## 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. 99 Figure 4.1 Figure 4.2 100 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. 101## 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. 102 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. 103 Figure 4.3 Figure 4.4 104 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 105## 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 . 106