This volume is concerned with the sculptured surfaces features
of the APT4 + Sculptured Surfaces system. Unlike the
unbounded analytic geometry of the APT system all sculptured
surfaces are bounded and are defined in terms of parametric
geometry in 3 dimensional space. This chapter introduces
some of the geometric concepts and explains the notation and
conventions which are adopted in the remaining chapters.
In 2 dimensional geometry a curve in the x,y-plane may have 3
alternative algebraic representations. The simplest of these,
which may not always be possible, is an explicit equation of
y = f(x).
The second alternative is an implicit equation
g(x,y) = 0
where g is a function of both x and y. Both the above are
referred to as analytic equations for the curve.
The third option is to express both x and y in terms of some
independent variable (or parameter ) t; the curve is then
described by the two parametric equations
x = x(t)
y = y(t).
As an example the circle centre (a,b) and radius r has
Explicit equation: y = b + sqrt(r*r-(x-a)*(x-a))
Implicit equation: (x-a)*(x-a)+(y-b)*(y-b)-r*r = 0
1) x = a + r(1-t*t)/(1+t*t) 2) x = a + r cos u
y = b + 2rt/(1+t*t) y = b + r sin u
In the parametric equations points in the first quadrant of
the circle correspond to values of t between 0 and 1 or to
values of the parameter u between 0 and PI/2. Both the
analytic and parametric representations have their advantages
and disadvantages. For solving intersection problems, such as
where the curve meets a given straight line the analytic
expressions are to be preferred. For plotting a segment of
the curve, which involves computing the coordinates of a
number of 'consecutive' points on the curve, the parametric
representation is preferable.
One important distinction between the representations is that
the analytic expression represents the entire curve, which may
in many cases (e.g. parabola) be unbounded; the parametric
representation will generally for a finite parameter range
represent only a segment of the curve.
In 3 dimensional space a single analytic implicit equation
f(x,y,z) = 0 will describe a surface; a curve can only be
described by two such expressions:
f1(x,y,z) = 0
and f2(x,y,z) = 0.
Strictly the curve is being defined as the intersection curve
of two surfaces.
Alternatively, a curve can be described by 3 parametric
x = x(u)
y = y(u)
z = z(u)
or, as a single parametric vector equation
r = r(u)
where r = xi + yj + zk is the vector from the origin 0 to a
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point on the curve and i, j, k are unit vectors along the
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In the Sculptured Surfaces system all 3 dimensional space
curves are parametric and the segment of the curve which
corresponds to the parametric range 0 <= u <= 1 is generally
considered. Figure 1.1 illustrates such a curve. In this
curve P0 corresponds to the parameter u = 0 and P1 to the
parameter u = 1. Points on the dotted section of the curve
to the left of P0 have negative parametric values and points
to the right of P1 have parametric values greater than 1.0.
As mentioned above a surface may be described by a single
analytic equation; alternatively it can be described
parametrically in terms of two independent parameters u and v.
Restricting these parameters to a finite range, such as
0 <= u <= 1, 0 <= v <= 1, will define a finite patch on the
surface. Figure 1.2 shows such a patch together with the
unit square in the u - v plane. The patch is a mapping of
this square such that each point in the unit square such as
P(u0,v0) corresponds to a point P'(x(u0,v0),y(u0,v0),z(u0,v0))
on the surface patch. The lines u = u0 or v = v0 in the
u,v-plane correspond to constant parametric curves on the
surface. These curves are represented by dotted lines in
In the Sculptured Surfaces system a sculptured surface
generally consists of a number of surface patches of this
type. Each patch will have a different parametric equation
but the geometric properties are generally related so that the
surface has a continuous slope across patch boundaries. For
each separate patch the range for the parameters u and v is 1
and the patch boundaries correspond to the constant parametric
curves u = 0, u = 1, u = 2, ... v = 0, v = 1 etc. Figure 1.3
shows a typical sculptured surface. As with many other
illustrations in this manual, each patch is illustrated by
depicting its boundaries and the constant parametric curves at
intervals of 0.5 in u and v.
Much of the input and output for the Sculptured Surfaces
system is expressed in terms of vector geometry. A vector is
simply defined as a quantity having magnitude and direction.
Once a reference axis system has been established a vector can
be uniquely described in terms of its 3 components parallel to
the x, y and z axes respectively:
v = v1i + v2j + v3k or v = (v1,v2,v3)
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where i, j, k are unit vectors parallel to the axes. The
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magnitude, or length, of this vector v is
sqrt(v1*v1 + v2*v2 + v3*v3).
A unit vector is a vector of length 1. Dividing v by its
magnitude gives a unit vector in the same direction as v.
