By Michael Beeson (auth.), Tetsuo Ida, Jacques Fleuriot (eds.)
ISBN-10: 3642406718
ISBN-13: 9783642406713
ISBN-10: 3642406726
ISBN-13: 9783642406720
This e-book constitutes the completely refereed post-workshop court cases of the ninth overseas Workshop on computerized Deduction in Geometry, ADG 2012, held in Edinburgh, united kingdom, in September 2012. the ten revised complete papers awarded including 2 invited papers have been rigorously chosen in the course of rounds of reviewing and development from the lectures given on the workshop. The convention represents a discussion board to switch rules and perspectives, to provide learn effects and growth, and to illustrate software program instruments on the intersection among geometry and automatic deduction; the scope of the ADG 2012 furthermore has been improved to hide themes in dynamic geometry.
Read or Download Automated Deduction in Geometry: 9th International Workshop, ADG 2012, Edinburgh, UK, September 17-19, 2012. Revised Selected Papers PDF
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Yang, D. Wang, and H. Hong - Fig. 1. Dot plots (left) and angular speeds (right) of two parameterizations of the same curve. The first row shows one parameterization with discontinuous angular speed and the second row shows the other with continuous angular speed. The red dots on the left-hand side correspond to the segment points. but its angular speed sometimes lacks continuity, causing sudden changes in the density of points in plotting (see Fig. 1). In this third paper, we address the problem of discontinuity by restricting piecewise M¨obius transformations to be C 1 .
99. 42 J. Yang, D. Wang, and H. 5 Algorithm Now we summarize the ideas and results discussed above into an algorithm. Algorithm 1 (C 1 Reparameterize) Input: p, a rational parameterization of a plane curve; δ, a real number greater than 1. Output: p∗ , a C 1 piecewise-rational reparameterization of p; u∗ , the uniformity of p∗ . 1. Compute ωp and μp using (1), up using (2) and ωp . 2. mnew ← Id; unew ← up . 3. Do mold ← mnew ; uold ← unew ; mnew , unew ← Improve(mold , ωp , ωp , μp ) Until unew /uold < δ.
The benchmark curves, chosen from [4, 14, 16], are all rational and their angular speeds are nonzero over [0, 1]. For the curves taken from [4], the involved parameters are specialized to concrete values. The list of the curves (in Maple format) is available from the authors upon request. Table 1. 01). In the table, u = Uniformity, N = Number of pieces, D Discontinuity (average), T = Time (seconds). 078 Table 1 presents the experimental results of reparameterization using standard M¨obius transformations, C 0 piecewise M¨ obius transformations and C 1 piecewise M¨ obius transformations.
Automated Deduction in Geometry: 9th International Workshop, ADG 2012, Edinburgh, UK, September 17-19, 2012. Revised Selected Papers by Michael Beeson (auth.), Tetsuo Ida, Jacques Fleuriot (eds.)
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