Lens Calculator

Name: Lens - Geometry Calculator
Author: Jürgen Kummer

Calculations at a convex spherical lens. This is made of two spherical caps, in between a cylinder. Those three have equal sized circular areas with the radius r_c. The focal length of a lense depends on its refractive index, which is determined by its material. Here, 1.5 is set, the value for normal glass.
Enter both sphere radiuses and the cylinder radius and choose the number of decimal places. Then click Calculate. The cylinder height can also be 0.

Formulas:

h_{1} = \frac{2 \cdot r_{1} - \sqrt{4 {r_{1}}^{2} - 4 {r_{c}}^{2}}}{2}

h_{2} = \frac{2 \cdot r_{2} - \sqrt{4 {r_{2}}^{2} - 4 {r_{c}}^{2}}}{2}

h = h_{1} + h_{2} + h_{c}

f = \frac{1}{{(n - 1) \cdot [\frac{1}{r_{1}} + \frac{1}{r_{2}} - \frac{h \cdot (n - 1)}{n \cdot r_{1} \cdot r_{2}}]}}

A = 2 \cdot π \cdot (r_{1} \cdot h_{1} + r_{2} \cdot h_{2} + r_{c} \cdot h_{c})

V = \frac{π}{3} \cdot [{h_{1}}^{2} \cdot (3 \cdot r_{1} - h_{1}) + {h_{2}}^{2} \cdot (3 \cdot r_{2} - h_{2})] + π \cdot {r_{c}}^{2} \cdot h_{c}

pi:

π = 3.141592653589793...

The refractive index has no unit, radius, heights and focal length have the same unit (e.g. centimeter), the area has this unit squared (e.g. square centimeter), the volume has this unit to the power of three (e.g. cubic centimeter). A/V has this unit ^-1.

The lens is of course better known as an optical instrument than as a geometric body. Such lenses have been known since ancient times, but were rarely used at that time because their manufacture was very difficult. Natural crystals such as quartz and beryl, but also emerald, were used, which could optically magnify objects when held at the right distance and angle. Lenses slowly became popular as a visual aid in the Middle Ages, initially only for the wealthy and educated classes. At the beginning of the 17th century, the idea of attaching two lenses one behind the other at a distance came up, thus creating the first telescopes. In 1609, Galileo Galilei finally had the idea of pointing such a telescope at the sky and thereby brought down an entire worldview, namely the geocentric model.
The name lens comes from the vegetable, Lens culinaris, whose seeds also have this shape, and was adopted into optics.

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