Concave Lens

Name: Concave Lens - Geometry Calculator
Author: Jürgen Kummer

Calculations at a concave lens. A concave lens is made of a cylinder, of which on each base side a matching spherical cap is removed. The spherical caps can be from spheres with different sizes, they must not intersect. So, both depths together may not be larger than the cylinder height.
Enter the radius and height of the cylinder, as well as the radiuses of the spheres defining the spherical caps. Choose the number of decimal places, then click Calculate.

Formulas:

i = s - \sqrt{s^{2} - r^{2}}

j = t - \sqrt{t^{2} - r^{2}}

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

A = 2 π \cdot (r h + s i + t j)

V = π r^{2} h - \frac{π}{3} \cdot [i^{2} \cdot (3 s - i) + j^{2} \cdot (3 t - j)]

pi:

π = 3.141592653589793 ...

Radiuses, height and depths have the same unit (e.g. meter), the area has this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit ^-1.

Concave means curved inward, as opposed to convex. In geometry, convex solids do not possess concave features such as sides or surfaces. Conversely, solids that do have concave features are described as non-convex. The concave lens that can be calculated here features two concave surfaces, making it a biconcave lens. The two spheres determining the curvature of the surfaces need not be of equal size, although in practice they often are. The counterpart to the concave lens is the convex lens. Lenses are solids of revolution.

In optics, concave lenses are also known as diverging lenses. They have a negative focal length and do not focus incident light but rather scatter it. A lens's focal length depends on its refractive index, which is determined by the material. A value of 1.5 is specified here for standard glass. Concave lenses are used in eyeglasses to correct nearsightedness. In optical instruments, such as certain telescopes, they are used in combination with other lenses. This shape is also found outside the field of optics, for instance approximately in red blood cells, the erythrocytes. For these, as in some technical applications, one reason for this shape is the larger surface area with the same volume of the convex hull, which in this instance is a cylinder. For such opaque objects, refractive index and focal length are of course irrelevant.

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