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In this answer I included the image below of a reflective X-ray telescope. It is made from two elements; the first is concentric shells of glancing (high incidence) angle paraboloidal surfaces, and the second element is the same except that they are hyperboloidal surfaces.

A "Classic" Cassegrain telescope is also a parabolic mirror followed by a hyperbolic mirror, except that the configuration looks very different since for visible light we can use near-normal incidence and are not restricted to high incidence angles.

Question: Despite looking so different, are these basically the same thing? Is the mathematical relationship between the parabolas and the hyperbolas the same in both cases?

Related in Space SE: How does this strangely-shaped horn at Honeysuckle Creek Tracking Station work?

X-ray telescope

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Short answer, yes, the x-ray grazing incidence mirrors set up like this still function like a Cassegrain to minimize off-axis aberration. However, it is still different in that a classical Cassegrain uses a fast positive paraboloid and a slow negative hyperboloid.

The Cassegrain design combines the paraboloid and hyperboloid to minimize off-axis aberrations effectively. In 1952, Hans Wolter outlined 3 layouts of paraboid-hyperboloid mirrors which could be used in a grazing-incidence for x-ray applications. The type 2 layout is most equivalent to a Cassegrain, using a positive paraboloid followed by negative hyperboloid, the application used here (in the XMM-Newton observatory) is a Type 1. As a result, the remaining aberrations are slightly different, though still minimized rather effectively. Also notable, is that the focal lengths of the paraboloid and hyperboloid in the Wolter designs are of similar focal lengths.

Type 1: enter image description here

Type 2: enter image description here

https://www.cosmos.esa.int/web/xmm-newton/technical-details-mirrors

https://en.wikipedia.org/wiki/Wolter_telescope

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    $\begingroup$ Wow, thank you for such a thorough answer! I'd always assumed that the the Cassegrain secondary was small to minimize lost and scattered light, and that the hyperbola was simply the correct shape to put the focus behind the primary. I didn't realize this actually corrected off-axis aberrations. So say an f/8 Cassegrain has better off-axis performance than an f/8 Newtonian? $\endgroup$
    – uhoh
    Commented Apr 11, 2021 at 20:06
  • $\begingroup$ you may find this interesting: Is it possible to have smaller solar sails and increase the amount of energy they receive with a constellation of mirrors? $\endgroup$
    – uhoh
    Commented Jun 11, 2021 at 10:57

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