Secant conic projections and NASA's GProjector

[Gallien]

Either I'm going mad (or simply extremely bad at geometry) or NASA's GPrjoector is bugged and doesn't produce the secant conic projections correctly.

Here is what I mean.
Screenshot from 2019-12-28 17-20-04.png

First, I decided that I want to use conic projection for my map.
The simple conic projection (tangent) has one standard parallel.

The 30* is the best choice, IMHO, because when you unravel the cone it becomes a half-circle. All longitudes are halved. (That's because sin30 = 1/2.) Every other standard parallel will give you irrational angles for your meridians. e.g. If your st.par. is 30* you have to rotate the map 45* to have your 90*E as a straight vertical line. If your st.par. is not 30* you'll have to rotate your map sqare root of x degrees to achieve the same result.

Then I decided that Secant projection is better overall - it spreads distortion more evenly across the map. Secant projections have 2 parallels, so I wanted to find out which st.par-s I need to have the same effect as the 30* in the example above.

The form of the unravelled cone depends only on the cone itself - it has nothing to do with the relative placement of the cone and the globe. So what I need is a cone that touches the globe at the 30th parallel and then move the cone a little south right through the globe.

This cone will cross the surface of the globe at 2 parallels that are equally distant from the parallel 30. It can be 20* and 40*, it can be the equator and 60*, it can be 30*S and the north pole. In all these cases the cone remains the same - it has the same form and it should always become a half-circle when unravelled.

But it seems that NASA's GProjector thinks otherwise. A tangent conic map with a st.par. at 30* is correctly represented as a half-circle... but not the secant conic projections with the standard parallels mentioned above. I've tried all 3 main conic projections - the equidistant, Lambert's conformal and Albers equal-area - but the result is the same. It is never a half-circle.

Is it me or is it the NASA guys who can't count?
Or is my version of the program broken?
Is there another program that does the maths correctly? FlexProjector doesn't have conic projections at all.

Here's an example of what GProjector gives me (st.par. 0* and 60*):
Screenshot from 2019-12-28 18-09-42.png