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CHAPTER  4

SCULPTURED SURFACES DEFINED IN TERMS OF

A SINGLE CURVE

The Sculptured Surfaces processor allows two kinds of surfaces
to  be  constructed  in terms of a single  previously  defined
synthetic curve.    The curve may be of the SPLINE,  CURSEG or
COMBIN type,  and a surface is swept out either by translating
it along a fixed direction in space or by rotating it about  a
fixed axis in space.   In the first case the surface generated
has much in common with a TABCYL,  the chief differences being
that  (i)  more  general  curves can now  be  used,  (ii)  the
surfaces generated are bounded, whereas a TABCYL is unbounded,
and (iii) the internal storage formats are different.

4.1  RULED CYLINDRICAL SURFACES

The  surface  obtained  by  translating a  curve  in  a  fixed
direction  in space is a ruled surface,  since every point  on
the  curve  traces  out a straight line  as  it  moves.    The
resulting  family of straight lines is the set of  rulings  of
the  surface,  and since they are all parallel the surface  is
cylindrical in the general sense of the word.

If  C is any previously defined synthetic curve,  the language
input for the definition of a ruled cylindrical surface  takes
either of the forms

S1  =  SSURF/RULED,C,AXIS,P1,P2

or

S2  =  SSURF/RULED,C,AXIS,V

Here  the 'axis' of the translational motion may be  specified
either  in  terms  of  two points P1  and  P2  whose  relative
positions define a direction, or more explicitly in terms of a
vector V.

The  surface  which  is generated by the method  described  is
bounded;   it  extends  for  10 units on either  side  of  the
original curve C.    If the curve is composite,  consisting of
more than one arc,  then each arc sweeps out a single  surface
patch.   A composite curve therefore gives rise to a composite
surface,  but if the curve has tangent continuity at the joins
between arcs then a smooth surface will result.

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Figure 4.1

Figure 4.2

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The following is a specific example:

P1  =  POINT/0,2,0
P2  =  POINT/1,2.5,0
P3  =  POINT/3,3,0
P4  =  POINT/5,2.5,0
P5  =  POINT/6.5,1.5,0
P6  =  POINT/8,2.5,0
C   =  SCURV/SPLINE,P1,P2,P3,P4,P5,P6
S   =  SSURF/RULED,C,AXIS,VECTOR,1,1,1

The  spline curve C is here a plane curve lying in the  x,y  -
plane,  though  it is equally possible to use a space curve in
the same way.    The curve is illustrated in Figure  4.1,  and
the  ruled  cylindrical surface defined by the  above  program
fragment is shown in Figure 4.2.

In  this Figure,  and in all subsequent surface illustrations,
the constant parameter lines corresponding to the values of 0,
0.5 and 1 are drawn for both u and v.    The surface in Figure
4.2 is made up of five patches, and the original profile curve
becomes  the  v = 0.5 constant parameter line of  the  overall
surface.

Two  important  restrictions  must  be  observed  in  defining
surfaces of the RULED type.   These are

i)  the  AXIS  direction must not be  parallel  to  the
tangent  vector to the curve at any point, as this
would lead to a surface with an undefined normal
vector along the rulings through those points;

ii) if the synthetic curve is a plane curve and the AXIS
is parallel to its plane of definition, then no straight
line in the axial direction may cross the curve more than
once.   The need to observe this restriction will rarely
arise in practice.   The restriction avoids the definition
of surfaces having multiple points corresponding to given
u,v parameter pairs.

Finally,  it  should be noted that any FLOW structure  imposed
upon  the definition of the original curve C is discarded when
the surface is constructed.    The surface parametrisation  in
the  sense of the curve therefore follows the PARAM or natural
parametrisation of the curve.   The transverse parametrisation
will  be proportional to geometric distance along the  surface
rulings.    The  constant parameter lines will be the  rulings
themselves and the family of parallel offsets of the  original
curve in the axial direction.

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4.2  SURFACES OF REVOLUTION

A  surface  of revolution is defined by rotating  a  synthetic
profile  curve  about  an axis in space.   As  for  the  RULED
surface of the last section, the synthetic curve may be either
a SPLINE,  a CURSEG or a COMBIN.    In practical cases it will
usually  be  a plane curve,  but a space curve may be used  if
desired.

The language defining a surface of revolution is as follows:

S = SSURF/REVOLV,C,AXIS,P1,V, CLW  , A,B                   -

CCLW

S = SSURF/REVOLV,C,AXIS,P1,P2, CLW  , A,B
CCLW

Here,  as  in  the last section,  C is  a  previously  defined
synthetic curve.    The axis of revolution is specified either
in terms of a point P1 lying on the axis and a vector V giving
its  direction  or by two points P1 and P2 lying on the  axis.
The  sense  of  rotation is specified as  clockwise  (CLW)  or
counter-clockwise  (CCLW),  while A,  B are the  starting  and
terminating  angles  of the rotational  motion,  expressed  in
degrees.

The  following  conventions  are observed in  determining  the
sense and extent of the rotation:

i)    The positive sense of the axis is the direction of the
vector V, or the direction of motion from P1 to P2.

ii)   A positive rotation is clockwise as viewed by an observer
looking along the positive axis direction.

iii)  The perpendicular line from the first defining point of
the profile curve onto the axis is the zero degree
reference line.   The starting and terminating angles A,
B are measured in a clockwise sense about the axis from
this line.   If the first defining point of the curve
happens to lie on the axis then the curve is searched for
a defining point which lies off the axis, and this is
used to establish the datum.

iv)   Specifying CLW will result in rotation in the positive
sense between A and B as defined in (iii), while CCLW
will give a rotation in the negative sense.   If A = 0
and B = 360 then a complete surface of revolution will
be generated.

