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authorsanine <sanine.not@pm.me>2023-02-12 23:53:22 -0600
committersanine <sanine.not@pm.me>2023-02-12 23:53:22 -0600
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-Here's an effort to document some of the academic work that was
-referenced during the implementation of cairo. It is presented in the
-context of operations as they would be performed by either
-cairo_stroke() or cairo_fill():
-
-Given a Bézier path, approximate it with line segments:
-
- The deCasteljau algorithm
- "Outillages methodes calcul", P de Casteljau, technical
- report, - Andre Citroen Automobiles SA, Paris, 1959
-
- That technical report might be "hard" to find, but fortunately
- this algorithm will be described in any reasonable textbook on
- computational geometry. Two that have been recommended by
- cairo contributors are:
-
- "Computational Geometry, Algorithms and Applications", M. de
- Berg, M. van Kreveld, M. Overmars, M. Schwarzkopf;
- Springer-Verlag, ISBN: 3-540-65620-0.
-
- "Computational Geometry in C (Second Edition)", Joseph
- O'Rourke, Cambridge University Press, ISBN 0521640105.
-
-Then, if stroking, construct a polygonal representation of the pen
-approximating a circle (if filling skip three steps):
-
- "Good approximation of circles by curvature-continuous Bezier
- curves", Tor Dokken and Morten Daehlen, Computer Aided
- Geometric Design 8 (1990) 22-41.
-
-Add points to that pen based on the initial/final path faces and take
-the convex hull:
-
- Convex hull algorithm
-
- [Again, see your favorite computational geometry
- textbook. Should cite the name of the algorithm cairo uses
- here, if it has a name.]
-
-Now, "convolve" the "tracing" of the pen with the tracing of the path:
-
- "A Kinetic Framework for Computational Geometry", Leonidas
- J. Guibas, Lyle Ramshaw, and Jorge Stolfi, Proceedings of the
- 24th IEEE Annual Symposium on Foundations of Computer Science
- (FOCS), November 1983, 100-111.
-
-The result of the convolution is a polygon that must be filled. A fill
-operations begins here. We use a very conventional Bentley-Ottmann
-pass for computing the intersections, informed by some hints on robust
-implementation courtesy of John Hobby:
-
- John D. Hobby, Practical Segment Intersection with Finite
- Precision Output, Computation Geometry Theory and
- Applications, 13(4), 1999.
-
- http://cm.bell-labs.com/who/hobby/93_2-27.pdf
-
-Hobby's primary contribution in that paper is his "tolerance square"
-algorithm for robustness against edges being "bent" due to restricting
-intersection coordinates to the grid available by finite-precision
-arithmetic. This is one algorithm we have not implemented yet.
-
-We use a data-structure called Skiplists in the our implementation
-of Bentley-Ottmann:
-
- W. Pugh, Skip Lists: a Probabilistic Alternative to Balanced Trees,
- Communications of the ACM, vol. 33, no. 6, pp.668-676, 1990.
-
- http://citeseer.ist.psu.edu/pugh90skip.html
-
-The random number generator used in our skip list implementation is a
-very small generator by Hars and Petruska. The generator is based on
-an invertable function on Z_{2^32} with full period and is described
-in
-
- Hars L. and Petruska G.,
- ``Pseudorandom Recursions: Small and Fast Pseurodandom
- Number Generators for Embedded Applications'',
- Hindawi Publishing Corporation
- EURASIP Journal on Embedded Systems
- Volume 2007, Article ID 98417, 13 pages
- doi:10.1155/2007/98417
-
- http://www.hindawi.com/getarticle.aspx?doi=10.1155/2007/98417&e=cta
-
-From the result of the intersection-finding pass, we are currently
-computing a tessellation of trapezoids, (the exact manner is
-undergoing some work right now with some important speedup), but we
-may want to rasterize directly from those edges at some point.
-
-Given the set of tessellated trapezoids, we currently execute a
-straightforward, (and slow), point-sampled rasterization, (and
-currently with a near-pessimal regular 15x17 grid).
-
-We've now computed a mask which gets fed along with the source and
-destination into cairo's fundamental rendering equation. The most
-basic form of this equation is:
-
- destination = (source IN mask) OP destination
-
-with the restriction that no part of the destination outside the
-current clip region is affected. In this equation, IN refers to the
-Porter-Duff "in" operation, while OP refers to a any user-selected
-Porter-Duff operator:
-
- T. Porter & T. Duff, Compositing Digital Images Computer
- Graphics Volume 18, Number 3 July 1984 pp 253-259
-
- http://keithp.com/~keithp/porterduff/p253-porter.pdf