LECTURE #6 Geometric modeling for engineering applications Geometric Modelling for Engineering Applications Introduction to modeling Geometric modeling Curve representation Hermite curve Bezier curve B-spline curve NURBS 1
Modeling Creating symbolic models of the physical world has long been a goal ofmathematicians, scientists, engineers,etc. Recently technology has advanced sufficiently to make computer modeling of physical geometry feasible. Engineering models are used in computer-based design, manufacturing and analysis. Geometric modeling simply means that design concepts are digitally inputted into software subsequently displays them in either 2-D or 3-D forms. a general term applied to 3-D computer-aided design techniques. Geometric models are computational (symbol) structures that capture the spatial aspects of the objects of interest for an application. 2
Modern Systems Permit the encoding of the mathematics of geometry in computer programs which hide most of the complexities of generation from the user. Allow for operations such as: creating specific shapes based upon input parameters (curve, rectangle, sphere) positions these entities within a model space combining basic entities to create more complex geometries. Geometric modeling Stress Analysis Geometric modeling Visualization Engineering Analysis Design Process Production planning Thermal Analysis Engineering drawing After-life Analysis CNC programming Geometric modeling is a basic engineering tool. Serves as the backbone of design Shadows the design process 3
Geometric modeling Geometric Modeling System Four primary components: (1) Symbol structures which represent solid objects (2) Processes which use such representations for answering geometric questions about the object (3) Input facilities (4) Output facilities and representations of results Computing Surveys, Vol 12 No 4 December 1980, 438-464 4
Geometry A typical solid model is defined by volumes, areas, lines, and keypoints. Volumes are bounded by areas. They represent solid objects. Areas are bounded by lines. They represent faces of solid objects, or planar or shell objects. Lines are bounded by keypoints. They represent edges of objects. Keypoints are locations in 3-D space. They represent vertices of objects. Volumes Areas Lines & Keypoints...Geometry - Preprocessing There is a built-in hierarchy among solid model entities. Keypoints are the foundation entities. Lines are built from the keypoints, areas from lines, and volumes from areas. This hierarchy holds true regardless of how the solid model is created. Not allow you to delete or modify a lower-order entity if it is attached to a higher-order entity. (Certain types of modifications are allowed discussed later.) Volumes Areas Lines Keypoints I ll just change this line Volumes Areas Lines Keypoints OOPs! Lines Keypoints Areas Volumes 5
Parametric line representation Parametric representation of a line. The parameter u, is varied from 0 to 1 to define all points along the line. X = X(u) Y = Y(u) P 2 u P 1 Parametric Line This means a parametric line can be defined by: L(u) = [x(u), y(u), z(u)] = A + (B - A)u where A and B and the line endpoints. e.g. A line from point A = (2,4,1) to point B = (7,5,5) can be represented as: x(u) = 2 + (7-2)u = 2 + 5u y(u) = 4 + (5-4)u = 4 + u z(u) = 1 + (5-1)u = 1 + 4u 6
Parametric definition Expanding the 2D parametric technique we used for a line to 3D, two parameters (u and v) are used. P 4 P 2 P 3 u P 1 v Parametric definition Points along edge P 1 P 2 have the form of P(u,0), along P 3 P 4, P(u,1) and so on. P 4 P(1,v) P(u,1) P 2 P 3 P(u,0) u P 1 v P(0,v) 7
Parametric definition By varying value of u and v, any point on the surface or the edge of the face may be defined. P 4 P 2 (u 1,v 1 ) P 3 u P 1 v Parametric cubic curves Algebraic form Geometric form: blending fn * geometric (boundary) conditions Blending function: p (u) = [ F1 F2 F3 F4 ] [ p(0), p(1), pu(0), pu(1) ] Magnitude and direction of tangent vectors Cubic Hermite blending function 8
Boundary conditions Blending functions 9
Curve Use in Design Engineering design requires ability to express complex curve shapes (beyond conics). examples are the bounding curves for: turbine blades ship hulls automotive body panels also curves of intersection between surfaces Curve Representation All forms of geometric modeling require the ability to define curves. Linear curves (1 st order) may be defined simply through their endpoints. Must have a means for the representation for curves of a higher order: conics free form or space curves 10
Curve Representation Curves may be defined using different equation formats. explicit Y = f(x), Z = g(x) implicit f(x,y,z) = 0 parametric X = X(t), Y = Y(t), Z = Z(t) The explicit and implicit formats have serious disadvantages for use in computer-based modeling Curve Representation Parametric: X = X(t), Y = Y(t), Z = Z(t); 0 t 1 (typ) Substituting a value for t gives a corresponding position along curve Overcomes problems associated with implicit and explicit methods Most commonly used representation scheme in modelers 11
Representing Complex Curves Typically represented a series of simpler curves (each defined by a single equation) pieced together at their endpoints (piecewise construction). Simpler curves may be linear or polynomial Equations for simpler curves based upon control points (data pts. used to define the curve) Use of control points General curve shape may be generated using methods of: Interpolation (also known as Curve fitting ) curve will pass though control points Approximation curve will pass near control points may interpolate the start and end points 12
