Applications of Laguerre geometry in CAGD

Abstract We briefly introduce to the basics of Laguerre geometry and then show that this classical sphere geometry can be applied to solve various problems in geometric design. In the present part, we focus on applications of the cyclographic model of Laguerre geometry and the cyclographic map. It relates the medial axis and Voronoi curves/surfaces to special surface/surface intersection and the corresponding trimming procedures to hidden line removal. Rational canal surfaces are treated as cyclographic images of rational curves in R 4. This leads to a simple control structure for rational canal surfaces. Its practical use is demonstrated at hand of modeling techniques with Dupin cyclides.

[1]  J. Oden,et al.  The Mathematics of Surfaces II , 1988 .

[2]  Christoph Mäurer.  Rationale Bézier-Kurven und Bézier-Flächenstücke auf Dupinschen Zykliden , 1997 .

[3]  Karl Strubecker.  Vorlesungen über darstellende Geometrie , 1914 .

[4]  Wilhelm Blaschke.  Untersuchungen über die Geometrie der Speere in der Euklidischen Ebene , 1910 .

[5]  Franz-Erich Wolter.  Cut Locus and Medial Axis in Global Shape Interrogation and Representation , 1995 .

[6]  Tom Lyche,et al.  Mathematical methods in computer aided geometric design , 1989 .

[7]  W. Blaschke.  Vorlesungen über Differentialgeometrie , 1912 .

[8]  Josef Hoschek,et al.  Fundamentals of computer aided geometric design , 1996 .

[9]  Evan C. Sherbrooke.  3-D shape interrogation by medial axis transform , 1996 .

[10]  W Lu.  RATIONALITY OF THE OFFSETS TO ALGEBRAIC CURVES AND SURFACES , 1994 .

[11]  C. Hoffmann.  Computer Vision, Descriptive Geometry, and Classical Mechanics , 1992 .

[12]  Rida T. Farouki,et al.  Analysis of the offset to a parabola , 1995, Comput. Aided Geom. Des..

[13]  K. Brauner.  Vorlesungen über Differentialgeometrie III , 1930 .

[14]  Wolfgang Böhm,et al.  On cyclides in geometric modeling , 1990, Comput. Aided Geom. Des..

[15]  M. J. Pratt,et al.  Cyclides in computer aided geometric design , 1990, Comput. Aided Geom. Des..

[16]  Rida T. Farouki,et al.  Computing Point/Curve and Curve/Curve Bisectors , 1992, IMA Conference on the Mathematics of Surfaces.

[17]  Hellmuth Stachel,et al.  Circular pipe-connections , 1988, Comput. Graph..

[18]  Andrew Zisserman,et al.  Viewpoint-invariant representation of generalized cylinders using the symmetry set , 1995, Image Vis. Comput..

[19]  Hwan Pyo Moon,et al.  MATHEMATICAL THEORY OF MEDIAL AXIS TRANSFORM , 1997 .

[20]  Helmut Pottmann,et al.  Computing Rational Parametrizations of Canal Surfaces , 1997, J. Symb. Comput..

[21]  Debasish Dutta,et al.  Intuitive procedure for constructing geometrically complex objects using cyclides , 1994, Comput. Aided Des..

[22]  Hyeong In Choi,et al.  New Algorithm for Medial Axis Transform of Plane Domain , 1997, CVGIP Graph. Model. Image Process..

[23]  Thomas E. Cecil.  Lie sphere geometry , 1992 .

[24]  Ivan Herman,et al.  Computer Graphics and Mathematics , 1992, Focus on Computer Graphics.

[25]  Helmut Pottmann,et al.  A Laguerre geometric approach to rational offsets , 1998, Comput. Aided Geom. Des..

[26]  Christoph Mäurer.  Rational curves and surfaces on Dupin ring cyclides , 1996 .

[27]  Y. L. Srinivas,et al.  Rational parametric representation of parabolic cyclide: formulation and applications , 1995 .

[28]  Gudrun Albrecht,et al.  Construction of Bézier rectangles and triangles on the symmetric Dupin horn cyclide by means of inversion , 1997, Comput. Aided Geom. Des..

[29]  Rida T. Farouki,et al.  Curves and surfaces in geometrical optics , 1992 .