This is a C++ math library containing classes for vectors, matrices, quaternions, and elements of projective geometric algebra. The specific classes are the following:
- Vector2D – A 2D vector (x, y) that extends to four dimensions as (x, y, 0, 0).
- Vector3D – A 3D vector (x, y, z) that extends to four dimensions as (x, y, z, 0).
- Vector4D – A 4D vector (x, y, z, w).
- Point2D – A 2D point (x, y) that extends to four dimensions as (x, y, 0, 1).
- Point3D – A 3D point (x, y, z) that extends to four dimensions as (x, y, z, 1).
- Matrix2D – A 2×2 matrix.
- Matrix3D – A 3×3 matrix.
- Matrix4D – A 4×4 matrix.
- Transform4D – A 4×4 matrix with fourth row always (0, 0, 0, 1).
- Quaternion – A convention quaternion xi + yj + zk + w.
- Bivector3D – A 3D bivector x e23 + y e31 + z e12.
- Bivector4D – A 4D bivector (line) vx e41 + vy e42 + vz e43 + mx e23 + my e31 + mz e12.
- Trivector4D – A 4D trivector (plane) x e234 + y e314 + z e124 + w e321.
- Motor – A 4D motion operator rx e41 + ry e42 + rz e43 + rw 𝟙 + ux e23 + uy e31 + uz e12 + uw.
Vector components can be swizzled using shading-language syntax as long as there are no repeated components. As an example, the following expressions are all valid for a Vector3D
object v
:
v.x
– The x component ofv
.v.xy
– A 2D vector having the x and y components ofv
.v.yzx
– A 3D vector having the components ofv
in the order (y, z, x).
Rows, columns, and submatrices can be extracted from matrix objects using a similar syntax. As an example, the following expressions are all valid for a Matrix3D
object m
:
m.m12
– The (1,2) entry ofm
.m.row0
– The first row ofm
.m.col1
– The second column ofm
.m.matrix2D
– The upper-left 2×2 submatrix ofm
.m.transpose
– The transpose ofm
.
All of the above are generally free operations, with no copying, when their results are consumed by an expression. For more information, see Eric Lengyel's 2018 GDC talk Linear Algebra Upgraded.
The ^
operator is overloaded for cases in which the wedge or antiwedge product can be applied between vectors, bivectors, points, lines, and planes. (Note that ^
has lower precedence than just about everything else, so parentheses will be necessary.)
The library does not provide operators that directly calculate the geometric product and antiproduct because they would tend to generate inefficient code and produce intermediate results having useless types when something like the sandwich product Q ⟇ p ⟇ ~Q appears in an expression. Instead, there are Transform()
functions that take some object p for the first parameter and the motor Q with which to transform it for the second parameter.
See Eric Lengyel's Projective Geometric Algebra website for more information about operations among these types.
There is API documentation embedded in the header files. The formatted equivalent can be found in the C4 Engine documentation.