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Complex \(\mathbb R_{0,1} \cong \mathbb C\)

Dual \(\mathbb R_{0,0,1} \cong \mathbb D\)

Quaternion \(\mathbb R_{0,2} \cong \mathbb H\)

Timespace \(\mathbb R_{1,3} \cong \mathbb M\)

3D GA \(\mathbb R_{3}\)

2D PGA \(\mathbb R_{2,0,1}\)

3D PGA \(\mathbb R_{3,0,1}\)

2D CGA \(\mathbb R_{3,1}\)

3D CGA \(\mathbb R_{4,1}\)

3D Mother Algebra \(\mathbb R_{4,4}\)

2D CSGA \(\mathbb R_{5,3}\)

3D CCGA \(\mathbb R_{6,3}\)

3D QCGA \(\mathbb R_{9,6}\)

Cubic 2D GA \(\mathbb R_{9,7}\)

Game

Basis and Metric

Ganja.js supports operator overloading and algebraic literals.

Operator Javascript Name \(a*b\) a*b Geometric Product \(a\wedge b\) a^b Outer Product \(a\vee b\) a&b Regressive Product \(a\cdot b\) a<<b Left Contraction \(a*b*\tilde a\) a>>>b Sandwich Product \(\tilde a\) ~a Conjugate \(\bar a\) !a Dual \(\bar{\bar a}\) a.Reverse Reverse \(a^{-1}\) a**-1 Inverse \(e^a\) Math.E**a Exponentiation \(a_{\langle b \rangle}\) a.Grade(b) Grade Extraction \(a+b\) or \(a-b\) a+b or a-b Multivector Addition/Subtraction \(4.2e_{12}\) 4.2e12 Blade Literals

Ganja.js supports vectors and matrices with multivector elements.

Operator Javascript Name \(\bold v = \begin{bmatrix} e_1 & 0 \end{bmatrix} \) v = [1e1,0]; Vector \(\bold A = \begin{bmatrix} 1 & 0 \\ 0 & e_{12} \end{bmatrix} \) A = [[1,0],[0,1e12]]; Matrix \(\bold v \cdot \bold w\) v*w Vector-Vector dot product. \(A \bold v\) A*v Matrix-Vector product. \(AB\) A*B Matrix-Matrix product. \(A^{HT}\) ~A Conjugate-Transpose Matrix

Ganja.js can graph 2D and 3D PGA and CGA elements.