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Bases (Q2/Q3)

pytanga.quadric linearises conics and quadrics by putting them in a projective quadric space — a geometric algebra whose vectors are the symmetric-matrix coefficients of a degree-2 hypersurface. Two bases are provided:

Basis Space Algebra Blades Represents
BasisQ2 conic space CA{6} Algebra(6, 0) b1 … b6 2D conics (symmetric 3×3)
BasisQ3 quadric space CA{10} Algebra(10, 0) b1 … b10 3D quadrics (symmetric 4×4) 
from pytanga.quadric import BasisQ2, BasisQ3

Q2 = BasisQ2()           # 2D conic space
Q3 = BasisQ3(opns=True)  # 3D quadric space (OPNS convention)

The dimension (6 or 10) is exactly the number of independent symmetric-matrix entries. opns= selects the outer-product (OPNS) convention (default True); pass opns=False for the inner-product (IPNS) convention.

Blade ↔ monomial correspondence

The named blades b1…bN correspond to the monomials of the quadratic form. BasisQ2.__init__ also sets b1…b6 and I (the pseudoscalar) as attributes (BasisQ3 sets b1…b10 and I):

blade Q2 (conic) Q3 (quadric)
b1, b2 x, y x, y
b3 1 (constant) z
b4 x² 1 (constant)
b5 y² x²
b6 xy y²
b7 — z²
b8 — xy
b9 — xz
b10 — yz

Blade bitmask IDs are powers of two: b1=1, b2=2, b3=4, ….

Euclidean rescaling

Perwass's original conic space uses a non-Euclidean basis e1…e6 with squared norms (1, 1, 2, 2, 2, 1) (Q2) and (1, 1, 1, 2, 2, 2, 2, 1, 1, 1) (Q3). pytanga.quadric rescales to a Euclidean basis b_i (b_i·b_i = 1) via b = e/√2 on the norm-2 blades, so the plain Algebra(6,0) / Algebra(10,0) metric can be used unchanged. The √2/2 factors that the rescaling introduces appear only in the point embedding and the matrix↔coefficient mapping — every other operation works with b1…bN.

Using a basis with Geometry

from pytanga.geometry import Geometry, Point

Q2 = BasisQ2()
geo = Geometry(Q2)

p = geo(Point(1.0, 2.0, 0.0))   # an embedded point (grade-1 blade)

Geometry(basis) wires the basis into the entity ↔ MV pipelines, so geo(Point(...)), geo(Rotor(...)), geo.analyze(mv), and geo.refine(...) all understand the quadric space.

See Also