Create Cayley Cubic surface in K3DSurf?

 
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clyde95



Joined: 03 Aug 2014
Posts: 4

PostPosted: Sun Aug 03, 2014 1:24 pm    Post subject: Create Cayley Cubic surface in K3DSurf? Reply with quote

Hi, I am beginner in K3DSurf.
First congratulations to Abderrahman on this wonderful and free application. Thank you.

How can I create Cayley Cubic surface in K3DSurf?
I am not even sure I found the right isosurface equation:

Code:
  -5*(x*x*y + x*x*z + y*y*x + y*y*z + z*z*y + z*z*x)
  +2*(x*y + x*z + y*z)


?

Thank you for the reply.

P.S.

Can the upper equation be converted to a parametric equations?
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nextstep
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Joined: 06 Jan 2007
Posts: 538

PostPosted: Sun Aug 03, 2014 11:37 pm    Post subject: Reply with quote

Hi Clyde95 and welcome to the forum,
I've added a new entry for the Cayley cubic surface (thanks for pointing me to this interesting surface). I used two implicit representations of the Cayley Cubic surfaces to obtain the attached images. You have to save the attached script as, for example, cayley.k3ds and import it into K3DSurf by using the "Load" button. You will then have two new implicit entry in the "Examples" list.
Quote:
Can the upper equation be converted to a parametric equations?

I don't think there is a parametric representation of the Cayley Cubic.
Quote:
I am not even sure I found the right isosurface equation

Cayley Cubic, Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface are all part of the Cubic surfaces family but the subject if far more complex than it first looks...for me too Wink
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clyde95



Joined: 03 Aug 2014
Posts: 4

PostPosted: Mon Aug 04, 2014 8:07 am    Post subject: Reply with quote

Hi Abderrahman,
This is gread!! Very Happy
Thank you so much for the quick reply and solution.

By the way, would you clarify a bit to me, which surfaces could use parametric equations, and which could not?
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nextstep
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Joined: 06 Jan 2007
Posts: 538

PostPosted: Tue Aug 05, 2014 3:00 am    Post subject: Reply with quote

Hi,
The Parametrization of an implicit equation is not always possible but , in some cases, this process can be carried out as an exact match (or as an approximation) to the parametric equation.
So, first of all, you should find out if your implicit form can be converted to a parametric form and after that, try to make the conversion itself.
There are some implicit forms that can be "easily" converted and to find out which one, you can do some google search with , for example, "how to convert implicit surfaces to parametric form?".
There are plenty of PDF manuals available for free but I'm afraid most of theme requires some good knowledge of the subject...
So, as you see, here too, the subject is more complex than it first look!Very Happy
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clyde95



Joined: 03 Aug 2014
Posts: 4

PostPosted: Tue Aug 05, 2014 6:45 am    Post subject: Reply with quote

Ah, yes this looks like something too complicated for my knowledge of mathematics.
Do you think Cayley Cubic surface's implicit equation could somehow be converted an parametric one? How would it look like?
Thank you.
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nextstep
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Joined: 06 Jan 2007
Posts: 538

PostPosted: Wed Aug 06, 2014 1:26 am    Post subject: Reply with quote

Hi,
Quote:
Do you think Cayley Cubic surface's implicit equation could somehow be converted an parametric one?


According to this document: http://www.cs.utexas.edu/~bajaj/cs384R10/reading/algebraicFEM-chap1.pdf , page 37, the parametrization of the Cayley surface is possible:

Quote:

7.4 Automatic Parametrization of Degree 3 Curves and Surfaces
Rational algebraic curves and surfaces can be represented by implicit or parametric equations. A
general algebraic curve of degree three (cubics) is represented implicitly by
C(x, y) = ax3 + by3 + cx2
y + dxy2 + ex2 + fy2 + gxy + hx + iy + j = 0,
All singular cubics are rational, or stated equivalently, they can also be represented by a pair of
rational parametric equations x = u(t)/w(t) and y = v(t)/w(t), where u, v and w are no
more than cubic polynomials. Non-singular cubics are not rational and the best one can achieve
is a parameteric representation with a single square root of rational functions. Next, a generalalgebraic surface of degree three (cubicoids) has an implicit equation given by
C(x, y, z) = ax3 + by3 + cz3 + dx2
y + ex2
z + fxy2 + gy2
z + hxz2 + iyz2 + jxyz
+ kx2 + ly2 + mz2 + nxy + oxz + pyz + qx + ry + sz + t = 0.
All singular cubicoids are rational as are all non-singular cubicoids except cones and cylinders with
non-singular cubic generating curves. Rational cubicoids can also be represented by a triad of
rational parametric equations x = u(s, t)/q(s, t), y = v(s, t)/q(s, t), and z = w(s, t)/q(s, t) with
again u, v, w and q being no more than cubic polynomials
. Also, non-rational cubicoids can be
represented by a parameteric representation with a single square root of rational functions


Nb: You probably just need to look for a software that can convert a 3D triangular mesh to a parametric representation.
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Abderrahman
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clyde95



Joined: 03 Aug 2014
Posts: 4

PostPosted: Wed Aug 06, 2014 9:48 am    Post subject: Reply with quote

Thank you.
So how would -5*(x*x*y + x*x*z + y*y*x + y*y*z + z*z*y + z*z*x)+2*(x*y + x*z + y*z) implicit equation look in a parametric form?
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