Why are circles projected onto circles under stereographic projection?
In 2021, I developed a geometrical proof I had not seen elsewhere.
I have started from the observation that the circle on the sphere and its image are both connected by some sort of cone. What is the base area of this cone? If the cone has a circular base, its intersection with a plane is an ellipse. If you gently squeeze the cone, you could turn that ellipse into a circle. Now the base is an ellipse. The projection rays form an elliptical cone.
What needs to be done to show that the image in the equatorial plane is also a circle? I look at the elliptical cone, viewing it from a direction perpendicular to the axis of the cone. The circle on the sphere appears as an ellipse, and the base of the cone – the ellipse – appears as a straight line. The image of the circle is also a circle if the ellipse it appears as has the same ratio of major and minor axis.
This is the same as stating that both circles (the one we know is a circle and the alleged one) have to be inclined with respect to the elliptical cone’s axis by the same angle. This equality of two characteristic angles is what my 2021 sketch (and explanations in the 2021 blog post) should illustrate. I am spotting a few angles and right-angled triangles, and I am applying a few basic geometrical rules to proof that the two inclination angles are the same.
In the 2021 blog post I had anticipated that the proof was either wrong or already known; fortunately, it was the latter option. I found a very similar proof on the website of a technical expert a few months later. However, this does not diminish the excitement of creating your own proof from scratch. The joy was in the discovery for myself.
As nearly any physics- or math-related art created in 2021, this has been inspired by Roger Penrose’s book The Road to Reality: his expositions of complex numbers and stereographic projection and his beautiful old-school hand-crafted drawings.
I tried to color the scan of the proof – this was done in MS Paint, with a mouse.
I have never been satisfied with this – neither with the drawing nor with the coloring.
Now – in 2024 – I am trying again. I might re-draw the whole proof some day, but I want to challenge myself by imposing these rules:
~ Re-use the old sketch and do not delete anything. Only cleaning of grey areas that should be white is allowed.
~ Add color but preserve the spirit of the original drawing.
As a first step, I inverted the drawing – to notice that “black” is actually grey. I have spent a ridiculous amount of time to…
~ Write slightly enhanced (?) letters over their 2021 versions, making them more cursive and consistent.
~ Clean up all the grey paper background to make it black.
~ Draw circles, lines, and ellipses over the original ones (using Procreate’s guides for these tasks).
~ Pick a color palette and develop a way of warped cross-hatching with a digital pencil.
This is the result! I’ve renamed it from Circles to Circles to Proven Cone.
The original image is 7000 x 5000 pixels (as the A4 paper was scanned at 600 dpi). Here is a close-up: A cut-out of a 1000 x 1000 pixels detail at the original resolution:
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Related blog posts:
2024:
Elliptical Collage. Stereographic Haiku
2021:
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