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Curves and Surfaces Introduction to Curve and Surface Geometry One of the biggest problems faced by vendors of Computer Aided Design products is the challenge of developing suitable software for drawing complex curves and surfaces. These notes provide an introduction to the topic and explain the major techniques that have been developed by mathematicians, engineers and programmers in order to make possible the construction of complex 2D curves and 3D surfaces. The ability to construct complex shapes, and subsequently manufacture products incorporating them has been a major aspect of competition in many industries. Shapes such as wing profiles and ship hulls are examples of complex functional shapes, but in many industries geometry is employed to produce aesthetically pleasing surfaces. It is not surprising that many of the major developments in curve and surface technology have occurred in the car industry where a combination of function and style are vital. Traditionally engineers drew curves using spline methods. These were techniques that employed pliable materials (such as lead strips) that could be curved around pins attached to pieces of wood to produce very smooth curves that could interpolate through control points (i.e. the pins). In the shipbuilding industry, very long curves were required and normally the only part of the building that was long enough to house this activity was the loft, so the practise of drawing spline curves became known as ‘lofting’. Today this expression is still used and the computer analogies of splines bear the same names. Splines that pass through chosen control points are called ‘interpolation splines’ and those that are merely pulled toward control points are called ‘approximation splines’. We will now consider the methods that have been used to construct computer splines. Simple Curves and Simple Geometry We all know from our early maths that the equation of a straight line is as follows y = ax + b Simple equation of a straight line This equation is based on the description of a line as a gradient and intercept. To describe a line segment we would specify two values of either x or y to bound the chosen line. This method of describing a line is simple, but does have drawbacks. Firstly it does not cope very well with vertical lines! We could of course revert to an explicit equation with x as a function of y but that only transfers the problem to horizontal lines. A better way to describe line segments may therefore be to use parametric curves, so our new straight line equation becomes: x = a1 + b1t y = a2 + b2t z = a3 + b3t Parametric equations of a straight line or in vector notation…………….. S(t) = A + Bt Parametric vector equations of a straight line This method allows us to easily draw robust straight lines. We can set the general nature of a straight line by changing the coefficients A and B. Straight lines are therefore easy, but now let us consider a simple curve. This can be represented by say a quadratic, cubic, quartic, quintic etc. polynomial like so: S(t) = A + Bt + Ct2 + Dt3 + ………… Power basis equation for a curve This method of describing curves uses as its 'basis', a series of powers of the independent variable and is therefore known as a 'power basis polynomial'. This may work well for the straight line where the meaning of the coefficients A and B is clear, but what is the meaning of C or D etc? If we want to draw higher order curves we might be better to use meaningful design variables. Spline curves are usually drawn using design variables that are meaningful to the designer, typically these are the positions of a set of control points through or near which the curve will pass. Interpolation Splines Perhaps the simplest way of describing curves for the designer is to consider a set of co-ordinates for each point to be interpolated, as with a parametric curve. These splines are clearly analogous to our early example of the techniques used by the shipbuilding industry. This is indeed a useful method, but given a set of m+1 points, we would need a curve of degree m to describe our shape. With many points we end up with very high degree curves. In order to plot the curves we have to find interpolating values of the polynomial coefficients and this involves the solution of potentially large sets of simultaneous equations. These curves can also easily become too 'squiggly' for the designer's liking and they offer no local control. In practice these problems are usually overcome by using only, say, cubic curve segments and which are joined or knotted together to form piecewise curves that can interpolate many points. The simple method of making piecewise curves can use Lagrangian interpolation methods to fit curves (typically cubic, ie degree m=3) through m+1 (typically therefore 4) control points. Another method of developing piecewise interpolation splines is to use Hermite interpolation. This involves design variables describing not just control points but also those describing the tangent of the curve at that control point. This has the advantage that local control over the curve is possible since any action taken at a control point will only affect the curve to either side of the point since on reaching a subsequent point the position and slope of the curve is set. Figure 1:Lagrange and Hermite interpolation used to construct a parametric cubic curve segment In order to make the construction of curves and surfaces easy and fast, another effective means is to use approximation splines. Approximation Splines There are many methods that have been developed