A New Approach to the Computer Aided Calculation of Points on 

the Envelope Helical Surface Characteristics

An envelope surface F is the envelope of a 1-parametric system of surfaces created by a continuous movement of a basic surface j .

Characteristics is a curve segment along which the envelope surface F touches the basic surface j . 

The same envelope surface F can be created by the continuous movement of characteristics. (Detailed description of envelope surfaces can by found in Velichová [3].)

Let us deal with an envelope helical surface F created by a helical movement (the movement is a class of geometric transformation concatenated from a revolution movement about the axis o and a translation movement in the direction of the vector collinear to the axis o of revolution) of a basic surface j .  An envelope rotational surface is a special type of the envelope helical surface with the helical movement pitch | z_v | (z_v is the translation vector corresponding to the angle of revolution equal to 2p) equal to zero.

In the paper (Szarková [2]), the helical and rotational surface F  created from the conical or cylindrical surface j  were discussed. In the present paper, the basic surface j will be a rotational surface created by revolving a plane curve segment located in the xz-plane about the coordinate axis x .  We will work in the creative space with the homogeneous coordinates (in correspondence with Qiulin [1]).

Any rotational surface j  with the axis ¹o  in general position can be transformed in such way that the axis of revolution will be in one of the coordinate axis and vice versa.

Let us create the basic surface in the basic position as the surface j´  with the axis ¹o´ generated from the basic curve c  located in the xz-plane, and defined by the vector function

applying a class of revolutions about the coordinate axis x defined by the matrix  T_R(v),

(2)

Let j´ is positioned in the space by the transformation T (3),

where T_oz  is the revolution about the coordinate axis z by the angle a , 

T_oy  is the revolution about the coordinate axis y by the angle b ,

T_p  is the translation with the direction vector  v = (x_p, y_p, z_p, 0) .

The basic surface is now moved from the basic position j´ and located to the general position j in the space determined by the chosen constant values a , b , v .

To simplify the results, the whole scene will be transformed by the transformation  T´ (in the transposed form) (4) so that we could work with the rotational surface j ´, with the axis ¹o´ congruent to the coordinate axis x:

T´ and  T  are inverse transformations.

The axis o of the helical movement located into the coordinate axis z is the axis of the envelope surface F . Let the axis o by defined by the point O(0,0,0,1) and the direction vector s(0,0, z₀,0) , where z₀  is the reduced pitch  z₀ = | z_v |/2p . 

 Fig. 2. 

The transformed axis o´ will be defined by the point O´(x₂, y₂, z₂,1) :

(x₂, y₂, z₂,1) = (0,0,0,1) . T´

and by the direction vector s´(x₁, y₁, z₁,0) : 

(x₁, y₁, z₁,0) = (0,0,z₀,0) . T´

The analytic representation of the parallel circle k on the surface j´ that is located in the plane parallel to the yz-plane and is incident to the point L(u₀,0) = r(u₀) = (x(u₀), 0, z(u₀),1) , u₀ Î [0, 1]  on the basic curve c  is in the form 

 where the radius of the parallel circle r_k = z(u₀) . 

The common tangent plane t  to the surface j ´ and the auxiliary tangent spherical surface  G  in the point  L on the basic curve  c  will be determined by the tangent vector  t_u  to the curve c  in the point L and the tangent vector t_v to the v-isoparametric curve on the surface j´ determined by the first derivative of the function (5) in the point L . Tangent plane direction vectors 

t_u = ¹r´(u₀) = (x´(u₀), 0, z´(u₀), 0),     where x´(u₀) 0

t_v = ²r´(0) = (0, 1, 0, 0) ,

define normal vector

n = t_u x t_v = (-z´(u₀), 0, x´(u₀), 0)

The normal line q  with the direction vector n

            x =  x(u₀) - z´(u₀). l

y = 0

z =  z(u₀) - x´(u₀). l     for  l Î [-, ] ,  u₀ Î [0, 1]

intersects the coordinate axis x in the point S´, which is the centre of the auxiliary tangent spherical surface G´:

The point S(x_S, y_S, z_S,1)  located on the axis ¹o  is defined by the transformation  T (3) : 

(x_S, y_S, z_S,1) = (0,0,0,1) . T

The tangent line to the helix, which is the trajectory of the point S movement, is parallel to the basic line on the direction conical surface located in the point P(y_S,-x_S,0,1). (The construction can by found in Velichová [3].) 