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Considerable   freedom  is  available  to  the  user  in   the
definition  of REVOLV surfaces.    For  example,  the  profile
curve may be defined as a complete closed curve,  and provided
this  does  not  intersect  the axis  a  generalised  toroidal
surface will result.

A true torus will be generated if the closed curve is a circle.

The  profile  curve  is  allowed  to  intersect  the  axis  of
revolution.   However, the rotation will in this case generate
a surface with a degeneracy at the point of intersection.   In
particular, there will not be a well-defined surface normal at
such a point and the processing of tool motion on the  surface
through such a point must be expected to fail.

Now  consider  a  specific example.    Using the  same  SPLINE
profile  curve as in Section 4.1,  a surface of revolution  is
defined by

P1  =  POINT/0,2,0
P2  =  POINT/1,2.5,0
P3  =  POINT/3,3,0
P4  =  POINT/5,2.5,0
P5  =  POINT/6.5,1.5,0
P6  =  POINT/8,2.5,0
C   =  SCURV/SPLINE,P1,P2,P3,P4,P5,P6
S   =  SSURF/REVOLV,C,AXIS,(POINT/0,0,0),   \$
(POINT/1,0,0),CLW,0,180

Here  the  curve  C lies in the x,y - plane and  the  axis  of
rotation  is  the  x  - axis.     The  resulting  surface   is
illustrated in Figure 4.3.

As regards the patch structure of REVOLV surfaces,  the number
of  patches in the u direction is equal to the number of  arcs
in  the  profile  curve.    The number of  patches  in  the  v
direction  is  governed  by  the  overall  angle  of  rotation         -

B-A,  and  is  determined by subdividing this angle  into  the
smallest  possible  number  of equal increments less  than  or
equal to 120 degrees.    Each increment gives rise to a patch,
and  each  patch therefore cannot span a rotational  range  of
more  than 120 degrees.    This is illustrated in Figure  4.3,
where the overall range of rotation is 180 degrees;    this is
divided  into  two  equal increments of 90  degrees   and  the
surface therefore has two patches in the v direction.

It should be noted that any FLOW structure associated with the
profile curve when originally defined is ignored by the system
in generating the surface.

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Figure 4.3

Figure 4.4

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If  the profile curve is a COMBIN which has points of  tangent
discontinuity   then  the  generated  surface   will   exhibit
corresponding lines of slope discontinuity.

For  a  second  example  of a  REVOLV  surface,  consider  the
following definition:

P1  =  POINT/1,3,0
P2  =  POINT/3,3,0
V1  =  VECTOR/0,1,0
V2  =  VECTOR/0,-1,0
C1  =  SCURV/CURSEG,P1,TANSPL,V1,P2
C2  =  SCURV/CURSEG,P2,TANSPL,V2,P1
C3  =  SCURV/COMBIN,C1,C2
S   =  SSURF/REVOLV,C3,AXIS,(POINT/0,0,0),   \$
(POINT/1,0,0),CCLW,0,130

The  surface is shown in Figure 4.4.  The following points are
illustrated:

i)  The profile curve is a closed COMBIN made up of two semi-
circular arcs to form a complete circle.   The surface is
therefore part of a torus, and there are two patches in
the u direction.

ii) The angle of rotation is 130 degrees: the system splits
this into two 65 increments, and hence the number of
patches in the v direction is also two.

Note that the axis of rotation is also shown in Figure 4.4.

4.3  DEFINITION ERRORS FOR RULED AND REVOLV SURFACES

If  we disregard syntax errors,  the most likely errors are as
follows:

DEFINITION ERROR 5353

THE SURFACE SIZE EXCEEDED A PROGRAMMED MAXIMUM IN REVOLV

DEFINITION ERROR 5354

THE AXIS VECTOR OF THE CYLINDER WAS NOT SPECIFIED CORRECTLY

DEFINITION ERROR 5355

ALL DEFINING POINTS OF THE CURVE LIE ON THE AXIS OF
REVOLUTION

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4.4  A SUGGESTED APPLICATION FOR REVOLV SURFACES

Surfaces  of this kind are very useful for defining mould  and
die cavities.    For example,  a die for a rotational part can
be  designed  straightforwardly in terms of a  REVOLV  surface
with  a range of rotation from 0 to 180 degrees.   It is  also
simple  to arrange for draft angles on cavities by  using  the
linear extensions of surfaces,  as explained in Section 7.1.5.
If  a horizontal plane profile curve is rotated between 5  and
175  degrees  then  the  linear extensions  of  the  resulting
surface  will  be uniformly tapered at an angle at  5  degrees
from the vertical.

4.5  VERIFICATION LISTINGS FOR RULED AND REVOLV SURFACES

RULED  and REVOLV surfaces are particularly simple to  define,
but as far as the Sculptured Surfaces system is concerned they
are  particular  cases  of  a  more  general  class  of  SMESH
surfaces.    These are covered in the next Chapter,  where the
interpretation of verification listings for all types of SMESH
surfaces is discussed .

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