Quadrics¶

pytanga.quadric represents 2D conics (symmetric 3×3 matrices) and 3D quadrics (symmetric 4×4 matrices) as grade-1 blades of a Euclidean-rescaled projective quadric space. Points embed as rank-1 matrices, conics/quadrics are reconstructed as joins of those embeddings, and the results are analysed back into concrete entities and rendered in the viewer.

Topics¶

Guide	What you will learn
Bases (Q2/Q3)	`BasisQ2` (conic space `CA{6}`) and `BasisQ3` (quadric space `CA{10}`) — blades, the Euclidean rescaling, point embedding
Conic space & visualization	Reconstructing conics/quadrics from points, `analyze`/`refine`, the rotation rotor, and how they render
Point tuples (7→8)	Point joins and their dual nets, and the Cayley–Bacharach 7→8 point effect

Quick start¶

from pytanga.geometry import Geometry, Point
from pytanga.quadric import BasisQ3

Q3 = BasisQ3(opns=True)
geo = Geometry(Q3)

# Nine points on a quadric; the join of their embeddings is the quadric.
p1 = geo(Point(2.0, 0.0, 0.0))
# … eight more …
quadric = p1 ^ p2 ^ p3 ^ p4 ^ p5 ^ p6 ^ p7 ^ p8 ^ p9   # grade-9 OPNS blade

viz.new(quadric)   # the visualizer analyzes the MV and draws the quadric

Examples¶

py/examples/ga/quadric/conic_demo.py — conic through 5 points, rotated by a slider
py/examples/ga/quadric/quadric3d_demo.py — quadric through 9 points, rotated by sliders
py/examples/ga/quadric/general_quadric.py — arbitrary quadrics from coefficients
py/examples/ga/quadric/point_tuples_demo.py, plane_pair_demo.py, quadric3d_raycast.py, quadric_intersection_demo.py

Background¶

The mathematical derivation lives in the developer docs: Conic & quadric space and dev/theory/quadric-{rotor,point-tuples,plane-pair}-derivation.md.