I have two functions, I'll call them A(t) and B(t) which renders a nice
curve on the 2D plane via x(t) = A(t), y(t)=B(t).

Now I want to extend them to a surface in 3D space:

x(u,v) = /* a simple term that combines A and B somehow */
y(u,v) = /* another term with these functions */
z(u,v) = /* a 3rd term */

Do you have some ideas for "nice" combinations that might result in
interesting surfaces?

I'll try to render them as povray UV patch and post the images. :-)

Lars R.

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> x(u,v) = /* a simple term that combines A and B somehow */
> y(u,v) = /* another term with these functions */
> z(u,v) = /* a 3rd term */
>
> Do you have some ideas for "nice" combinations that might result in
> interesting surfaces?

Without knowing what A and B actually are, it's difficult to suggest 
what might look nice. This might work:

x = A(u)*sin(v)
y = B(u)*sin(v)
z = cos(v)


Terms like A(B(u)) might also be interesting?

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On 03/31/2014 03:39 PM, scott wrote:>> x(u,v) = /* a simple term that
combines A and B somehow */
>> y(u,v) = /* another term with these functions */
>> z(u,v) = /* a 3rd term */
>>
>> Do you have some ideas for "nice" combinations that might result in
>> interesting surfaces?
>
> Without knowing what A and B actually are, it's difficult to suggest
> what might look nice. This might work:

Okay, you're right. I just thought there are some "common approaches" to
do so, regardless of the functions A and B.

I found something interesting:

  https://en.wikipedia.org/wiki/Superformula

there is also a 3D extension that only maps the 2D figure to a sphere.
It looks nice, but not sooo interesting, I think.

The Fresnel integrals S(t), C(t) give the Euler spiral in 2D. I'd like
to have it in 3D, somehow. :-)

> Terms like A(B(u)) might also be interesting?

I'll try it.

Lars R.

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Always a good question.  I was playing around with just some simple spiral math,
and I thought that a '3D spiral' would look cool.  I tried rotating around y at
the same time I was creating the spiral around z, and ... it kinda looked like
something the cat got into.  :D

You might try starting with the simple combinations of addition, subtraction,
multiplication, and division.  a of g and b of a sound like they could yield
interesting results.  The regular functions like sin have great potential.
Maybe multiply sin by a across one axis, and by b across the other, while doing
something to give it width in the third dimension.  or use a and b as the width
functions.

a^b,  b^a, ....

Thanks for the superfunction link.

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For the record, I found an interesting page that shows some 2D formulæ
incl. graphs:

http://www.mathematische-basteleien.de/kurven.htm

(The page is in German, but mathematical formulæ and their graphs are
international, I think)

I'll try to "extend" some of them into 3D space…

Lars R.

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