Theory of Piecewise Circular Curves of T.F. Banchoff - Case Study Example

In the plane a smooth piecewise circular curve is a curve made from arcs of circles such that at points where two arcs join the tangent lines agree. Properties of such curves are considered in this paper." PIECEWISE CIRCULAR CURVES Banchoff and Giblin define the piecewise circular curve thus: “A PC curve is given by a finite sequence of circular arcs or line segments with the endpoint of one arc coinciding with the beginning point of the next.” Jaroslaw R. Rossignac and Aristides A. G Requicha view PCCs as “composed of line and arc segments and exhibit first-degree geometric continuity (GI), i.e., they have continuous unit tangent directions.” These piecewise circular curves or PC curves have according to Banchoff and Giblin a number of ‘special and attractive properties’ While they are almost as easy to define as polygons, PCCs are more versatile as they have a well defined tangent at every point. They are said to be ‘smooth’ if ‘the directed tangent line at the end of one arc coincides with the directed tangent line at the beginning of the next.’ Also no arc degenerates to a single point in a smooth PCC. When PCCs are used to approximate smooth curves, not only is the approximation point wise close but also the tangent lines at the points of the smooth curve are approximated by the tangent lines of the PCC. Piecewise circular curves occur naturally as the solutions of a number of variational problems related to isoperimetric problems. A classical example is finding the shortest length enclosing a fixed area. The result here is a circle. When the curve is required to surround a fixed pair of points, then the curve of shortest length enclosing a given area will be either a circle or a lens. The Reuleaux "triangle" is a three-arc PC curve enclosing an equilateral triangle, with each radius equal to the length of a side of the triangle. PC ellipse If it is required that a curve of fixed length L surround a given pair of discs of the same radius, then, for a certain range of values of L, the curve that encloses the greatest area is a smooth convex PC curve consisting of two arcs on the boundary circles of the discs and two arcs of equal radius tangent to both discs. Such four-arc convex PC curves have long been used in engineering drawing for approximating ellipses, and we call such a curve a PC ellipse. One special PC ellipse is the convex envelope. Banchoff and Giblin establish some natural properties of PCCs that find application in computer applications: The parallel curves of a PCC are PCCs since the parallel curves of circular arcs are circular arcs. This is especially important in computer graphics. Parallel curves are also called offset curves as they are obtained by moving away fro the curve. However, if the radii are decreased a set of inner parallel curves will be obtained. Depending on whether the number of cusps is even or odd, closed PC curves with n arcs and a given evolute polygon fall into two classes. If the number of cusps is even, there is always a relation between the side-lengths of the polygon of the form and the radii can be varied to give a family of parallel PC curves. If the number of cusps is odd, there is no restriction on the sides. A closed curve is obtained with a given evolute polygon and fixed radii. A remarkable example of a symmetry set of a PCC is seen in a transition called a moth in the study of symmetry sets of one-parameter families of plane curves. Bitangent circles either grow out or come together and disappear when a four-arc PCC is perturbed. The Piecewise Circular Curve is an elementary class of curve for which all general phenomena that appear in the study of plane polygons are present. PIECEWISE CIRCULAR CURVES for SOLID MODELERS To meet the challenges of the modern world, computer software engineers work constantly towards providing designers with systems that allow them to see as efficiently as possible graphic images of their designs, to selectively edit them and check their efficiency during the design process itself. Solid models as the digital representations of existing or imagined physical objects are called, are used in almost every sphere of life today from a great variety of design and manufacturing activities like computer-aided design and engineering analysis to animation, medical testing, visualization of scientific research and so on. Because its importance is only increasing, solid modeling is tending towards becoming increasingly user-friendly thus making the layman more productive. To create efficient solid models requires an efficient modeling system that is essentially a computer programme that provides facilities for storing and manipulating data that represent the geometry of objects or a combination of objects. These can be created by a person through a graphic user interface (GUI) or by the application of software through an application programming interface (API). Given the huge demand for efficient geometric modeling systems, solid modeling systems or solid modelers therefore need to be more versatile than ever before to be able to represent a wide range of objects, as quickly and efficiently as well as easily as possible. The most widely used models in the Geometric Modeling Systems are: Constructive Solid Geometry (CSG) and Boundary Representations (BRep). However, these have their limitations too. Boolean operations are cumbersome in Boundary Representations, while realistic visualizations and volumetric properties computation is complex when the Constructive Solid Geometry model is used. With the result, a number of hybrid systems have evolved, which support both schemes and perform operations in the best suited model. Yet , algorithms for the boundary evaluation of CSG trees are complex, and still little is known on algorithms for the inverse conversion, from BRep to CSG. Other representation schemes as the octtrees work well for Boolean operations but require a large memory. Navazo, Fontdecaba, Brunet have studied the Extended Octtree model as an intermediate tool in the