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https://github.com/jeremyong/klein

P(R*_{3, 0, 1}) specialized SIMD Geometric Algebra Library
https://github.com/jeremyong/klein

3d 3d-graphics algebra animation dual-quaternions geometric-algebra inverse-kinematics pga projective-geometry quaternion-algebra quaternions simd sse

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P(R*_{3, 0, 1}) specialized SIMD Geometric Algebra Library

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# [Klein](https://jeremyong.com/klein)



[![License: MIT](https://img.shields.io/badge/License-MIT-blueviolet.svg)](https://opensource.org/licenses/MIT)

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👉👉 [Project Site](https://jeremyong.com/klein) 👈👈



## Description



Do you need to do any of the following? Quickly? _Really_ quickly even?



- Projecting points onto lines, lines to planes, points to planes?

- Measuring distances and angles between points, lines, and planes?

- Rotate or translate points, lines, and planes?

- Perform smooth rigid body transforms? Interpolate them smoothly?

- Construct lines from points? Planes from points? Planes from a line and a point?

- Intersect planes to form lines? Intersect a planes and lines to form points?



If so, then Klein is the library for you!



Klein is an implementation of `P(R*_{3, 0, 1})`, aka 3D Projective Geometric Algebra.

It is designed for applications that demand high-throughput (animation libraries,

kinematic solvers, etc). In contrast to other GA libraries, Klein does not attempt to

generalize the metric or dimensionality of the space. In exchange for this loss of generality,

Klein implements the algebraic operations using the full weight of SSE (Streaming

SIMD Extensions) for maximum throughput.



## Requirements



- Machine with a processor that supports SSE3 or later (Steam hardware survey reports 100% market penetration)

- C++11/14/17 compliant compiler (tested with GCC 9.2.1, Clang 9.0.1, and Visual Studio 2019)

- Optional SSE4.1 support



## Usage



You have two options to use Klein in your codebase. First, you can simply copy the contents of the

`public` folder somewhere in your include path. Alternatively, you can include this entire project

in your source tree, and using cmake, `add_subdirectory(Klein)` and link the `klein::klein` interface

target.



In your code, there is a single header to include via `#include `, at which point

you can create planes, points, lines, ideal lines, bivectors, motors, directions, and use their

operations. Please refer to the [project site](https://jeremyong.com/klein) for the most up-to-date

documentation.



## Motivation



PGA fully streamlines traditionally used quaternions, and dual-quaternions in a single algebra.

Normally, the onus is on the user to perform appropriate casts and ensure signs and memory layout

are accounted for. Here, all types are unified within the geometric algebra,

and operations such as applying quaternion or dual-quaternions (rotor/motor) to planes, points,

and lines make sense. There is a surprising amount of uniformity in the algebra, which enables

efficient implementation, a simple API, and reduced code size.



## Performance Considerations



It is known that a "better" way to vectorize computation in general is to arrange the data in an SoA

layout to avoid unnecessary cross-lane arithmetic or unnecessary shuffling. PGA is unique in that

a given PGA multivector has a natural decomposition into 4 blocks of 4 floating-point quantities.

For the even sub-algebra (isomorphic to the space of dual-quaternions) also known as the _motor

algebra_, the geometric product can be densely packed and implemented efficiently using SSE.



## References



Klein is deeply indebted to several members of the GA community and their work. Beyond the works

cited here, the author stands of the shoulders of giants (Felix _Klein_, Sophus Lie, Arthur Cayley,

William Rowan Hamilton, Julius Plücker, and William Kingdon Clifford, among others).



[1]

Gunn, Charles G. (2019).

Course notes Geometric Algebra for Computer Graphics, SIGGRAPH 2019.

[arXiv link](https://arxiv.org/abs/2002.04509)



[2]

Steven De Keninck and Charles Gunn. (2019).

SIGGRAPH 2019 Geometric Algebra Course.

[youtube link](https://www.youtube.com/watch?v=tX4H_ctggYo)



[3]

Leo Dorst, Daniel Fontijne, Stephen Mann. (2007)

Geometric Algebra for Computer Science.

Burlington, MA: Morgan Kaufmann Publishers Inc.