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