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.
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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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.
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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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.
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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