conversions between CSG trees and BRep. As the focus of this paper is on the properties of Piecewise Circular Curves, the efficiency of their application in fundamental geometric computations and the advantages of their use in solid modelers are addressed here. Taking the search for more efficient solid modeling systems further Rossignac and Requicha presented a study reporting an increase in the geometric coverage of modelers supporting Boolean operations. Using the peculiar properties of Piecewise-Circular Curves, they were able to propose an alternative scheme for approximating intersections that involve toroidal or blending surfaces. Modern solid modelers must be able not only to represent a wide range of objects but also support Boolean operations on solids. These operations are very useful for defining solids through CSG (Constructive Solid Geometry), for identifying spatial interferences, and for modeling physical processes such as machining and fabrication. Computing boundaries of solids defined through Boolean operations requires algorithms for the intersection of two surfaces or a curve and a surface. Conventionally, modelers employ closed-form parametric expressions for the curves of intersection and compute their intersections by finding the roots of low-degree polynomials. However, the curves that result from intersections involving complex surfaces usually cannot be expressed in closed form and so are approximated by cubic splines that interpolate points lying on the true intersections. Cubic splines are efficient in that they exhibit second-degree continuity. However, they are expensive to process in solid modeling computations. Rossignac and Requicha give up the second degree continuity available when dealing with cubic splines to get computational simplicity. They present a method for interpolating three dimensional points and associated unit tangent vectors by smooth space curves composed of straight line segments and circular arcs. These curves or PCCs (Piecewise-Circular Curves) have continuous unit tangents and can be used in efficient algorithms for performing fundamental geometric computations, such as determining the minimum distance from a point to a curve or the intersection of a curve and a surface. Current solid modelers use either of the two following techniques to support Boolean operations on objects bounded by natural surfaces: 1. All surfaces are approximated by planes and these are used for calculating intersections. While this is acceptable as a representation of the object it is not accurate enough for numerical calculations. 2. Curves of intersection of natural quadrics are expressed through the parametric equations x = x (t), y = y (t), z = z (t) of a curve which are substituted in the implicit equation F(x, y, z) = 0 which is solved for t. this is fairly accurate but not applicable to toroidal or blending surfaces. Here the curves of intersection are approximated with cubic splines. This results in complex high-degree equations that need to be solved numerically. This is unwieldy and inefficient. Thus even for a solid of moderate complexity thousands of calculations are necessary after determining the curves of intersection and classifying them according to their segments which are inside, outside or on the boundary of the objects. The alternative method proposed by Rossignac and Requicha employs curve approximation of surface intersections. Curve approximation has been employed for quite a while now with Bezier and B-spline parametric curves being most popular. These allow for parametric formulation, second degree continuity and local control. Seeing parametric formulation as the most important feature and trading second degree continuity for ‘computational simplicity’, Rossignac and Requicha favor PCCs over cubic splines for the following reasons: 1. PCCs are composed of line and arc segments and exhibit first-degree geometric continuity (GI). In other words they have continuous unit tangent directions. 2. PCC intersections with any natural surface, including tori, result in fourth-degree equations which can be solved analytically. 3. Piecewise-Circular Curves compare well with piecewise-cubic curves as regards the computational complexity of the algorithms. However, the bi-arc spans of the piecewise-circular curve are better for approximating space curves than the cubic spans. 4. Speed-ups based on the convex-hull properties of piecewise-cubic curves may be used equally well with PCCs Thus Piecewise-Circular Curves have many applications in geometric modeling. They lend themselves well to approximation of intersections in solid modelers that support Boolean operations, leading to simple algorithms thus allowing for ease of computation. Furthermore, Piecewise-Circular Curves also allow solid modelers to easily incorporate valid sweeping, blending, growing, and shrinking operations. References: Thomas Banchoff and Peter Giblin, On the Geometry of Piecewise Circular Curves, The American Mathematical Monthly, Vol. 101, No. 5 (May, 1994), pp. 403-416, 29 January, 2009. http://poncelet.math.nthu.edu.tw/disk5/js/linkage/13.pdf Rossignac,J.R. and Requicha, A. A. G, Piecewise-Circular Curves for Geometric Modeling, IBM Journal of Research and Development (1987) p296-313, 29 January, 2009. http://www.research.ibm.com/journal/rd/313/ibmrd3103D.pdf Jarek R. Rossignac, GVU Center, College of Computing,. Georgia Institute of Technology, Atlanta and Aristides A.G. Requicha, Computer Science Department, University of Southern Californai, Los Angeles, SolidModeling, 2 February, 2009. http://www.gvu.gatech.edu/~jarek/papers/SolidModelingWebster.pdf Isabel Navazo, Josep Fontdecaba, Pere Brunet, Extended Octrees, Between Csg Trees And Boundary Representations, Eurographics’87, Elsevier Science Publishers B.V. (North-Holland), © Eurographics Association, 1987, 3 Feb 2009. https://www.eg.org/EG/DL/Conf/EG87/papers/EUROGRAPHICS_87 pp239-247_abstract.pdf Piecewise Circular Curve, © 1996-9 Eric W. Weisstein,1999-05-25, 29 January 2009. http://bbs.sachina.pku.edu.cn/stat/mathworld/math/p/p314.htm Read More