I'm busy at the moment, but:

define a sphere with radius 0.5
pow(x,2) + pow(y,2) = 0.5

so pow(y,2) = 0.5 - pow(x,2)

and y = sqrt ( 0.5 - pow(x,2))

translate y*0.5

y = sqrt ( 0.5 - pow(x,2)) +0.5

make your function equal to the sin of that times pi/2 so that the function
smoothly increases from 0 to 1

add the smallest number POV will work with to that (1E-whatever)

Multiply that by a line equation, the slop of which is the tightness

I will give this some more thought

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