design vertical curves using either parabola or circular curves

Oleg,

It's mostly true that there are no visual differences. One of the differences isn't in what you can see, but in how they behave in combination with other elements.
In theory you can design a vertical loop using a circular curve, but when you use a parabolic curve that will never happen.

I suggest you do an internet search for the differences and why a parabola is the preferred curve type.

Jan

Oleg,

It's mostly true that there are no visual differences. One of the differences isn't in what you can see, but in how they behave in combination with other elements.
In theory you can design a vertical loop using a circular curve, but when you use a parabolic curve that will never happen.

I suggest you do an internet search for the differences and why a parabola is the preferred curve type.

Jan

Thanks Jan, even if both the circular and parabolic curves are symmetrical, there are fundamental differences between the two. But to make sure  I have made a consultation with both online academia and Chat GPT Wolfram engine - they both concur with my premise of a small difference that should be obvious in vertical alignment design. 

Oleg

Oleg,

I never claimed there are no differences, it's just that they aren't visible for the small section of the parabola that is displayed.

Jan

Took me a bit of time to confirm it but I think your assumption is not correct and differences between the two curves are significant, which I believe has a far reaching consequences... You can see image of comparison below. Happy to be corrected.  

Which tool are you using to test? Profile Between Points shows a clear difference between the 2 curve types.

Same with the Curve from Element tool