Lens Math for a Semicircle and Vesica Piscis?

[shintashi]

Hi, I'm trying to find the way that light (sunlight?) distorts through a lens with a diameter of 1, if the lens has the shape of a semicircle (width of 1/2), and if the lens has the shape of a Vesica Piscis (width of 1/√3).

if you need units, yards works.

I'm also curious about how that light changes if the semicircle becomes flatter, turning into a circle segment instead of a full semi circle, and the same for the Vesica Piscis Lens.

I saw the equations on wiki for lenses but they don't seem to interface well with optometry calculators so I wasn't able to check my work or see any patterns. For now, I'd just like to know what light does passing through a lens with either a flat + curve, or with curve + curve.

I understand if the light source is close like a candle or lamp it turns upside down. I'm trying to understand more how sunlight focuses, since it approximates an ideal straight line before converging or diverging through a lens.

-τħαηκς

 

[Drakkith]

A lens with a spherical shape (a semicircle would be a plano-convex lens and a Vesica Piscis would be a bi-convex lens whose surfaces are both spherical) has several inherent aberrations that have to be taken into account when designing optical systems. The primary aberration is known as spherical aberration, where the light passing through the edges of the lens is brought to a focus closer to the lens than light passing through near the center. In addition to spherical aberration, a spherical lens also suffers from coma and astigmatism if my memory serves, and, like all lenses, it also suffers from chromatic aberration.

A list of common aberrations can be found here: https://en.wikipedia.org/wiki/Optical_aberration

 

[Drakkith]

I'm also curious about how that light changes if the semicircle becomes flatter, turning into a circle segment instead of a full semi circle, and the same for the Vesica Piscis Lens.

As long as the lens surfaces remain spherical, the aberrations remain. However, a more strongly curved surface generates aberrations of a larger magnitude than a less strongly curved surface. For lenses and mirrors whose focal lengths are about 10x more than their diameter some of these aberrations become slight enough to ignore. For example, a spherical mirror is often used in small reflecting telescopes since the effects of diffraction limit the image quality more than spherical aberration.

Bending the surface to another type of conic section, such a paraboloid, hyperboloid, or ellipsoid, is often used to correct for various aberrations. I have a telescope which has two non-spherical mirrors that, together, greatly decrease most of the major aberrations that telescopes often suffer from.