to produce approximation splines, e.g. Bézier curves, B-splines (uniform, non-uniform or periodic), Beta-splines and NURBS (Non-Uniform, Rational B-Splines) to name a few. We will now try to look at the most important of these in order to give an introduction to them. Bézier Curves The first application of approximation splines is credited to the French mathematician Pierre Bézier who worked for the Renault motor company. He proposed a method of drawing curves that would provide an intuitive feel for automotive body designers, allowing shapes to be manipulated by pulling out the curves. These curves interpolated end points, but approximated intermediate points. This, it is argued, makes the curves easy to join together and simple to calculate as well as manipulate. A Bézier curve for a simple cubic curve (degree 3) will have 4 control points as we might expect, and might look like this: P3 P1 P2 P0 Figure 2: Possible Bézier curves of degree 3 The most convenient equation used to describe a Bézier curve uses 'blending functions' which are used to 'blend' the effect of the control points on each part of the curve. A Bézier curve then is described by the following equation: where The equation of a Bezier curve Perhaps the easiest way to understand how a curve might be constructed is to use the de Casteljau construction of a Bezier curve. This is based on a physical analogy of points travelling in straight lines along tracks. Consider a simple quadratic curve with 3 control points as shown below: P0 P1’ P0’’ P0’ P0 P0 Figure 3: The de Casteljau construction The curve generated from the above can be represented by the following set of linear equations thus: P0’ = (1-t)P0 + tP1 P1’ = (1-t)P1 + tP2 P0’’ = (1-t)P0’ + tP1’ By substitution we get, S = P0’’ = (1-t)2 P0 + 2(1-t)t P1 + t2 P2 Thus the basis of the polynomial is not a power series, but a set of functions as follows: (1-t)2 2(1-t)t t2 The basis function describes the blending of the point ‘attraction’ along the curve. We can plot these to observe the effect that each control point has. 1 0 0 1 B2,2 B1,2 B0,2 Figure 4: The Bernstein basis blending functions for a Bézier curve of degree 2 As already mentioned the Bézier curve passes through its first and last control points. This property derives from the Bernstein basis function since at the endpoints (t=0,1) the functions are zero except for the first and last functions e.g. degree 3 – B0,3 and B3,3. At t=0 the curve is tangential to the first segment of the control polygon, i.e. the line from control point 1 to control point 2. Similarly at t=1 the curve is tangential to the last segment of the control polygon. It is possible to create higher order Bézier curves to deal with more complex shapes, however, these can become unwieldy and expensive to compute. Perhaps the greatest problem with this is that small changes in the position of control points can make large deviations in the shape of the curve and often not locally. Most CAD designers would not like this and prefer to be able to edit curves using control points that exert local control only. One solution to this is to use many lower order curves joined together and these are known as piecewise curve segments. Initially this is made relatively easy since the first and last points are interpolated, however, the problem becomes significant when we require smooth curve joins. Continuity The notion of continuity refers to the degree of smoothness between curve segments that are used to make up a piecewise curve. Consider figure 5: Figure 5: No continuity (the dots on the curves represent t between 0 and 1 in steps of 0.01) If we wish to make 2 curves meet we can do so by ensuring that the last control point on the first curve coincides with the first control point on the second curve, e.g. see figure 6: Figure 6 G0, C0 continuity The curves in figure 6 join but not smoothly. If these were train tracks a train could not get over this join. Since the curves are tangential to their first and last control polygon segments we can make the tangents on either side of the join equal by making the last two control points of the first curve and the first two control points of the second curve lie along a common straight line, e.g. see figure 7. Figure 7 G1 continuity Observing this plot we can see that we now have a smooth join between two curves. However if we imagine that the curves describe the motion of an object, say a car, then although we can imagine driving such a route we can also see that there would be an impossible step change in speed as the route shape changes from curve 1 to curve 2. Thus the tangents at the curve ends have the same direction but different magnitudes. These curves have geometric continuity (denoted as G0, G1 etc), ie their geometric derivatives are equal. If we want to have continuity of both direction and speed we have to ensure that the parametric derivatives are equal. See figure 8: Figure 8 G1 and C1 continuity Now not only do the 4 central control points lie along a line but the distances between the last 2 control points of the 1st curve and the first 2 control points of the 2nd curve are equal. These are said to have parametric continuity (denoted by C1). We can also define conditions for higher continuities for 2nd and 3rd derivatives to achieve, e.g. C2 and C3. E.g. see figure 9. Figure 9 G2, C2 continuity In general two curves have C or parametric continuity if the their parametric derivatives are equal at the join whilst they have G or geometric continuity if their parametric derivatives are proportional to each other. These examples illustrate continuity for 2D curves