"Mirrored" Isosurfaces

 
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nextstep
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Joined: 06 Jan 2007
Posts: 538

PostPosted: Fri Sep 07, 2007 5:03 am    Post subject: "Mirrored" Isosurfaces Reply with quote

Hi all,
You can use this formula to obtain a mirrored view of an isosurface F() :
Quote:
G(x,y,z) = F(x,y,z)* F(x - 2*P(x,y,z)*a/R,
y - 2*P(x,y,z)*b/R,
z - 2*P(x,y,z)*c/R)
Where :
P(x,y,z) = ax+by+cz+d (Equation of the Plan)
R = (a^2 + b^2 + c^2)

Maybe for future use in minimal surface? Wink
Enjoy Razz
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Abderrahman


Last edited by nextstep on Sun Oct 26, 2014 7:13 am; edited 1 time in total
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inode



Joined: 27 Jan 2007
Posts: 127
Location: Austria

PostPosted: Fri Sep 14, 2007 11:44 pm    Post subject: Reply with quote

can you publish an example?

e.g. hyperbolic paraboloid x^2-y^2-z
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nextstep
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Joined: 06 Jan 2007
Posts: 538

PostPosted: Sat Sep 15, 2007 6:05 am    Post subject: Reply with quote

Hi all,
Here it is Cool . I added the plan to the equation for better understanding. We can probably make interesting shapes by using this formulas especially for minimal surfaces...
Quote:
/*
Hyperbol and it's mirrored image to the plan defined
by : y = 4 (or y-4 =0)
*/
Name: Mirror_1
F(): (x^2-y^2-z) *(x^2-(y-2*(y-4))^2-z) :
y-4:
[x]: -4 , 4
[y]: -4 , 12
[z]: -4 , 4
;


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nextstep
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Joined: 06 Jan 2007
Posts: 538

PostPosted: Sun Oct 26, 2014 7:16 am    Post subject: Reply with quote

Hi all,
For a Torus ((sqrt(x*x+y*y)-3)^2 + z*z -1) and a plan (P=y-z+4), this is what you get :
Code:
{
    "Iso3D": {
        "Name": [
            "MirroredTorus"
        ],
        "Description": [
            "Mirrored Torus"
        ],
        "Component": [
            "TorusIso",
            "Plan"
        ],
        "Const": [
            "a=0",
            "b=1",
            "c=-1",
            "d=4",
            "R=(0^2 + (1)^2 + (-1)^2)"
        ],
        "Funct": [
            "Torus = ((sqrt(x*x+y*y)-3)^2 + z*z -1)",
            "P = (a*x+b*y+c*z+d)"
        ],
        "Fxyz": [
            "Torus(x,y,z+1,t)*Torus(x-2*P(x,y,z,t)*a/R,y-2*P(x,y,z,t)*b/R,z+1-2*P(x,y,z,t)*c/R,t)",
            "y-z+4"
        ],
        "Xmax": [
            "4",
            "4"
        ],
        "Xmin": [
            "-4",
            "-4"
        ],
        "Ymax": [
            "8",
            "8"
        ],
        "Ymin": [
            "-6",
            "-6"
        ],
        "Zmax": [
            "9",
            "9"
        ],
        "Zmin": [
            "-4",
            "-4"
        ]
    }
}

MirroredTorus by taha_ab, on Flickr
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Abderrahman
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nextstep
Site Admin


Joined: 06 Jan 2007
Posts: 538

PostPosted: Sun Oct 26, 2014 5:19 pm    Post subject: Reply with quote

Hi all,
This is a better script for the mirrored Torus, however, it should not work with MathMod previous versions... Hope to publish it as soon, stay tuned Smile
Quote:

{
"Iso3D": {
"Component": [
"TorusIso",
"Plan"
],
"Const": [
"a=0",
"b=1",
"c=-1",
"d=4",
"R=(a^2 + b^2 + c^2)"
],
"Description": [
"Mirrored Torus"
],
"Funct": [
"Torus = ((sqrt(x*x+y*y)-3)^2 + z*z -1)",
"P = (a*x+b*y+c*z+d)"
],
"Fxyz": [
"Torus(x,y,z+1,t)*Torus(x-2*P(x,y,z,t)*a/R,y-2*P(x,y,z,t)*b/R,z+1-2*P(x,y,z,t)*c/R,t)",
"a*x + b*y +c*z + d"
],
"Name": [
"MirroredTorus"
],
"Xmax": [
"4",
"4"
],
"Xmin": [
"-4",
"-4"
],
"Ymax": [
"4",
"4"
],
"Ymin": [
"-6",
"-6"
],
"Zmax": [
"8",
"8"
],
"Zmin": [
"-2",
"-2"
]
}
}

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Abderrahman
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denisc



Joined: 24 Apr 2013
Posts: 92

PostPosted: Thu Oct 30, 2014 11:24 am    Post subject: Reply with quote

hello next step

thanks
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