How to get a 2D net of a 3 D figure?

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So basically if a take a hyperbola from a low and high y value then spin that curve around the y axis I get a hyperbolic funnel (aka hyperbolic horn or hyperbolic coin fountain). Lets say the funnel has negligible width say 0.5mm and is made of sheet metal

Now say I slice this funnel as one might slice a cake (though here its not a solid like a cake). I would presume each slice could be laid flat and this would yield 2D nets of the 3 D shape? (Maybe Im wrong? If so please explain why) If Im not wrong then Im wondering how to precisely know what those nets look like . And a side question would be does the angle of the "cake slices" have any relevance? Preferably Id like to have a way to generate a function for the net curves based on "slice" angle.

Purpose is to generate a hyperbolic funnel from 2D nets (that can be welded together) as getting a 3D funnel fabricated is very advanced machine work and the few places that sell them charge a fortune.

This has been bothering me for a long time. Thank you for any help you can provide.