Eric Lengyel, Ph.D.Founder, President, and Chief Technology Officer
Terathon Software LLC Eric Lengyel (pronounced LENG · el, with no Y) is the founder of Terathon Software. He holds a Ph.D. in Computer Science from the University of California, Davis, and a Masters Degree in Mathematics from Virginia Tech. Eric is the creator of the following technologies:
Eric is the author of the new book series Foundations of Game Engine Development, and he is the editor for the Game Engine Gems series. Almost 20 years ago, he wrote the first book focused on mathematics in games, the bestselling Mathematics for 3D Game Programming & Computer Graphics. Eric has also been a longtime member of the editorial board for the Journal of Computer Graphics Techniques, and he has been a major contributor to the successful Game Programming Gems series. Eric is an active researcher in the field of projective geometric algebra. The educational resources that he has created for this subject can be found on projectivegeometricalgebra.org. Eric previously worked in the advanced technology group at Naughty Dog and the graphics and imaging department at Apple. Before those, he was the lead programmer for the fifth installment of Sierra Studio's popular RPG adventure series Quest for Glory. When he's not at his desk, Eric can usually be found running somewhere in the mountains of northern California. 
The first volume of Foundations of Game Engine Development discusses the mathematics needed by
engineers who work on games or other types of virtual simulations. The book begins with conventional treatments
of topics such as linear algebra, transforms, and geometry. Then, it introduces Grassmann algebra and geometric
algebra to provide a much deeper understanding of the subject matter and highlight the places where traditional
arithmetic with vectors, matrices, quaternions, etc., isn't quite correct.

Motion Blur and the
VelocityDepthGradient Buffer Motion blur can be simulated using a velocity buffer in conjunction with a postprocessing shader to render a directional blur for
pixels belonging to moving objects. A basic implementation of this technique produces adequate results for some applications, but it also produces
a fuzzy halo artifact. This gem discusses an improvement to this motion blur technique that eliminates halo artifacts without also affecting cases
where motion blur would be correctly rendered, producing images of much higher quality than is possible with previous techniques.
Moments of Inertia for Common Shapes
Instead of laboriously evaluating a complicated integral to derive the moment of inertia for a particular shape, one may choose to
look it up, but existing references can be difficult to find, and those that do exist are sometimes inaccurate or incomplete. This gem provides the
derivations of the moments of inertia for a variety of common shapes and summarizes them in a handy reference table.


The second volume in the Foundations of Game Engine Development series explores the vast subject of realtime rendering in
modern game engines. The book provides a detailed introduction to color science, world structure, projections, shading, light sources, shadows,
fog, and visibility methods. This is followed by extensive discussions of a variety of advanced rendering techniques that include volumetric effects,
atmospheric shadowing, ambient occlusion, motion blur, and isosurface extraction. Emphasis is placed on practical implementation, and code is included.

A JitterTolerant Rigid Body Sleep Condition
All physics engines exhibit some jitter no matter how good the constraint solver is. This chapter discusses a simple condition
that can be used to determine when it is the proper time to put a rigid body to sleep, and it is highly tolerant to jitter.
Bit Hacks for Games
Game programmers have long been known for coming up with clever tricks that allow various short calculations to be performed more efficiently.
The techniques usually employ some kind of logical bit manipulation, or “bit twiddling”, to obtain a result in a roundabout way with the goal of reducing
the number of instructions, eliminating expensive instructions like divisions, or removing costly branches. This chapter describes a variety of interesting bit hacks
that are likely to be applicable to game engine codebases.


This book, now used as a text in computer graphics courses at many universities
around the world, illustrates the mathematics that a programmer would need to develop
a professionalquality 3D engine. Although the book is geared toward applications
in game development, many of the topics appeal to general interests in 3D graphics.
It starts at a fairly basic level in areas such as vector geometry and linear algebra,
and then progresses to more advanced topics in 3D game programming such as
illumination, visibility determination, and collision detection. Particular attention
is given to derivations of key results, ensuring that the reader is not forced to endure
gaps in the theory. The book assumes a working knowledge of trigonometry
and calculus, but also includes sections that review the important tools
used from these disciplines, such as trigonometric identities, differential
equations, and Taylor series.

The Open Game Engine Exchange Format
This chapter provides a comprehensive overview of the OpenGEX file format.
Fog with a Linear Density Function
The chapter discusses the mathematical details about rendering techniques and visibility culling for halfspace fog with
a linear density function.
Smooth Horizon Mapping
The chapter describes a highquality horizon mapping technique for applying soft shadows to normalmapped surfaces.