Interpolating Curve Piecewise linear Linear segments used to approximate smooth shape Segments joints known as KNOTS Requires too many datapoints for most shape approximations Representation not flexible enough to editing Piecewise linear Piecewise polynomial (composite curves) Segments defined by polynomial functions Again, segments join at KNOTS Most common polynomial used is cubic (3 rd order) Segment shape controlled by two or more adjacent control points. Piecewise linear Interpolation curve (cubic) 13
Curve continuity concern is continuity at knots (where curve segments join) continuity conditions: point continuity (no slope or curvature restriction) tangent continuity (same slope at knot) curvature continuity (same slope and curvature at knot) Composite curves: continuity Point continuity C 0 : continuity of endpoint only, or continuity of position Tangent continuity C 1 : tangent continuity or first derivative of position Curvature continuity C 2 : curvature continuity or second derivative of position 14
What is Spline? A spline is a numeric function that is piecewise-defined by polynomial functions, and that possesses a sufficiently high degree of smoothness at the places where the polynomial pieces connect (known as knods) The most commonly used splines are cubic spline, i.e., of order 3 in particular, cubic B-spline. It is commonly accepted that the first mathematical reference to splines is the 1946 paper by I.J. Schoenberg. Synthetic Curves Analytical curves are insufficient for designing complex machinery parts and, therefore synthetic curves are used. Synthetic curves are commonly used when interpolation curves are needed and it is easy to modify these curves locally. 15
Synthetic Curves CAD\CAM systems have got 3 types of synthetic curves such as Hermite cubic splines, Bezier curves B-spline curves. Cubic splines are interpolating curves. Bezier and B- splines are approximating curves. On some cases B- splines can be interpolating. Synthetic Curves Both Bézier curves and B-splines are polynomial parametric curves. Polynomial parametric forms can not represent some simple curves such as circles. Bézier curves and B-splines are generalized to rational Bézier curves and Non-Uniform Rational B-splines, or NURBS for short. Rational Bézier curves are more powerful than Bézier curves since the former now can represent circles and ellipses. Similarly, NURBS are more powerful than B-splines. NURBS B-spline Bezier Rational Bezier 16
Hermite Cubic Spline A cubic Hermite spline is a spline where each piece is a third-order polynomial specified in Hermite form: that is, by its values and first derivatives at the end points of the corresponding domain interval. Hermite splines are named after Charles Hermite. They are named in his honor. He was a French mathematician who did research on orthogonal polynomials. One of his students was Henri Poincare. Bezier Curve Pioneering work was done in France by Renault engineer Pierre Bezier and Citroen s physicist and mathematican and Paul de Cateljau. They worked nearly parallel to each other, but because Bézier published the results of his work, Bézier curves were named after him, while de Casteljau s name is only known and used for the algorithms. 17
Bezier Curve The mathematical basis for Bézier curves the Bernstein polynomial has been known since 1912. Bézier curves were widely publicized in 1962 by the French engineer Pierre Bezier. The study of these curves was however first developed in 1959 by mathematician Paul de Casteljau using de Casteljau s algorithm, a numerically stable method to evaluate Bezier curves at Citroen. Bezier Curve From 1933 to 1975 Bézier worked for Renault In 1960 when he devoted a substantial amount of his time working on his UNISURF system Between 1968 and 1979 when he became Professor of Production Engineering at the Conservatoire National des Arts et Metiers. Pierre Etienne Bézier 1910-1999 18
De Casteljau's algorithm De Casteljau s algorithm is a recursive method to evaluate polynomials in Bernstein form of Bezier curves. It can also be used to split a single Bezier curve into two Bezier curves at any arbitrary parameter value. Although the algorithm is slower for most architectures when compared with the direct approach, it is more numerically stable. Paul de Casteljau Bezier Curve The formula can be expressed explicitly as follows: where are the binomial coefficients. 19
Bezier Curve Some terminology is associated with these parametric curves. We have where the polynomials are known as Bernstein* basis polynomials of degree n. *These polynomials were first defined by the Russian mathematician Sergei Natanovich Bernstein around 1910. Linear Bézier curves Given points P 0 and P 1, a linear Bézier curve is simply a straight line between those two points. The curve is given by and is equivalent to linear interpolation. The t in the function for a linear Bézier curve can be thought of as describing how far B(t) is from P 0 to P 1. For example when t=0.25, B(t) is one quarter of the way from point P 0 to P 1. As t varies from 0 to 1, B(t) describes a straight line from P 0 to P 1. 20
