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4349 | Serge | 1 | Here's an effort to document some of the academic work that was |
2 | referenced during the implementation of cairo. It is presented in the |
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3 | context of operations as they would be performed by either |
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4 | cairo_stroke() or cairo_fill(): |
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5 | |||
6 | Given a Bézier path, approximate it with line segments: |
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7 | |||
8 | The deCasteljau algorithm |
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9 | "Outillages methodes calcul", P de Casteljau, technical |
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10 | report, - Andre Citroen Automobiles SA, Paris, 1959 |
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11 | |||
12 | That technical report might be "hard" to find, but fortunately |
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13 | this algorithm will be described in any reasonable textbook on |
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14 | computational geometry. Two that have been recommended by |
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15 | cairo contributors are: |
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16 | |||
17 | "Computational Geometry, Algorithms and Applications", M. de |
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18 | Berg, M. van Kreveld, M. Overmars, M. Schwarzkopf; |
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19 | Springer-Verlag, ISBN: 3-540-65620-0. |
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20 | |||
21 | "Computational Geometry in C (Second Edition)", Joseph |
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22 | O'Rourke, Cambridge University Press, ISBN 0521640105. |
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23 | |||
24 | Then, if stroking, construct a polygonal representation of the pen |
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25 | approximating a circle (if filling skip three steps): |
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26 | |||
27 | "Good approximation of circles by curvature-continuous Bezier |
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28 | curves", Tor Dokken and Morten Daehlen, Computer Aided |
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29 | Geometric Design 8 (1990) 22-41. |
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30 | |||
31 | Add points to that pen based on the initial/final path faces and take |
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32 | the convex hull: |
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33 | |||
34 | Convex hull algorithm |
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35 | |||
36 | [Again, see your favorite computational geometry |
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37 | textbook. Should cite the name of the algorithm cairo uses |
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38 | here, if it has a name.] |
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39 | |||
40 | Now, "convolve" the "tracing" of the pen with the tracing of the path: |
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41 | |||
42 | "A Kinetic Framework for Computational Geometry", Leonidas |
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43 | J. Guibas, Lyle Ramshaw, and Jorge Stolfi, Proceedings of the |
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44 | 24th IEEE Annual Symposium on Foundations of Computer Science |
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45 | (FOCS), November 1983, 100-111. |
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46 | |||
47 | The result of the convolution is a polygon that must be filled. A fill |
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48 | operations begins here. We use a very conventional Bentley-Ottmann |
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49 | pass for computing the intersections, informed by some hints on robust |
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50 | implementation courtesy of John Hobby: |
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51 | |||
52 | John D. Hobby, Practical Segment Intersection with Finite |
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53 | Precision Output, Computation Geometry Theory and |
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54 | Applications, 13(4), 1999. |
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55 | |||
56 | http://cm.bell-labs.com/who/hobby/93_2-27.pdf |
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57 | |||
58 | Hobby's primary contribution in that paper is his "tolerance square" |
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59 | algorithm for robustness against edges being "bent" due to restricting |
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60 | intersection coordinates to the grid available by finite-precision |
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61 | arithmetic. This is one algorithm we have not implemented yet. |
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62 | |||
63 | We use a data-structure called Skiplists in the our implementation |
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64 | of Bentley-Ottmann: |
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65 | |||
66 | W. Pugh, Skip Lists: a Probabilistic Alternative to Balanced Trees, |
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67 | Communications of the ACM, vol. 33, no. 6, pp.668-676, 1990. |
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68 | |||
69 | http://citeseer.ist.psu.edu/pugh90skip.html |
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70 | |||
71 | The random number generator used in our skip list implementation is a |
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72 | very small generator by Hars and Petruska. The generator is based on |
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73 | an invertable function on Z_{2^32} with full period and is described |
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74 | in |
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75 | |||
76 | Hars L. and Petruska G., |
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77 | ``Pseudorandom Recursions: Small and Fast Pseurodandom |
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78 | Number Generators for Embedded Applications'', |
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79 | Hindawi Publishing Corporation |
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80 | EURASIP Journal on Embedded Systems |
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81 | Volume 2007, Article ID 98417, 13 pages |
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82 | doi:10.1155/2007/98417 |
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83 | |||
84 | http://www.hindawi.com/getarticle.aspx?doi=10.1155/2007/98417&e=cta |
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85 | |||
86 | From the result of the intersection-finding pass, we are currently |
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87 | computing a tessellation of trapezoids, (the exact manner is |
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88 | undergoing some work right now with some important speedup), but we |
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89 | may want to rasterize directly from those edges at some point. |
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90 | |||
91 | Given the set of tessellated trapezoids, we currently execute a |
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92 | straightforward, (and slow), point-sampled rasterization, (and |
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93 | currently with a near-pessimal regular 15x17 grid). |
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94 | |||
95 | We've now computed a mask which gets fed along with the source and |
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96 | destination into cairo's fundamental rendering equation. The most |
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97 | basic form of this equation is: |
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98 | |||
99 | destination = (source IN mask) OP destination |
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100 | |||
101 | with the restriction that no part of the destination outside the |
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102 | current clip region is affected. In this equation, IN refers to the |
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103 | Porter-Duff "in" operation, while OP refers to a any user-selected |
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104 | Porter-Duff operator: |
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105 | |||
106 | T. Porter & T. Duff, Compositing Digital Images Computer |
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107 | Graphics Volume 18, Number 3 July 1984 pp 253-259 |
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108 | |||
109 | http://keithp.com/~keithp/porterduff/p253-porter.pdf |