂.

Curves k₁ and k₂ are lines, k₁ || x₁ and k₂ || x₂. Let the curve m become a circle defined by the vector function

(6)          . 

The surface defined in this way has the following parametric representation according to equation (4)

(7)          

In Fig. 2a are shown curves k₁, k₂ and m are shown. In Fig. 2b is shown a surface segment, which is called the surface of an elliptic movement is shown.

3. The Section of the surface  by the plane x₁x₂

 If the curve k₃ is the section of the surface  by the plane x₁x₂ (x₃ = 0), then we get its parametric representation from (4)

(8)           . 

 If for any t   I is

(9)          G(t) = F(t), 

then the corresponding generating line p || x₁x₂ and its intersection point with the plane x₁x₂ is a point at infinity.

The section of the surface of elliptic movement by the plane x₁x₂ (from the example 1 according to (8)) has the following parametric representation

(10)           . 

This section is the ellipse k₃ with the centre in the origin of the coordinate system, values of the semiaxes are

and .

In the case, when the equation (9) expresses identity for the interval I , then all generating lines of the surface  are parallel to the plane x₁x₂ and the section of the surface by the plane x₁x₂ cannot be described by equations (8).

 If we choose the vector functions (1) as

(11)          

and the curve m is a line parametrized by the vector function

(12)           x(t) = (t, t, 0) ,   t  R ,

then F(t) = G(t) = f(t).

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

(13)          

and it is a cylindrical surface. Its generating lines are parallel to the plane x₁x₂. The curves k₁ and k₂ are congruent. Revolving the curve k₁ about axis x₃ by the angle 90° we would get the curve k₂ .

The section k₃ of the cylindrical surface (13) by the plane 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 .

Example 2: Circular cylindrical surface

Curves k₁ and k₂ are semicircles with centres S₁x₁ , S₂x₂ , with the same radius r and |OS₁| = |OS₁| = p. The semicircles are parametrized by the vector functions

The surface is a half of the circular cylindrical surface, which has the following parametric representation according to (13)

(14)           

Fig. 3a illustrates curves k₁, k₂ , m, and the section of the surface by the plane x₁x₂ which is created by lines l₁and l₂ . In Fig. 3b the segment of the circular cylindrical surface is shown.

4. Developable and skew surfaces

The generating line of the surface is defined by choice of the parameter t  I. When we substitute the parametric representation (2) of the curve m to the vector functions (1) and differentiate the vector functions (1) according to argument t, then we get:

These vector functions for chosen t  _I are defining the direction vectors of tangent lines of curves k₁ and k₂. To make the generating line of the surface  torsal, vectors y₁'(t) , y₂'(t) and (3) must be linearly dependent. From this condition we get equation:

(15)          .   

If the equation (15) is identity on the interval I, the surface  is created by torsal lines only, and it is a developable surface. If the equation (15) is not identity, the surface  is a skew surface, on which torsal generating lines can exist.

The equation (15) of the surface of an elliptic movement has the form:

.

Then the lines for parameters t = 0,  /2,  ,  3/2  are torsal lines located in the planes x₁x₃ and x₂x₃.

In the case, when the cylindrical surface has the parametric presentation (13) we can simply verify that the equation (15) is an identity and the cylindrical surface will be a developable surface. The intersection points K₁ and L₁ of the line l₁ with the semicircles k₁ and k₂ from the example 2 (Fig. 3a) are examples of points in which derivative of the functions f and g is improper. The tangent lines of the curve k₁ and k₂ in the points K₁ and L₁ are parallel with the axis x₃. Analogously for the line l₂ .

5. Continuity between the surfaces φ and skew surfaces

Continuity between the mentioned ruled surfaces and skew surfaces, which are defined by three basic curves, is clearly seen on the surface of an elliptic movement. If the section of the surface by plane x₁x₂ is the curve k₃, then it is possible to define the surface by basic curves k₁, k₂ and k₃. The generating lines of the surface are lines intersecting the basic curves.

6. Envelope of orthographic views of the ruled surface generating lines in the plane x₁x₂

Generating lines of the ruled surface are orthogonally projected to the plane x₁x₂, and parametric representation of these orthographic views can be given by the first two equations in (4) without the parameter u:

(16)          . 

