Gerald Farin

A survey of curve and surface methods in CAGD

B~zier Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4. History . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5. Polynomial curve forms . . . . . . . . . . . . . . . . 6 6. Bernstein polynomials . . . . . . . . . . . . . . . . . 7 71 The de Casteljau algori thm . . . . . . . . . . . . . . 8 8. Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 9 9. Degree elevation . . . . . . . . . . . . . . . . . . . . . 9 10. Subdivision . . . . . . . . . . . . . . . . . . . . . . . . . 10 11. Composi te B6zier curves . . . . . . . . . . . . . . . 10 12. Visual continuity . . . . . . . . . . . . . . . . . . . . . 11 13. Smooth interpolat ion by piecewise cubics . . . . 12 14. Nonparametr ic B6zier curves . . . . . . . . . . . . 14 15. Integration . . . . . . . . . . . . . . . . . . . . . . . . . 14

Triangular Berstein-Be´zier patches

4. Hermite Interpolants . . , . . . . . . . . . . . . . 104 4.1. The Co nine parameter interpolant ...... 104 4.2. C’ quintic interpolants 104 4.3. The general case 106 5. Split Triangle Interpolants . . . . . . . . . 107 5.1. The C’ Clough-Tocher interpolant . . _ . . 108 5.2. Limitations of the Clough-Tocher split . 110 5.3. The C’ Powell-Sabin interpolants . . . . 111 5.4. C’ Split square interpolants . . . . . . . . . 112

Surface/surface intersection

Abstract Finding the intersection of two surfaces is important for many Computer Aided Design tasks concerned with surface modeling. An adaptive algorithm is developed for finding the intersection curve(s) of pairs of rectangular parametric patches which are continuously differentiable. The balance between robustness and efficiency of the algorithm is controlled by a set of tolerances. A suite of examples concludes the paper.

Cubic Spline Interpolation

In this chapter, we discuss what is probably the most popular curve scheme: C2 cubic interpolatory splines. We have seen how polynomial Lagrange interpolation fails to produce acceptable results. On the other hand, we saw that cubic B-spline curves are a powerful modeling tool; they are able to model complex shapes easily. This “modeling” is carried out as an approximation process, manipulating the control polygon until a desired shape is achieved. We will see how cubic splines can also be used to fulfill the task of interpolation, the task of finding a spline curve passing through a given set of points. Cubic spline interpolation was introduced into the CAGD literature by J. Ferguson [183] in 1964, while the mathematical theory was studied in approximation theory (see de Boor [112] or Holladay [264]). For an outline of the history of splines, see Schumaker [423]. Because of the subjects importance, we present two entirely independent derivations of cubic interpolatory splines: the B-spline form and the Hermite form.

Automatic fairing algorithm for B-spline curves

Abstract An algorithm is presented for locally fairing B-spline curves. The algorithm is based on repeatedly removing and reinserting knots of the spline. These knots are selected automatically by means of a fairness criterion. The proposed scheme involves a new knot removal algorithm that compares favourably to existing ones.

3D face authentication and recognition based on bilateral symmetry analysis

We present a novel and computationally fast method for automatic human face authentication. Taking a 3D triangular facial mesh as input, the approach first automatically extracts the bilateral symmetry plane of the facial surface. The intersection between the symmetry plane and the facial surface, namely the symmetry profile, is then computed. Using both the mean curvature plot of the facial surface and the curvature plot of the symmetry profile curve, three essential points of the nose on the symmetry profile are automatically extracted. The three essential points uniquely determine a Face Intrinsic Coordinate System (FICS). Different faces are aligned based on the FICS. The symmetry profile, together with two transverse profiles, composes a compact representation, called the SFC representation, of a 3D face surface. The face authentication and recognition steps are finally performed by comparing the SFC representations of the faces. The proposed method was tested on 382 face surfaces, which come from 166 individuals and cover a wide ethnic and age variety. The equal error rate (EER) of face authentication on scans with variable facial expressions is 10.8%. For scans with normal expression, the ERR is 0.8%.

Algorithms for rational Bézier curves

Abstract A recursive algorithm for the evaluation of rational Bezier curves is presented; it consists of a construction that works with a constant cross ratio. This geometric principle is carried over to other algorithms.

Curvature and the fairness of curves and surfaces

The use of curvature plots for the design of curves that have to meet aesthetic requirements is discussed. The aim is to emphasize the usefulness of curvature as a measure for curve fairness. A local method to optimize the curvature plot of a given cubic spline curve is presented. It automatically determines where the curve is to be faired and can be applied repeatedly. A straightforward generalization to surfaces is easy to formulate.<<ETX>>

Fairing cubic B-spline curves

Abstract Algorithms are presented for locally fairing a B-spline curve in order to produce a pleasant curvature plot. The methods are based on inversions of the knot insertion algorithm for B-spline curves.

Discrete Coons patches

We investigate surfaces which interpolate given boundary curves. We show that the discrete bilinearly blended Coons patch can be defined as the solution of a linear system. With the goal of producing better shape than the Coons patch, this idea is generalized, resulting in a new method based on a blend of variational principles. We show that no single blend of variational principles can produce “good” shape for all boundary curve geometries. We also discuss triangular Coons patches and point out the connections to the rectangular case.