Why does the optical design of the Webb mirror have three different types of panel?

Question

I read that there are A, B, and C type mirror segments on Webb. Why are there three types, and why are they distributed around the way they are?

I assume the segments have to overall form exact sections of a parabola in their respective positions, so I naively thought each section would be more symmetrical, not having just three optical formulas.

The question boils down to, why are there B and C segments?

Here is the official NASA page on this. And below is a Flickr image from NASA that maps out the different segments.

Answer 1

The James Webb Space Telescope is a three-mirror anastigmat featuring an ellipsoidal primary, hyperboloidal secondary, and ellipsoidal tertiary.

Contreras, James W.; Lightsey, Paul A. (22 October 2004). "Optical design and analysis of the James Webb Space Telescope: optical telescope element". Conference Proceedings of the SPIE. 5524. doi:10.1117/12.559871. It's paywalled there but available here

As @uhoh answered they vary by necessity from the main axis.

Answer 2

One might also ask Why do telescopes use hexagonal mirror pieces instead of pie slice shaped ones? since in that case all segments would be identical and perhaps allow for interesting alternatives in fairing-packing and unfolding pattern.

Of course it would pose other challenges.

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I know this one because I spend all day every day thinking about hexagonal tilings. (1, 2, 3, 4)

1. Tiling the "sphere" and keeping uniform gaps between elements means the "hexagonal" shapes are not hexagonal and must differ

It turns out that you can not tile a sphere (or a parabola or ellipsoid or other primary mirror figure¹) with hexagons. For a sphere you need at least regular 12 pentagons plus 0 or more regular hexagons.

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¹hat tip to @leftaroundabout for setting me straight

2. The shapes of the surfaces differ as well.

The primary of the JWST is not spherical (and it seems not even parabolic¹), so mirrors at different distances from the axis will have different shapes.

The mirror has two hexagonal rings, the center or (0, 0) hexagon is completely missing. It could have been included with a hole through the center (Cassegrain) but that would have added a fourth kind of mirror so it was a lot more complexity for a small increase in light (and small reduction in the complexity of the point spread function due to diffraction by the hole).

However the 2nd ring has six corner and six side elements, which differ not only in their distance from the axis but in their orientation. For the corner mirrors the line from the mirror center to the optical axis passes through an edge, while for the "side" mirrors it passes through a corner.

So you have three different off-axis parabolic shapes of mirrors that have to be made:

1. first ring (six)
2. second ring corners (six)
3. second ring sides (six)

From this answer to How would NASA confirm the James Webb Space Telescope is undamaged after the clamp release incident?:

Three kinds of off-axis parabolic hexagons (A, B, C):

From JWST.NASA.gov's Webb's Mirrors