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Reflection at parabola with GA

Hello!

I’m 1/2 year into GA now, amazed, overwhelmed, confused (GA, PGA, CGA?). GA highly inspires me, and besides Infinitesimals/Hyperreal Numbers/Surreal Numbers, it is one of the things in Math that I always missed in school and university, that just felt like it had to be there, but noone could tell or point to; and that I finally found.

I could think of a gazillion problems that seem to be easy with or lending themselves to GA, but as a starter I want to tackle a single problem to learn to do “real” problems with GA.

I wonder how one would model the reflection of lines going through a point (“source”) at a parabola.

First I would like to solve the the obvious case: Source at focus of the parabola.

Then I’d like to look into other source points, going on with other curves that act as reflector. Then I’d like to look into the “density” of lines, and even something like incidence angle (at an intersection), occlusion (Heaviside function), variation of parameters, 3D case…

Uses could be modeling a lamp like a Poul Henningsen PH 5, or a Wolter telescope.

Things where I am stuck/unsure and need help:

  • Which GA is best suited (for the 2D case, later maybe 3D)
  • How do I describe a curve or function in GA? Parabola, standard form, rotation?
  • How do reflect at a curve? Would of course need the tangent at the point of reflection…
  • The source is a pencil of lines, which has a known representation in some GA’s, and if it is at the focus of the parabola, the outcome has to be a pencil of lines though some point at infinity (parallel) lines. But I’m not sure if the density is uniform, and how to analyse that. Of course in a more general case the outcome is more complicated.
  • How do I analyse/model different densities of pencils?

Any pointers to papers, videos, presentation, books etc. are highly welcome, along of course explanations of actual steps.

I find tons of theoretic material regarding basics and properties of GA, but miss the actual practical examples.

@Egbert This is an interesting question!

I think that 2D euclidean PGA is a good tool for solving it.

I’ve written a ganja demo that carries out the reflection of a line pencil in a parabola in this context.

Here’ll I’ll give a sketch of the ideas behind the code. You can look at the source code for the details.

For simplicity assume the parabola C is the standard one y = x^2. Parametrize it as C(t) = e_{12} + t e_{20} + t^2 e_{01}. For future reference, compute the derivative \dot{C}(t) = e_{20} + 2t e_{01}.

Let m be a line in the line pencil centered at P (or "the line pencil in P"). Let M be an intersection of m with C. Given M, it is easy to find the unique t satisfying C(t) = M. Then the tangent line t_M at M is given by the join of the point with the direction of movement at the point: t_M = C(t) \vee \dot{C}(t) (= -2t e_{1} + e_2 + t^2 e_{0}).

With t_M in hand, the reflection of m in t_M is given by the sandwich t_M m t_M, and ganja does the rest.

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June 7, 2021: I’ve added color coding to the demo. Each line in the pencil is assigned a color between red and blue. Then the geometry associated to that line (intersection points with parabola, their tangent lines, and the reflected lines) are drawn with the same color.

In general, such a reflected line pencil is called a caustic curve with respect to the reflecting curve. This curve is naturally in line-wise form (as described above). But like every well-behaved curve, it also has a point-wise form. I’ve implemented an approximation to this point-wise caustic, obtained by intersecting consecutive lines of the line-wise caustic. The exact point-wise curve is obtained by the wedge product r(t) \wedge \dot{r}(t) where r(t) is the line-wise form. In general r(t) \wedge \dot{r}(t) is the instantaneous center of rotation of a line-wise curve r(t), just as C(t) \vee \dot{C}(t) is the instantaneous line of motion of a point-wise curve C(t).)

Exercise: Use the automatic differentiation feature of euclidean PGA to replace the approximate caustic curve with the exact caustic curve.

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