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nextstep

Joined: 06 Jan 2007
Posts: 539 Posted: Wed Oct 22, 2014 5:31 am    Post subject: Thick parametric surfaces Hi,
As for the implicit surfaces, we can give some thickness to a parametric surface (F) by defining a second parametric surface (G) with the formula:

"Gx": ["Fx(u,v,t) + T*n1(u,v,t)/R(u,v,t)"],
"Gy": ["Fy(u,v,t) + T*n2(u,v,t)/R(u,v,t)"],
"Gz": ["Fz(u,v,t) + T*n3(u,v,t)/R(u,v,t)"]

Where T = Thickness value;
"n1 = (a2(u,v,t)*b3(u,v,t)-a3(u,v,t)*b2(u,v,t))",
"n2 = (a3(u,v,t)*b1(u,v,t)-a1(u,v,t)*b3(u,v,t))",
"n3 = (a1(u,v,t)*b2(u,v,t)-a2(u,v,t)*b1(u,v,t))",
"R = sqrt(n1(u,v,t)^2 + n2(u,v,t)^2 + n3(u,v,t)^2)"

" a1 = dFx/du",
" a2 = dFy/du",
" a3 = dFz/du",
" b1 = dFx/dv",
" b2 = dFy/dv",
" b3 = dFz/dv"

With the parametric Torus, you obtain this:

 Quote: { "Param3D": { "Name": [ "Torus" ], "Component": [ "Torus_1", "Torus_2" ], "Const": [ "T=1.3" ], "Funct": [ "Fx = (1+0.5*cos(u))*cos(v)", "Fy = (1+0.5*cos(u))*sin(v)", "Fz = 0.5*sin(u)", "a1 = (-0.5*sin(u)*cos(v))", "a2 = (-0.5*sin(u)*sin(v))", "a3 = (0.5*cos(u))", "b1 = (-(1+0.5*cos(u))*sin(v))", "b2 = ((1+0.5*cos(u))*cos(v))", "b3 = 0", "n1 = (a2(u,v,t)*b3(u,v,t)-a3(u,v,t)*b2(u,v,t))", "n2 = (a3(u,v,t)*b1(u,v,t)-a1(u,v,t)*b3(u,v,t))", "n3 = (a1(u,v,t)*b2(u,v,t)-a2(u,v,t)*b1(u,v,t))", "R = sqrt(n1(u,v,t)^2 + n2(u,v,t)^2 + n3(u,v,t)^2)", "Gx = Fx(u,v,t) + T*n1(u,v,t)/R(u,v,t)", "Gy = Fy(u,v,t) + T*n2(u,v,t)/R(u,v,t)", "Gz = Fz(u,v,t) + T*n3(u,v,t)/R(u,v,t)" ], "Fx": [ "Fx(u,v,t)", "Gx(u,v,t)" ], "Fy": [ "Fy(u,v,t)", "Gy(u,v,t)" ], "Fz": [ "Fz(u,v,t)", "Gz(u,v,t)" ], "Umax": [ "2*pi", "2*pi" ], "Umin": [ "0", "0" ], "Vmax": [ "5*pi/3", "5*pi/3" ], "Vmin": [ "0", "0" ] } } Torus by taha_ab, on Flickr

Enjoy _________________
Cheers,
Abderrahman   nextstep

Joined: 06 Jan 2007
Posts: 539 Posted: Mon Oct 05, 2015 12:24 pm    Post subject: This is the first implementation of the algorithm for building "Thick" parametric surfaces. Here you can see the result when applied to the "Richmond polar" parametric surface: MathMod script: { "Param3D": { "Description ": ["Thick Richmond polar Parametric surface by Abderrahman Taha 04/10/2015"], "Name": ["ThicknessParametric"], "Component": ["richmondpolar1", "richmondpolar2"], "Const": ["cu=0.0000000000001", "cv=0.0000000000001"], "Funct": ["Fx=-cos(t+v)/(2*u)-u^3*cos(t-3*v)/6", "Fy=-sin(t+v)/(2*u)+u^3*sin(t-3*v)/6", "Fz=u*cos(t-v)", "DFxu=((Fx(u,v,t)-Fx(u+cu,v,t))/cu)", "DFxv=((Fx(u,v,t)-Fx(u,v+cv,t))/cv)", "DFyu=((Fy(u,v,t)-Fy(u+cu,v,t))/cu)", "DFyv=((Fy(u,v,t)-Fy(u,v+cv,t))/cv)", "DFzu=((Fz(u,v,t)-Fz(u+cu,v,t))/cu)", "DFzv=((Fz(u,v,t)-Fz(u,v+cv,t))/cv)", "n1=(DFyu(u,v,t)*DFzv(u,v,t)-DFzu(u,v,t)*DFyv(u,v,t))", "n2=(DFzu(u,v,t)*DFxv(u,v,t)-DFxu(u,v,t)*DFzv(u,v,t))", "n3=(DFxu(u,v,t)*DFyv(u,v,t)-DFyu(u,v,t)*DFxv(u,v,t))", "R=sqrt(n1(u,v,t)^2+n2(u,v,t)^2+n3(u,v,t)^2)", "Tickness=.3", "Gx=Fx(u,v,t)+Tickness(u,v,t)*n1(u,v,t)/R(u,v,t)", "Gy=Fy(u,v,t)+Tickness(u,v,t)*n2(u,v,t)/R(u,v,t)", "Gz=Fz(u,v,t)+Tickness(u,v,t)*n3(u,v,t)/R(u,v,t)"], "Fx": ["Fx(u,v,t)", "Gx(u,v,t)"], "Fy": ["Fy(u,v,t)", "Gy(u,v,t)"], "Fz": ["Fz(u,v,t)", "Gz(u,v,t)"], "Umax": ["1.3", "1.3"], "Umin": ["0.3", "0.3"], "Vmax": ["2*pi", "2*pi"], "Vmin": ["0", "0"] } } richmond2 by taha_ab, on Flickr Double Nib1 by taha_ab, on Flickr Bat Suit by taha_ab, on Flickr_________________Cheers, Abderrahman   nextstep

Joined: 06 Jan 2007
Posts: 539 Posted: Tue Oct 06, 2015 1:27 pm    Post subject:  Twisted Clifford Torus by taha_ab, on Flickr Bumpy Klein by taha_ab, on Flickr_________________Cheers, Abderrahman   Display posts from previous: All Posts1 Day7 Days2 Weeks1 Month3 Months6 Months1 Year Oldest FirstNewest First
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