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nextstep
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Joined: 06 Jan 2007
Posts: 539
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 Posted: Wed Oct 22, 2014 5:31 am Post subject: Thick parametric surfaces |
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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 |
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nextstep
Site Admin
Joined: 06 Jan 2007
Posts: 539
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| Posted: Mon Oct 05, 2015 12:24 pm Post subject: |
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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
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Abderrahman |
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nextstep
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