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 1 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. 3 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. 5 Figure 1.3 6

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. - - - - 7 Figure 1.4 8

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. - 9 Figure 1.5 10 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. 11 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. 12