The PDF documents at our site are copyrighted but not subject to the liberal terms of the Boost License . You may download them for your personal use and you may set hyperlinks to them; however, the documents may not be posted online by anyone other than us. These are living documents that are updated over time. Unauthorized copies posted to the Internet run the risk of not being updated by the poster when we update ours.

Updates to the PDF Documents
The PDF Documents

Updates to the PDF Documents.

July 23, 2016. A new document that describes the robust algorithm for computing the eigenvalues and eigenvectors of a 2x2 symmetric matrix. The source code is in GteSymmetricEigensolver2x2.h but is also listed in this document.
May 31, 2016. This is a major revision of the document. The new version shows how to estimate vertex tangents and vertex normals for a parameterized mesh (or mesh with texture coordinates) at each vertex using the entire set of triangles that share the vertex. The algorithm is useful for constructing the vertex normals and tangents whose interpolated values are used in a pixel shader for tangent-space normal mapping. An implementation of these algorithms will be posted with the update GTEngine version 2.6. This construction may be used as a replacement for the one-triangle (biased) estimate in the document Estimating a Tangent Vector for Bump Mapping. The bump mapping document has been removed but the link to it is now redirected to the new version of this document.
BumpMapping.pdf ("Estimating a Tangent Vector for Bump Mapping")

May 31, 2016. The pseudocode listed header files for GTEngine but the body of main had code style and references to Wild Magic 5 code. Also, the file GteIntpAkimaNonuniform1.h is referenced in the pseudocode but is missing from the GTEngine distribution. The source file has been uploaded to the website but will be included in the update for GTEngine version 2.6.
April 24, 2016. Removed the references to Wild Magic 4 code and added pseudocode for the actual fitting algorithm. After posting the revision, a minor change was added in the pseudocode (changed an occurrence of circles[] to centers[]).
April 7, 2016. Fixed a typographical error in the T(s) equation on page 4 [θ(t) needed to be θ(s)].
April 2, 2016. At the request of a reader, the document was rewritten again, this time eliminating the discussion of the min-heap speed up that was interleaved in the algorithm. The discussion is easier to follow and it is clearer now how the nesting of cycles is handled. More pseudocode was added. The GTEngine source code (GteMinimalCycleBasis.h) was rewritten to match the document. Without the min-heap support, the performance is still reasonable. Running on an Intel i7-4790 CPU (3.6 GHz), a graph with 32K vertices and 32K edges required 12 seconds to complete, including the validation that the graph is planar. And this run used exact rational arithmetic.
March 14, 2016. Fixed a broken link to source code and a minor typographical error.
March 7, 2016. This is a complete rewrite of the document The Minimal Cycle Basis for a Planar Graph. The PDF name is the same but the title is new (using correct terminology). The Wild Magic 5 sample application was ported, but testing showed that the implementation was incorrect. The GTEngine implementation has been tested more heavily, and code was added to test whether the input graph is planar (requires exact arithmetic).
December 3, 2015. Fixed broken links to source code.
November 1, 2015. Provided more details for the construction of the Green's functions. Fixed some typographical errors and incorrect equation references. Added comments about the unboundedness of the Green's functions for dimensions 4 and larger.
September 20, 2015. Fixed the pseudocode regarding intersections and overlaps. the constructions.
August 16, 2015. Fixed several typographical errors and added some clarification to the constructions.
August 1, 2015. Fixed several typographical errors and added some clarification to the constructions.
July 25, 2015. A new document describing how to compute the minimum-volume box that contains a set of 3D points.
June 23, 2015. Fixed some typographical errors.
May 31, 2015. Major revision of the content of the documents for computing the distance from a 3D circle to a point, line, or circle. The documents have been combined into a single new document. The source code for the distance queries was updated to match the document, and unit tests were added internally to verify the results. The old documents have been removed, but the links to them are now redirected to the new document.
DistancePoint3Circle3.pdf ("Distance Between Point and Circle or Disk in 3D")
DistanceLine3Circle3.pdf ("Distance Between Line and Circle or Disk in 3D")
DistanceCircle3Circle3.pdf ("Distance Between Two Circles in 3D")
DistanceCircle3Disk3.pdf ("Distance Between a Circle and a Disk in 3D")

May 17, 2015. Major revision of the document. The previous version described only the exhaustive O(n2) algorithm for computing the minimum-area rectangle bounding a convex polygon. The revised version describes the O(n) rotating calipers algorithm. The GTEngine source code was modified to match the document, and it now runs nearly twice as fast as the previous code (that did not do an efficient job of maintaining the supporting vertex information).
May 13, 2015. Removed the document FastInverseSqrt.pdf ("Fast and Accurate Inverse Square Root"). Enough is enough for wondering where the magic number came from (see the Wikipedia page for details about fast inverse square root). Added the following document for approximating the inverse square root function using a minimax approach. The algorithm is of interest because it establishes a mathematical pattern that can be used for other function approximations. Its application to inverse square root is of interest only for hardware that does not have support for the operation (current generation floating-point units do have support).
April 25, 2015. Removed documents that were not accessed frequently, are not particularly important, need significant rewriting, or have been rewritten and the old links redirected to the new files.
Bisection.pdf ("Bisection in 1D, 2D, and 3D")
CompressedUnitVectors.pdf ("Compressed Unit Vectors")
CordicMethods.pdf ("CORDIC Methods")
CollinearityTest.pdf ("A Collinearity Test Independent of Point Order")
CommandLineParsing.pdf ("Command Line Parsing")
DistancePointToQuadratic.pdf ("Distance from Point to a General Quadratic Curve or a General Quadric Surface")
EigenSymmetric3x3.pdf ("Eigensystems for 3x3 Symmetric Matrices (Revisited)")
EigenSymmetricNxN.pdf ("Eigensystems for Symmetric Matrices")
GeometricInvariance.pdf ("Geometric Invariance (vector fields, Lie algebra, prolongations)")
LeftHandedToRightHanded.pdf ("Conversion of Left-Handed Coordinates to Right-Handed Coordinates")
OdeNumericalMethods.pdf ("Numerical Methods for Ordinary Differential Equations")
PdeNumericalMethods.pdf ("Numerical Methods for Partial Differential Equations")
PointFromDistances.pdf ("Reconstructing a Point from Distances")
Polysolids.pdf ("Polysolids and Boolean Operations")
SpecialFunctions.pdf ("Special Functions")
SphericalHarmonics.pdf ("Spherical Harmonics")
SubdivideParabola.pdf ("Subdivision of a Parabolic Segment by Arc Length")

