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File:Mug and Torus morph.gif

Mug_and_Torus_morph.gif(240 × 240 pixels, file size: 497 KB, MIME type: image/gif, looped, 58 frames, 12 s)


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Contents

Summary

A coffee mug morphing into a torus, a popular example in topology.

Rendered using POV-Ray.

Türkçe: Bir kahve bardağının simide sürekli deformasyonunu gösteren bir homeomorfizma animasyonu.

POV-Ray source code

/* 
Torus to mug morphing animation, by Lucas V. Barbosa - January 17, 2007 - Coded in: POV-Ray 3.6
License:

Info
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Author user page: http://en.wikipedia.org/wiki/User:Kieff
The final animation at Wikimedia Commons: http://commons.wikimedia.org/wiki/Image:Mug_and_Torus_morph.gif


Notes
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Several modifications were done for each part of the animation and I didn't keep them all here.
So if you wish to reproduce the animation, you'll have to play with this code quite a bit. :)

*/


background { colour rgb 1 }

global_settings { assumed_gamma 1.5 }

camera {
        orthographic
        location <0,4,-5>*0.8
        look_at <0,0,0>
        //rotate -45*y
}

#declare dist = 0.2;
light_source {
        <0,0,0>, 1
        //translate <-3,5,-5>*1
        translate 5*y
        translate -5*z
        translate 2*x
}


#declare a = 1;

// Cylinders for the mug shape
#declare C_a = function { pow(x / a,2) + pow(z / a,2)-1.5 }
#declare C_b = function { pow(x / a,2) + pow(z / a,2)-1.2 }
#declare B_b = function { (y+1.3) }


// Torus (handle and the morphing target)
#declare R = 1; #declare r = 0.22;
#declare T_a = function { pow(R - sqrt(pow(x,2) + pow(y,2)),2) + pow(z,2) - pow(r,2) }
#declare R2 = 1; #declare r2 = 0.5;
#declare T_b = function { pow(R2 - sqrt(pow(x,2) + pow(y,2)),2) + pow(z,2) - pow(r2,2) }

// Hollow mug (concave)
#declare Mug = function { min(min(max(C_a(x,y,z),B_b(x,y,z)),max(C_a(x,y,z),-C_b(x,y,z))),max(T_a(x-1,y,z),-C_a(x,y,z))) }

// Solid mug (cylinder with handle, convex)
#declare Mug2 = function { min(C_a(x,y,z),max(T_a(x-1,y,z),-C_a(x,y,z))) }


#declare c = 0.5;

// Linear to smooth interpolation
#declare Int = pow(sin(clock*pi/2),2);


// Functions for the mug parts, left as backups.
// max(C_a(x,y,z),B_b(x,y,z)) bottom
// max(C_a(x,y,z),-C_b(x,y,z)) sides
// min(max(C_a(x,y,z),B_b(x,y,z)),max(C_a(x,y,z),-C_b(x,y,z))) bottom+sides
// max(T_a(x-1,y,z),-C_a(x,y,z)) handle
// min(min(max(C_a(x,y,z),B_b(x,y,z)),max(C_a(x,y,z),-C_b(x,y,z))),max(T_a(x-1,y,z),-C_a(x,y,z))) full mug
        

// Render as isosurface or CSG.
// Isosurface is used in the transitional states.
#declare RenderIsosurface = 0;
#if (RenderIsosurface)
union {
        isosurface {
                function {
                        // Add previously defined object functions using Int and (1-Int) as to generate the smooth transition
                //Mug2(x,y,z)*(1-Int) +
                        Mug(x,y,z)
                        //T_b(x-1,y,z)*Int
                }
                contained_by { box { -<3,1.5,3>, <3,1.5,3> } }
                accuracy 0.001
                max_gradient 5 // 10
                //rotate -90*x
        }
        cylinder {
                <0,-1.3+Int*2.8,0>, <0,-1.5,0>, sqrt(1.5)
        }
        pigment {
                colour rgb <0.6,0.8,1> transmit 0.1
        }
        finish {
                specular 0.5
                roughness 0.01
                ambient 0.2
        }
}
#else
union {
        difference {
                cylinder {
                        -1.5*y, 1.5*y, sqrt(1.5)
                }
                cylinder {
                        -2*y, 2*y, sqrt(1.2)
                }
        }
        difference {
                torus {
                        R, r
                        rotate 90*x
                        translate x
                }
                cylinder {
                        -1.5*y, 1.5*y, sqrt(1.5)
                }
        }
        cylinder {
                <0,0-0.25+Int*(1.5+0.25),0>, <0,-1.5,0>, sqrt(1.5)
        }
        pigment {
                colour rgb <0.6,0.8,1> transmit 0.1
        }
        finish {
                specular 0.5
                roughness 0.01
                ambient 0.2
        }
}
#end

/* END OF FILE */

Updated January 12, 2007

  1. New colors: blue hue, brighter, softer shadows
  2. Changed light angle
  3. Non-linear interpolation: looks smoother and less boring
  4. Faster frame rate
  5. Cropped to relevant area
  6. Overall, smaller size for a better-looking image.

Updated January 20, 2007

  1. Removed dithering. Image now has visible colour bands, but at least thumbnails won't look excessively grainy.

Updated March 1, 2007

  1. More saturation of colors

Licensing

Public domain I, the copyright holder of this work, release this work into the public domain. This applies worldwide.
In some countries this may not be legally possible; if so:
I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

See also

The following pages on Schools Wikipedia link to this image (list may be incomplete):

Background information

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