The equation (16) is the equation of a one-parametric line system and its envelope can be found by differentiating of the equation (16) according to parameter t:

(17)           

and from the equations (16) and (17) we get:

(18)           . 

If an envelope exists and it is a curve marked as m’, then the equations (18) are its parametric representation. The points, for which x(t)y'(t)-x'(t)y(t)=0, do not have to be necessarily troublesome points. This problem will be not investigated here.

The envelope m’ depends only on the curve m, what is evident from the equations (18) and the geometric view, too.

If the curve m is a circle parametrized by function (6), then according to (18), the envelope m’ has the parametric representation:

(19)          

the curve m’  is an asteroid (Fig. 4a).

Fig. 4a.	Fig. 4b.

Orthographic views of the cylindrical surface generating lines (example 2) in the plane x₁x₂ are examples for the one-parametric system of lines, which has not any envelope.

7. Modification of the method for creation of ruled surfaces

The idea described above allows us to define and create ruled surfaces by a method, which we could call dual for defining and creating the surfaces . The surface is defined by the curves k₁ and k₂ , which are parametrized by functions (1). Let the curve m’ be without singular points in the plane x₁x₂ . Tangent lines of the curve m’ create a one-parametric system of lines. Let K₁ and L₁ be intersections of one tangent line (which is intersecting line with the axes x₁ and x₂ too) of the one-parametric system with the axis x₁ and x₂. We can construct the surface generating line p by means of points K₁ and L₁ with the same method as in the introduction (see Fig. 1).

Example 3: Let the surface be defined by curves k₁ a k₂ , which are parametrized by vector functions (5) and the curve m’  x₁x₂  that is a parabola expressed by parametric representation

.

 The vector function

(20)           

of parameter v describes the system of tangent lines to the parabola m’  for any value of parameter t  R (Fig. 4b).

The intersections of tangent lines with axes x₁ and x₂ are the points K₁ and L₁   with the following coordinates

(21)          . 

The coordinates of the points K  and L are

.

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

(22)          

The set of points M, which are projected orthogonally to the points K₁ and L₁ on the axes x₁ and x₂ can be  parameterized according to (21) by the function

which is the generatrix of the ruled surface constructed by the method described in the first part.

The both modifications of the presented method are illustrated in Fig. 4b showing the construction of the surface projected orthogonally to the plane x₁x₂, and in Fig 4a, where the orthogonal projection of the construction of the surface determined by circle m and asteroid m´ can be seen. Description of the surface construction is shown in Fig.5a. A segment of the surface is shown in Fig. 5b.

The section of the surface by the plane x₁x₂ has the following parametric representation according to (8)

,

the curve k₃ is parabola.

 The equation (15) has the form

,

and therefore the surface has only one torsal basic line correspondent to the parameter t = 0.

Now we will show some examples of the surfaces .

Example 4: Conical surface

The vector functions (1) are

The curve k₁ is a line, the curve k₂ is a parabola. Let the curve m be a line parallel to the axis x₂ , which is defined by the vector function

(23)          x(t) = (k, t, 0),   t  R,  k is a nonzero constant from R (Fig. 6a).

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

It is evident that the surface 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 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 nothing new for us, it is a classical construction of conical surface generating lines.

Example 5: Frezier’s cylindroid

The vector functions (1) are

 The curves k₁ and k₂ are semicircles as in the example 2 for the cylindrical surface, but the circle k₂ is translated by the translation vector (0, 0, q). The curve m is a line defined by the vector function (12), Fig. 7a.

 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

and this is fulfilled only for the points  , in which derivative of the functions f and g is improper. Orthographic 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.

At the end of this paper there are illustrated two complicated surfaces (see Figs 8 and 9).

The segment of the surface demonstrated in Fig. 8 is defined by curves k₁, k₂ and m, where the curve k₁ is Witch of Agnési, k₂ is a parabola and the curve m is an epicycloid. In Fig. 9 is a segment of the surface for which the curve k₁ is Witch of Agnési, the curve k₂ is a parabola and the curve m is a circle with its centre in the origin.

Fig. 8.	Fig. 9.