April 25, 2015. Slight clean-up of the document and a fix to several equation references.
April 24, 2015. Expanded the document to include more NURBS representations. In particular, there is a degree-3 representation for a full circle that uses an internal repeated knot to splice together two half-circles. There is a tensor product NURBS surface, degree-3 in each parameter, that uses the same idea of an internal repeated knot to splice together two hemispheres.
April 23, 2015. Major rewrite of the document on converting between coordinate systems. The original version of this document was entitled Conversion of Left-Handed Coordinates to Right-Handed Coordinates and was written to handle the conversion of LightWave coordinate systems (left-handed) to Wild Magic coordinate systems (right-handed). The process was specific to LightWave's choice of representing rotations using Euler angles, and the discussion included how to deal with cameras, lights, and transformation hierarchies. The new version is the general process of converting between any two coordinate systems. An implementation is provided that automates the process.
April 20, 2015. Added significant improvements to the document for the test-intersection and find-intersection queries for ellipses and to the document for computing the area of intersection of ellipses. The source code implementation for the test-intersection and find-intersection queries were updated to use the details described in the document. Implemented the query for area of intersection of ellipses.
April 4, 2015. Rewrote the document for computing the roots of low-degree polynomials. The new document goes into great detail about the classification of roots (real or non-real, multiplicities) and how to use exact rational arithmetic to correctly classify the roots in a program. This leads to more robust root finding using the closed-form expressions for polynomials of degrees 2, 3, and 4. The motivation for the revisions was based on trying to compute intersections of ellipses, and the nonrobust root finder for quartic polynomials created many problems numerically.
April 2, 2015. Fixed the algorithm description for conversion from a BSRational object to a floating-point type. When the round-up step causes a carry-out, so to speak, from the trailing significand, a block of code was executed to set w to 1 and adjust p-q. This was incorrect and, in fact, not necessary because w is not used as the trailing significand in the conversion.
March 29, 2015. Added some definitions for variables used in the equations in the section Separation Tests Involving Other Directions. Converted the verbatim pseudocode to use lstlisting format.
March 10, 2015. Fixed typographical errors and added some clarification.
January 26, 2015. Fixed an error in the description for region 2 and added some clarification about the sign tests for the partial derivatives of R(s,t).
January 5, 2015. Added comments about fitting lines y = a*x + b and planes z = a*x + b*y + c, mentioning that the numerical problem for solving the linear equations in the coefficients is ill conditioned unless you compute the averages of the data points first and subtract those averages from the data points before solving the linear system.
January 4, 2015. A new document describing the test-intersection query for boxes and cones.
December 28, 2014. Modified the document on line-cone intersection to give greater detail. The source code was updated accordingly.
December 12, 2014. The document was rewritten to add more details about the algorithm. The source code was modified accordingly.
December 7, 2014. A new document describing a variation of an iterative eigensolver for symmetric 3x3 matrices. A source code implementation and a sample application are provided. The GTEngine code now uses only the iterative solvers for symmetric matrices; we also have an iterative implementation for the singular value decomposition (SVD).
November 27, 2014. The pseudocode for computing the fitted cylinder subtracted the input point average for numerically stable computations. The returned center needed the average added to it.
November 25, 2014. Overhauled the arbitrary precision library to improve the performance and to improve readability. The unsigned integer arithmetic was factored out of BSNumber into two classes, one for arbitrary precision with storage of type std::vector and one for user-selected fixed precision with storage of type std::array. Both classes share code for the arithmetic logic unit. Many computations are now performed in-place to avoid expensive allocation, deallocation, and memory copies. A new PDF is posted that greatly expands on the library compared to the discussion in the GPGPU Programming for Games and Science book. The document serves as a discussion about the design of the library and a reference for how to use it. Examples are provided for using BSPrecision to determine the template parameter of UIntegerFP<N> that represents the maximum number of bits required to compute the exact results for a sequence of expressions.
November 6, 2014. Implemented a robust algorithm for computing the distance between line segments in any dimension. Revised the PDF for computing distance between segments in 3D to describe the new algorithm. A GPU implementation is available in the sample application.
October 20, 2014. The volume equation for a dodecahedron was in error.
August 15, 2014. Fixed a typographical error in Equation (14). Replaced the LaTeX verbatim commands with lstlisting commands for more readable pseudocode.

The PDF Documents. The hierarchical organization matches that of the GTEngine.vcxproj project file.

Mathematics/Computational Geometry:
Mathematics/Curves, Surfaces, and Volumes:
Mathematics/Geometric Primitives:
Mathematics/Numerical Methods: