forked from KolibriOS/kolibrios
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
|
|
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
|