Conical constant
The conical constant , also known as the Schwarzschild constant , is one of two parameters, along with the radius, with which a conic section can be specified in terms of size and shape. The representation of conic sections using conical constant and apex radius plays a role in specifying the shape of aspherical lenses and optical mirrors.
A conic section line symmetrical to the x-axis through the zero point of the coordinate system can be given by the following formula:
- R = radius of curvature at zero point
- k = conical constant
The conical constant determines the shape of the line. For k = 0 the result is a circular line which, when rotated around the x-axis, corresponds to a spherical (spherical) surface. If k is different from zero, there are curves or surfaces that deviate from the circular or spherical shape:
| constant | Curve type | Area type |
|---|---|---|
| k <-1 | hyperbole | Hyperboloid |
| k = -1 | parabola | Paraboloid |
| -1 < k <0 | Ellipse (high) | prolate ellipsoid of revolution |
| k = 0 | circle | Bullet |
| k > 0 | Ellipse (wide) | oblate ellipsoid of revolution |
If the conical constant k is less than or equal to zero, it is related to the numerical eccentricity of the conic section line as follows:
The display of the conic lines in this form has the advantage for the calculation of optical surfaces that by varying k surfaces with different characteristics can be selected without changing the intersection of the curve with the x-axis. It shares this property with the vertex equation of the conic sections. In contrast to the apex equation, this formula can be used to treat not only flat but also (with positive k) also high ellipses, and thus not only prolate, but also oblate rotational ellipsoids .
