Cousin Ricky <ric### [at] yahoocom> wrote:
However, I have a quibble about
> UW-Madison's terminology.  Those mirrors are what I consider *concave*,
> not convex.

SO many people have a hard time with one vs the other.  Dunno why.

> This macro implements a concave paraboloid in 3 dimensions:
>
> #macro Concave_Paraboloid (yFocus)
>   poly { 2, <-1, 0, 0, 0, 0, 0, yFocus * 4, -1, 0, 0> }
> #end

Nice.  I was using a parabaloid isosurface.

> The vertex is at the origin, and the concave surface faces upward for a
> positive yFocus.  You would need to intersect this with a limiting
> object of appropriate size.  I used poly{2} rather than quadric{}
> because the latter is buggy.

This might be the way, since when I was modeling the reflections with trace ()
to make sure my dimensions were right, with some camera angles and clipped_by
dimensions, my whole isosurface disappeared.

The most important bit of information is that the focal length should be twice
its "thickness" so that the lower mirror projects the image through the hole in
the upper mirror.

To do that, I was going by:

https://en.wikipedia.org/wiki/Parabola#In_a_Cartesian_coordinate_system

to get:
y = 1/(Factor*focal_length) * (pow (x, 2) + pow (z, 2))

So I had
#declare focal_length = 2;
#declare Factor = 4.0;

and I had 1-unit high slices stacked on top of one another, with the top
clipped_by making the hole in the top as well.

I can see the cube I placed in the bottom, but it doesn't project a real image
out the top.  Your poly seemed to be too concave, and I needed to "flatten it
out" to get similar results - still no illusory image even tweaking the focal
length over a wide range.   It gets close, but never "pops out" of the hole.

- BW

I thought this was interesting, but it didn't really help me:
https://spie.org/etop/2007/etop07methodsV.pdf

Patent:
https://patents.google.com/patent/US3647284

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