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ufoace
Joined: 11 Mar 2013 Posts: 46


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Furan
Joined: 05 Oct 2010 Posts: 64 Location: Prague, Czech Republic

Posted: Mon Apr 08, 2013 6:51 am Post subject: 


It's pretty easy once you get the main curve. Follow the same technique as for the slanted cylinder:
Get a parametrical representation of the knot.
X=X(t)
Y=Y(t)
Z=Z(t)
Derivate according to t. You'll get the tangencial vector u.
cross u with (0,0,1) to get one orthogonal vector B
cross B with u to get the second orthogonal vector C.
Normalize.
Parametric surface
x = X(t) + R * (bx*cos(phi) + cx*sin(phi))
y = Y(t) + R * (by*cos(phi) + cy*sin(phi))
z = Z(t) + R * (bz*cos(phi) + cz*sin(phi))
t = 0...2pi usually
phi=0...2pi
You may find some useful segments of code here:
http://www.fractalforums.com/theory/ageneralizationoftriplexzpplusc/ 

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Furan
Joined: 05 Oct 2010 Posts: 64 Location: Prague, Czech Republic

Posted: Mon Apr 08, 2013 10:51 am Post subject: 


I completely forgot I had this all along:
Code:  a=3
Trifold knot curve:
x = a*sin(t)+2*a*sin(2*t)
y = a*cos(t)2*a*cos(2*t)
z = a*sin(3*t)
t=0..2*pi
Inner functions:
u1(u) = 2*cos(u)+8*cos(2*u)
u2(u) = 2*sin(u)+8*sin(2*u)
u3(u) = 6*cos(3*u)
v1(u) = u2(u)
v2(u) = u1(u)
v3(u) = 0 // omitted
w1(u) = 12*cos(3*u)*(cos(u)+4*cos(2*u))
w2(u) = 12*cos(3*u)*(sin(u)4*sin(2*u))
w3(u) = 6832*cos(3*u)
normv(u) = sqrt(32*cos(3*u)+68)
normw(u) = sqrt(8)*sqrt(795+36*cos(9*u)+217*cos(6*u)+652*cos(3*u))
xp(u,v) = v1(u)*cos(v)/normv(u)+w1(u)*sin(v)/normw(u)
yp(u,v) = v2(u)*cos(v)/normv(u)+w2(u)*sin(v)/normw(u)
zp(u,v) = w3(u)*sin(v)/normw(u)
Trifold Knot Surface:
r=1
x = a*sin(u)+2*a*sin(2*u)+r*xp(u,v)
y = a*cos(u)2*a*cos(2*u)+r*yp(u,v)
z = a*sin(3*u)+r*zp(u,v)
u=0..2*pi
v=0..2*pi



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ufoace
Joined: 11 Mar 2013 Posts: 46

Posted: Tue Apr 09, 2013 10:35 am Post subject: 


Thank you Furan, that is extremely helpful! I will be attempting to write an isosurface version so that I can multiply the spirographs more easily with different functions. I am working on an interactive isosurface platform (see video that end, contact anyone if you want a copy  i was going to mix the spirographs with perlin noise function bended around inside the spirograph for an exploration area.
what is confusing me is the mathematical complexity of bending an isosurface cylinder around the line made by a circle or a parabole, if the cylinder runs along x, it' s easy, and if it runs along x*x, parabolic shape, I don' t how the formula copes with the changing vectors.... hmm maybe from this solution also: http://k3dsurf.s4.bizhat.com/viewtopic.php?t=108&postdays=0&postorder=asc&start=75
Here is the interactive platform I am working on, it lets you walk on /fly around isosurfaces. i can t do parametric on it:
http://youtu.be/ygeCClSxrhU 

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Furan
Joined: 05 Oct 2010 Posts: 64 Location: Prague, Czech Republic

Posted: Wed Apr 10, 2013 4:54 am Post subject: 


Well, I am struggling with the very same problem. The thing is, the spirograph isn't really a function, because for given [x,y,z] you cannot directly define unique position t. Therefore I had to use the "Initial guess" and find it with NewtonRhapson.
So one solution, that I just didn't have time to implement, is to enable the engine to cope with it:
For the ith integration step on the ray, run the initial guess and compare it with (t, phi, r)_i1 If it's on the same level, you can start iterating from i1 values.
The initial guess also may be a function of (t, phi, r)_i1 and Jacobian J_i1 and just transform the ray vector to the local coordinates t,phi,r. They are curved, but still you can get a good approximation, especially if you take smaller steps at high curvatures.
Ok, now I have seen your video. Nice work!!! I would love to shoot some zombies in a semifractal world Well, that changes things. Now I'd suggest to leave it in parametrical coordinates and following the process I had just described, calculate the isofunction on a very fine cubic grid (1024x1024x1024). So for every [x,y,z] you will get deltaR(r,t,phi) as the shortest outer distance from the knot surface. (Negative for points inside, or the other way). Apply perlin noise and other features on this discrete deltaR(x,y,z) function. Than by linear interpolation or a more sophisticated method find points deltaR=0 and generate polygonal mesh. 

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