Quadratic Bézier curves A quadratic Bézier curve is the path traced by the function B(t), given points P 0, P 1, and P 2, which can be interpreted as the linear interpolant of corresponding points on the linear Bézier curves from P 0 to P 1 and from P 1 to P 2 respectively. Rearranging the preceding equation yields: Quadratic Bézier curves For quadratic Bézier curves one can construct intermediate points Q 0 and Q 1 such that as t varies from 0 to 1: Point Q 0 varies from P 0 to P 1 and describes a linear Bézier curve. Point Q 1 varies from P 1 to P 2 and describes a linear Bézier curve. Point B(t) varies from Q 0 to Q 1 and describes a quadratic Bézier curve. 21
Cubic Bezier Curve Writing B Pi,Pj,Pk (t) for the quadratic Bézier curve defined by points P i, P j, and P k, the cubic Bézier curve can be defined as a linear combination of two quadratic Bézier curves: The explicit form of the curve is: For some choices of P 1 and P 2 the curve may intersect itself, or contain a cusp. Cubic Bezier Curve For cubic curves one can construct intermediate points Q 0, Q 1, and Q 2 that describe linear Bézier curves, and points R 0 & R 1 that describe quadratic Bézier curves: 22
Quartic Bezier Curve For fourth-order curves one can construct intermediate points Q 0, Q 1, Q 2 & Q 3 that describe linear Bézier curves, points R 0, R 1 & R 2 that describe quadratic Bézier curves, and points S 0 & S 1 that describe cubic Bézier curves: Fifth Order Bezier Curve For fifth-order curves, one can construct similar intermediate points. 23
Bezier curve defined by 4 points Pull by coincident control points Bezier curve defined by 4 points Closed Bezier curve Influence of point position 24
B-Splines A fundamental theorem states that every spline function of a given degree, smoothness, and domain partition, can be uniquely represented as a linear combination of B- splines of that same degree and smoothness, and over that same partition. B-splines were investigated as early as the 19th century by Nikolai Lobachevskj. The term B-spline was coined by Isaac Jacob Schoenberg. B-Spline Synthetic Curves B-spline curves are specified by giving set of coordinates, called control points, which indicates the general shape of the curve. B-splines can be either interpolating or approximating curves. Interpolation splines used for construction and to display the results of engineering. 25
B-Spline Synthetic Curves Approximation B-spines defined as linear and 2nd degree and the flexibility is provided by the basic functions with (n+1) control points, B-splines are defined as P : set of control points u : knot vector k : spline s degree B-Spline Synthetic Curves P is the set of control points as shown in Figure, N is the B-spline blending functions and they are defined as, 26
Effect of curve order The range of u is related to the number of control points and the knot vectors. The effect of the degree of the B-spline curves on the shape of the curve is shown in Figure. B- spline curve lays in the control polygon. Basic Functions Basic functions defined from the knot vector U={u 0...u m } and for i=0, m-1, u u i+1. u i knot vector, U is the set of knot vectors. Specifically, a B-spline order p (or degree p-1) basic functions are defined as N i,p (u). The derivative of the B-spline curve and the k th derivative of the function is, 27
B-Spline Synthetic Curves With (m+1) control points, there are always (n=m+p-1) basic functions. The basis functions are 1, at the end points of the curve defined as a and b. If there s no other definition, then a=0 and b=1. {P i } the set of the control points forms the control polygon from Figure. P 3 P 2 Control Polygo n P 1 B-spline curve N 0,p (a)=1 N n,p (b)=1 P 4 Linear, quadratic, cubic B-spline 28
Influence of control point position Blending functions for linear B-spline 29
Quadratic B-spline blending fn (k=3) Non-uniform rational B-spline NURBS is a mathematical model commonly used in computer graphics for generating and representing curves and surfaces. It offers great flexibility and precision for handling both analytic and modeled shapes. NURBS are commonly used in CAD, CAM and CAE and are part of numerous industry standards, such as IGES, STEP, ACIS, and PHIGS. 30
Non-uniform rational B-spline Real-time, interactive rendering of NURBS curves and surfaces was first made commercially available on Silicon Graphics workstation in 1989. In 1993, the first interactive NURBS modeller for PCs, called NURBS, was developed by CAS Berlin Today most professional computer graphics applications available for desktop use offer NURBS technology, which is most often realized by integrating a NURBS engine from a specialized company. Non-uniform rational B-spline Ø Most modern CAD systems use the NURBS curve representation scheme. Ø Uniformity deals with the spacing of control points. Ø Rational functions include a weighting value at each control point for effect of control point.ü Ø very popular due to their flexibility in curve generation. 31
B-spline curves Piecewise Polynomials Approximating Splines B 0,1 B 1,1 B 2,1 B 3,1 B 4,1 B 5,1 B 6,1 B 0,2 B 1,2 B 2,2 B 3,2 B 4,2 B 5,2 B 0,3 B 1,3 B 2,3 B 3,3 B 4,3 B 0,4 B 1,4 B 2,4 B 3,4 NURBS Non-uniform rational B-spline NURBS permit definition of surfaces from ratios of polynomials. (Rational functions permit much better control over the derivatives of curves, hence the surface curvature, than polynomials alone.) The sphere primitive is shown to the left in wire frame and shaded views. 32
Why NURNS is used Flexibility to create sculptural shapes Tension to keep surfaces smooth and taught Alignment to create smooth, invisible joins 33