For a unit vector u = u1i + u2j + u3k the three components
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u1,u2,u3 are equal to the direction cosines cos(A1), cos(A2),
cos(A3) of the vector. A1 is the angle between the vector
and the X axis, A2 and A3 have similar interpretations.
Two vectors v and w are said to be orthogonal if the angle
between their directions is 90 degrees. A simple test for
this is to evaluate their scalar product
v.w = v1*w1 + v2*w2 + v3*w3
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If v and w are orthogonal then v.w = 0.
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Once an origin and axis system have been established each
point P on a parametric curve can be identified by its
position r which is the vector OP. r = r(u) is then a vector
- - -
function of the parameter u. The three components of r are
x(u), y(u) and z(u) respectively.
Associated with each point of the curve are three unit
vectors; these are the tangent T, the normal N and binormal B.
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T is the direction of the vector derivative dr/du .
N lies in the instantaneous plane of the curve and is
orthogonal to T. The centre of curvature of the curve lies
along N at a distance equal to the radius of curvature.
B the binormal is in a direction orthogonal to both N and T.
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For a plane curve the direction of B is constant and is in
fact the normal to the plane of the curve.
Figure 1.4 shows a simple parametric curve and these
The position vector r of a point P on a parametric surface is
dependent upon the two surface parameters u and v. Thus
r = r(u,v). If either of these parameters is fixed, such as
u = u0, the position vector of P is then r = r(u0,v); this is
then a parametric curve on the surface, different points on
the curve corresponding to different values of v. At any
point along this curve the partial derivative dr/dv will be in
the direction of the tangent vector to this curve.
Similarly the constant parametric curve v = v0 will lie on the
surface and have a tangent in the direction of dr/du.
In the Sculptured Surfaces system the constant parametric curves
u = u0 are referred to as spline curves and the curves v = v0 are
called cross-splines. Both dr/du and dr/dv lie in the tangent
plane to the surface; their common perpendicular defines the
direction of the normal to the tangent plane, or surface normal,
at this point.
The second derivative d(dr/dv)/du is related to the rate of
change of dr/du as the surface is traversed in the direction
of the parameter v increasing; this is generally referred to
as the 'twist vector'. The sculptured surfaces verification
listing normally gives the twist vector at each corner of a
surface patch. Figure 1.5 shows a sculptured surface patch
together with the vectors dr/du , dr/dv and the surface
normal at the point u = 0.5, v = 0.5 of the surface. In this
figure, as in the sculptured surfaces system, dr/du is
referred to as the TANSPL and dr/dv as the CRSSPL.
All input data for the sculptured surface system is expressed
in terms of cartesian coordinates. In this system the
position of a point is specified by defining 3 real numbers
which are respectively the x, y and z coordinates referred to
a standard set of axes. In order to provide an exact
parametric representation of a conic curve some of the
internal geometric data, and some of the output data, is
converted to a system of homogeneous coordinates.
In the homogenous coordinate system each point has a non-
unique representation by a set of 4 coordinates X,Y,Z, and W.
These can be related to the cartesian coordinates (x,y,z) by
x = X/W
y = Y/W
z = Z/W.
In the simplest case, if W = 1.0 then X,Y and Z are
numerically equal to x,y and z. However for any other value
of W the values will differ.
For each point the cartesian coordinates are unique but many
alternative sets of homogeneous coordinates are possible, for
example the point with cartesian coordinates (1,2,3) has
homogeneous coordinates: (1,2,3,1) or (2,4,6,2) or (0.5,1.0,
1.5,0.5) etc. In the homogeneous coordinate system it is
only the ratio of the coordinates which is significant. The
advantage of using homogeneous coordinates is the extension of
the types of curve which have simple parametric
For example, the simplest cartesian parametric representation
of a circle of radius 1 centred at the origin in the XY plane
x = (1-t*t)/(1+t*t), y = 2t/(1+t*t), z = 0.
An equivalent, but simpler, homogeneous representation is
X = 1-t*t, Y = 2t, Z = 0, W = 1+t*t
All part program examples in this manual have been processed
using the SSV1 processor on a DEC VAX 11/780 computer. When
using a different computer the results obtained may differ in
detail because of differences in compiler and computer word
length; these differences should however be confined to the
last few significant digits.
In order to make the examples self contained, simple direct
methods have been used to define points and vector quantities.
The sculptured surfaces system allows, as an alternative, the
use of any of the APT definitions for points and vectors
described in Volume 1. In addition indirect definitions
using the INTOF statement are also acceptable. The examples
have been chosen to illustrate the features of the system
rather than as realistic samples of the types of part which
can be designed and manufactured with the system. The default
option for print out with the system is SSPRT,ON. This gives
header tables for all curves and surfaces and a summary of
their geometric properties. Where a full verification
listing is quoted this has been obtained using the statement
PRINT/SSTEST,ON which gives full details of curves and