There was an article in Antique Radio Classified some years ago that
made me aware of capacitors with tapered plates.

You can look up that patent online www.uspto.gov, click on search
under patents, click on patent number search, put in the number,
and when it tells you click on images. You have to have a tiff viewer
plugin in your web browser to read them. That particular patent seems
to apply to a capacitor with flat plates, so perhaps there was a pending
patent on the tapered plates.

A variable capacitor with semicircular plates and the axis of rotation
going through the middle and flat plates gives of course a linear
relationship between capacitance and angular rotation. By altering the
plate shape a bit and moving the axis of rotation off-center you can
get a capacitor that tunes straight-line wavelength. Wavelength is
proportional to the square root of capacitance. In the early days of
radio wavelength was often specified instead of frequency.

You can make a variable capacitor that is straight-line frequency, which
rotation has to be proportional to the reciprocal of the square root of
capacitance. With flat plates this requires a fairly radical plate
shape, with the axis of rotation very far off center. As a result there
has to be a lot of empty space for those long narrow plates to swing around.
I'll post a picture of one of these to that binaries group.

So a way to get around the odd plate shape and the space it occupies
is to use plates of tapered thickness. It would be nice if you got
straight-line something, but even if you don't you have something
where the stations are more spread out than they are with straight
line capacitance.

I have an old TRF receiver of the vintage of multiple knobs and 01A tubes
that uses capacitors like this.
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jhhaynes at earthlink dot net