The tangent line to the helix in the point S can be determined by the direction vector t(-y_S, x_S, (-1)ⁱz₀, 0). Constant value  i = 1  is valid for the clockwise and  i = 2  for the anticlockwise helical movement.

 By the transformation  T´ (4) of the vector  t  we can obtain the coordinates of the direction vector t´(x_t, y_t, z_t, 0) of the tangent line to the helix of the point S´ with axis o´ 

(x_t, y_t, z_t,0) = (-y_S, x_S, (-1)ⁱz₀, 0) . T´

Centre S´ is incident to the plane  x  perpendicular to the vector  t´, and is determined implicitly by the equation 

x_t(x - x_G) + y_t y + z_t z = 0

Plane x  intersects the parallel circle k (6) located in the plane m´// m  (7) 

in the points ¹M´, ²M´ of the constructed characteristics e´ on the surface j ´. Their coordinates are defined as follows 

(8)

The parameter ^jt can be calculated from the quadratic equation (9) that is the solution of the equations (6) - (8) 

(9)

.

for the value y_t 0.

is a negative number, the points of the characteristics do not exist.

For D 0 

It is necessary to eliminate the case of the tangent line to the helix of the point S´ determined by the direction vector  t´ that is located in the xz-plane.

 By transformation  T (3) of the point M´(x´, y´, z´,1) of the characteristics e´ on the surface j ´ we can obtain the coordinates of the point M(x_M, y_M, z_M, 1) of the characteristics e 

(x_M, y_M, z_M, 1) = (x´, y´, z´,1) . T .

The shape of the envelope surface can be better comprehended by its meridian section than by the characteristics, which is usually a space curve segment. Coordinates of the point M* = (x*, y*, z*,1) located on the meridian section in the xz-plane can be obtained from the coordinates of the point M = (x_M, y_M, z_M, 1) on the characteristics, as the solutions of the following equations 

where w is the directed angle of the revolution about the axis o to the xz-plane oriented in the helical movement direction and constant value  i = 1  is valid for the clockwise and  i = 2  for the anticlockwise helical movement. 

If the surface j is a rotational conical or cylindrical surface, then we can create the envelope surface F by the presented method, but also by the method published in the paper (Szarková [2]). Graphical processing of the helical envelope surfaces created from the rotational surfaces by the two given different methods can be compared on illustrations in figures 3 and 4 presented in the paper (Szarková [2]) and figures 4 and 5 presented in this paper.

In the paper (Szarková) [2] the created envelope helical surface F  had been represented as the net of parts of characteristics in the Fig.3, resp. as the net of meridian section segments in the Fig.5. In this paper the same helical envelope surface F  is represented as the net of helices that are trajectories of some points on characteristics in the Fig.3 and on meridian section in the Fig.4. 

Fig. 3	Fig. 4

Fig. 5.

In the Fig.5 the characteristics and the meridian section of the envelope helical surface F determined by the basic rotational surface j  with the basic curve in a cissoid is illustrated. The envelope helical surface created from the basic rotational surface j with the basic curve in a Witch of Agnési is represented as the net of helices that are trajectories of the points on characteristics in the Fig.6. 

Fig. 6.

The formulas for the calculations are a bit complicated, but the computer processing is very fast. The general advantage of the computer approach to the given problematic dwells in its creative access. The interactive access available for the constructor helps him to realise the design of the elaborating machines in a short time, as it provides the possibility to fix the generating surface of the created envelope surface into the desired position. It is possible to exclude efficiently those positions, in which the envelope surface cannot be created (does not exit) or is of an unsuitable design.

Presented figures are examples of the characteristics, the meridian section and the basic surface j  of the envelope surface F projected in the Monge projection method and Axonometry and visualised as outputs of the programme (written by the author) on the screen.

References