but the same priciples apply to 3D curves and to surfaces. Bézier curves also have a number of other useful properties. It can be seen from figure 4 that the basis functions always sums to 1. This ensures that the curve is always contained within its convex hull a fact useful for intersection computations and the like. Bézier curves are also said to be affinely invariant with respect to their control points. This means that a Bézier curve is transformed simply by transforming its control points. This means curves can simply spun or scaled e.g., simply by spinning or scaling the control points. Another useful property is that the curve is variation diminishing, indicating that the curve intersects any plane only as often as its control polygon. B-Splines B-splines consist of curve segments that have many similarities with single Bézier curves. The major advantage of B-splines over Bézier curves is that we achieve continuity in piecewise curves by sharing control points in the formulation. The number of points shared is dependent on the degree of continuity required. This means that local changes to the curve are possible. A B-spline curve can be defined in a similar way to a Bézier: where {Pi: =0,1,2….,n} are the control points, k is the order of the polynomial segments of the B- spline curve and Bi,k are the normalised B-spline blending functions. The points where curves are joined together are called 'knots' and to generate a B-spline we must specify a 'knot vector' which is a list of the positions of these knots. These are usually normalised along the parameter space to give values between 0 and 1 and may or may not be uniformly spaced. There are m – 1 knots in a curve since the start and end points are considered as knots. Where the knots are uniformly spaced the curve’s blending functions have equal weight. We can set a cubic B-spline with any number of control points since we can link any number of curve segments together and apply conditions for continuity between these. The algorithm for generating the curve will then have to set the required number of curve segments and the continuity between them. A cubic B-spline with C2 continuity then approximates a series of n+1 control points where n = 3 and there are n - 2 curve segments. Rational Splines One drawback with Bézier curves and simple B-splines is that they cannot describe exactly conic sections such as a parabola or circle. A means to overcome this is the use of rational parametric curves. Rational curves are defined on the basis of homogeneous co-ordinates, which represent three-dimensional points in terms of four dimensions with an additional w-axis co-ordinate. To draw the curve in three-dimensional space the first three co-ordinates of each point are divided by the fourth. This essentially uses one spline as the numerator and another as the denominator. Non-Uniform Rational B-Splines (or NURBS) Non uniform spacing allows the weight of the functions to vary in their ‘pull’ along the curve and can therefore offer greater control. Where the curve is required to interpolate a particular point, multiple control points can be placed at one position (for a cubic this is 3). This approach can be used with Bézier curves but also with non-uniform B-splines to produce NURBS. This fourth dimension does come at some cost as calculations become larger and more draining on computer resources. However most modern CAD systems do use NURBS, fortunately much of the work is already done by these systems leaving the designer to think about the shape and as little as possible about the maths. Surfaces The major advantage of parametric methods of representing curves is that they can be generalised into surface representation. Both the Bézier curve and the B-Spline can be extended into 3D to create surface patches. Bézier Surface Patches If each point on a Bézier curve were moved along a curve of its own a Bézier Surface patch would be traced out. The patches are defined by the network of points that form the control polyhedron. Many of the properties of the patch are born from those belonging to the Bézier curve. The surface approximates all of the points except the four corner points that are interpolated, they are tangent to the start and end segments, contained within the convex hull of its control points and invariant under affine transformations. The general equation is obtained by extending the curve equation with another parametric variable, normally u and v are used. Bézier surface patches suffer the same problems as Bézier curves. The primary problem being that the number of control points is directly related to the order and continuity. Again it is preferable to create a surface using a series of patches. Also as with the curves the continuity requirement prevents the ability to make local manipulations. For this reason B-Splines (and particularly NURBS) form the basis of most modern surface modelling packages. B-Spline Surface Patches We move to B-Spline surfaces for the same reason as we moved from Bézier curves to B-Spline Curves as again the surface properties are analogous to the curve properties. Importantly this gives us a surface with which the continuity is independent of the number of control points and can be locally manipulated. The rest of the useful B-Spline properties also remain in tact. Rational Parametric Surfaces These too are analogous to their curve roots and are defined by the equations of the Bézier or B- spline surfaces written in homogeneous co-ordinates. This results in surface representation that is capable of modelling spheres and cylinders as well as more abstract surfaces.