A Guided Tour of the Plane-Based Geometric Algebra PGA | Hacker News Reader

A Guided Tour of the Plane-Based Geometric Algebra PGA

bivector.net

45 pointsenkimute4 comments

When it comes to representing geometric objects like lines and planes using CGA/PGA, is it not possible to think of the representations as exploiting fixed points? So in 3D, a line is the fixed point of some rotation, and a plane is the fixed point of some reflection. I was thinking this might be easier than using vectors, bivectors, trivectors and all that.

To make the idea concrete, one could do plane geometry with the dual quaternions. One could use dual quaternions to represent rotations and reflections in the plane. A point object can thus be represented as a 180-degree rotation about a point, and a line objects can be represented as a reflection about a line. There's therefore no need to think in terms of bivectors.

This is one of the ideas presented in this Chapter. (i.e. to identify the geometry with the invariant of the transformation). In 3D if x represents a reflection, it also represents the plane left invariant by that reflection. If x represents a rotation/translation it also represents the euclidean/infinite line left invariant by that rotation/translation. And if it represents a point reflection, it also represents the 1 point left invariant by that point reflection.

Framing it in terms of vectors/bivectors is what allows us to quantify these ideas. (i.e. have to write down your planes/lines/points with numbers at some point).

But you got the essence! The abstract/coordinate free/geometric/group theory ways of thinking about this are most insightful.

So in PGA, vectors are reflections, other isometries are combinations of reflections, and the geometry are the associated invariants. (including planes, lines, points and screw axi).

A fun extension is to replace the reflection by an inversion (reflection in a sphere). Vectors now become inversions (leaving spheres invariant), the rotors become the conformal group (composition of two inversions), leaving circles invariant, etc .. the corresponding Clifford Algebra is CGA (R(4,1), with a proper parametrisation).

Given a "Conformal Geometric Calculus" is not feasible due to the distance metric not being preserved, a "[Hyper-Plane] Geometric Calculus" should be; is this to be included a future version of this document, the book, or some other endeavor?

The WebGL stuff works in Chromium but not in Firefox.

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