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CHAPTER 1

INTRODUCTION

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.

1.1  PARAMETRIC GEOMETRY

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
the form

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

Parametric equations:

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

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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
equations:

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
-      -    -    -
point  on the curve and  i, j, k  are unit vectors  along  the
-  -  -
coordinate axes.

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Figure 1.1

Figure 1.2

4

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
Figure 1.2.

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.

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Figure 1.3

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1.2  VECTOR GEOMETRY

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)
-     -     -     -    -

where i,  j,  k are unit vectors parallel to the  axes.    The
-   -   -
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
-     -     -     -

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
- -                                                   -

If v and w are orthogonal then v.w = 0.
-     -                     - -

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Figure 1.4

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1.2.1  VECTOR GEOMETRY OF PARAMETRIC CURVES

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.
-             -              -
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.
-                                                    -      -
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
associated vectors.

1.2.2  VECTOR GEOMETRY OF PARAMETRIC SURFACES

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.
-

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Figure 1.5

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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.
-

1.3  HOMOGENEOUS COORDINATES

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
the equations:

x  =  X/W

y  =  Y/W

z  =  Z/W.

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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
representations.

For example,  the simplest cartesian parametric representation
of a circle of radius 1 centred at the origin in the XY  plane
is

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

1.4  EXAMPLES IN THIS MANUAL

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
surfaces.

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