Mathematical Concepts
Introduction to Game Development, Second Edition,
Cengage Learning, 2010.
Mathematics has become an essential component of modern game development. As both
the main processors and graphics processors in our gaming hardware become more powerful,
the complexity of the mathematics used to model realistic environments and physical
simulations increases without bound. This chapter provides an introduction to several
fields of mathematics that are vital to today's game engines.
Trigonometry is a ubiquitous tool used extensively by game programmers and serves as this chapter's opening topic and prerequisite for the indisputably important topic of linear algebra. The bulk of this chapter discusses vectors and matrices, the indispensable tools of linear algebra with which every 3D game developer needs to be familiar. We also introduce mathematical representations of geometrical entities, such as lines and planes, and describe how to perform certain routine calculations with them. 
The Open Game Engine Exchange (OpenGEX) format is a textbased file format designed to facilitate the transfer of complex scene data
between applications such as modeling tools and game engines. The OpenGEX format is built upon the data structure concepts defined by the Open Data
Description Language (OpenDDL), a generic language for the storage of arbitrary data in humanreadable format. This specification provides a description
of the data structures defined by OpenGEX, and it includes an OpenDDL reference as an appendix.


Tweaking a Vertex's Projected Depth Value
Game Programming Gems,
Charles River Media, 2000.
The goal of this article is to find a way to offset a polygon's depth in a scene without
changing its projected screen coordinates or altering its texture mapping perspective. Most 3D
graphics libraries contain some kind of polygon offset function to help achieve this goal.
However, these solutions generally lack fine control and usually incur a pervertex performance
cost. This gem presents an alternative method which modifies the projection matrix to achieve
the depth offset effect.
A Fast CylinderFrustum Intersection Test
Game Programming Gems,
Charles River Media, 2000.
Before attempting to render a complex object, many games first determine whether a geometrically
simple volume bounding that object is visible. Due to their computational efficiency, spheres
and boxes are commonly used as bounding volumes, but it is sometimes the case that objects are
naturally suited to be bounded by a cylinder. Although we will not be able to achieve the speed at
which a sphere or box could be tested, this gem presents a quick algorithm for determining whether
an arbitrary cylinder potentially intersects the view frustum (and thus whether it is visible).

Oblique View Frustums for
Mirrors and Portals Game Programming Gems 5,
Charles River Media, 2005.
Techniques for rendering mirrors and portals displaying a remote part of the scene require that
an extra clipping plane be used to prevent geometry seen in the mirror or portal from crossing
into the local scene. This gem discusses a technique that modifies the projection matrix in such
a way that the conventional near plane of the view frustum is repositioned to serve as the generally
oblique boundary clipping plane.


Applying Decals to Arbitrary Surfaces
Game Programming Gems 2,
Charles River Media, 2001.
Many games need to render special effects such as scorch marks on a wall or footprints on the
ground that are not an original part of a scene, but are created during gameplay. These effects
are commonly implemented by creating a new object, which we will call a decal, that coincides
with an existing surface and rendering it using some kind of depth offset technique. Applying a
decal to the interior of a planar surface is simple, but difficulties arise when applying decals
to the more complex surfaces used in today's games to represent curved objects and terrain patches.
This article presents a general method for applying a decal to an arbitrarily shaped surface and
concurrently clipping the decal to the surface's boundary.

Charles River Media, 2003.
ISBN: 1584502940 This book once provided a much needed resource and concentrates
specifically on 78 of the extensions most important to developing modern 3D games.
The book is laid out in an intuitive fashion, discussing groups of extensions that modify
or augment similar components of the base OpenGL architecture. In addition, the text
focuses mainly on operational and implementation issues, discussing the underlying
mathematics of an extension only when it is critical to understanding that
extension's functionality.


TJunction Elimination and Retriangulation
Game Programming Gems 3,
Charles River Media, 2002.
This gem describes how to detect possible sources of seams in complex 3D scenes and how to
modify static geometry so that visible artifacts are avoided. Since Tjunction elimination adds
vertices to existing polygons (that are not necessarily convex), this article also discusses a
method for triangulating arbitrary concave polygons.
Also published in Best of Game Programming Gems, 2008. 
VoxelBased Terrain for
RealTime Virtual Simulations By Eric Lengyel, University of California Davis, 2010.
Website: Transvoxel Algorithm

Hyperreal Structures Arising from
an Infinite Base Logarithm By Eric Lengyel, Virginia Tech, 1996.

My email address is my last name at this web page's domain. I get a huge amount of email, and I do
read every bit of it. Because of the volume of email that I receive, it can sometimes take a while for me to respond.

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