Lame Klein bottle

[DarkBeam]

  I am very lame, was thinking about making a Klein Bottle using four half-toruses. The problem is that one of them should have a variable radius, that closes the bottle in itself smoothly, and I am not able to do this

Was thinkin about a function of atan(y,x) to do this;

Code:

#info DarkBeam's Klein Bottle
#include "DE-Raytracer.frag"
#group KBottle

#define PI 3.14159

//Radius of the tubes
uniform float tubeRadius; slider[0,0.3614,100.]

//Radius of the goup of tubes
uniform float groupRadius; slider[0,0.85,5.]

//Radius of the whole object (because it looks like a torus (-:)
uniform float objectRadius; slider[0,.7,1.7]

uniform float ggg; slider[0,.7,1.7]
uniform float hhh; slider[0,3.,12.]
float Tor(vec3 p, float R, float tR, float Tube){
      float j;
      float a = p.x,b = p.y,c = p.z;
	j =sqrt(a*a+b*b)-R ;
	j =sqrt(j*j+c*c);
	return abs(j-tR)-Tube;
}

float ETor(vec3 p, float R, float tR, float Tube, float el ){  // THIS IS WRONG
      float j;
      float a = p.x,b = p.y,c = p.z;
	j =sqrt(a*a+b*b);
      a = (atan(b,a))/hhh;
       j = j*(ggg-tubeRadius*a*a*a)-R;
	j =sqrt(j*j+c*c);
	return abs(j-tR)-Tube;
}

float YinYang(vec3 p, float R, float tR, float Tube){
      float j;
      float a = p.x,b = p.y,c = p.z;
      if (a>0) {b+=R;} else {b-=R;}
	j =sqrt(a*a+b*b)-R ;
	j =sqrt(j*j+c*c);
	return abs(j-tR)-Tube;
}

float DE(vec3 p) {
	float d1,d2,d3;
       d1=1000;//Tor(p,1.,.3,.1);
       if (p.z<0.0 &&1) {
           	return .1-p.z;
		} 
       if (p.x>0) {
             p.y +=objectRadius;
           	d1=ETor(p/groupRadius,2.*1.,.2,.02,-tubeRadius);
             p.y -=objectRadius;
		} else {
             p.y +=2.;
		d1=Tor(p.yzx,.2,.1,.02);
             p.y -= 2.;
		}
       d3=YinYang(p,1.,.3,.02);
       d1=min(d1,d3);
	return d1;
}
#preset default
FOV = 0.62536
Eye = -0.684574,2.97917,-0.56351
Target = 1.28819,-5.52322,0.968231
Up = -0.474302,-0.261493,-0.84063
AntiAlias = 1
Detail = -3
DetailAO = -1.57143
FudgeFactor = 0.66265
MaxRaySteps = 64
BoundingSphere = 4 Locked
Dither = 0 Locked
NormalBackStep = 1
AO = 0,0,0,0.90123
Specular = 4.4304
SpecularExp = 16
SpotLight = 0.435294,0.737255,1,0.5
SpotLightDir = 0.65626,0.5
CamLight = 1,0.941176,0.898039,1.0566
CamLightMin = 0
Glow = 1,1,1,0
GlowMax = 20
Fog = 0
HardShadow = 0
ShadowSoft = 12.9032
Reflection = 0
BaseColor = 0.701961,0.701961,0.701961
OrbitStrength = 0
X = 0.411765,0.6,0.560784,0.41748
Y = 0.666667,0.666667,0.498039,-0.16504
Z = 1,0.258824,0.207843,1
R = 0.0823529,0.278431,1,0.82352
BackgroundColor = 0.607843,0.866667,0.560784
GradientBackground = 0.3261
CycleColors = true
Cycles = 4.04901
EnableFloor = false
FloorNormal = 0,0,0
FloorHeight = 0
FloorColor = 1,1,1
tubeRadius = 0.07534
groupRadius = 0.29851
objectRadius = 0.43077
Rotations = 3
RotDenominator = 3
RotNumeratorX = -2
RotNumeratorY = -4
#endpreset

The problem is that the cut plane does not work well and I did not found a good mangling for that atan()

Thanks for your endless patience

[DarkBeam]

If somebody finds the answer I will donate the Figure-8 immersion formula, that looks so cool

Yes it's still a Klein Bottle but does not look like

[blob]

Completely off-topic but I was wondering if an hyperbolic tesselation transform for dIFS was in the realm of the possible?

[hobold]

A torus with a varying radius? Here: http://en.wikipedia.org/wiki/Dupin_cyclide

[fractower]

From my second favorite math related sight http://paulbourke.net/geometry/klein/

Paul rules!!

[DarkBeam]

Thanks but I found a workaround for my function - here it is.
For some values it goes wrong but hey it's normal

Code:

#info DarkBeam's Klein Bottle
#include "DE-Raytracer.frag"
#group KBottle

#define PI 3.14159

//Radius of the tubes
uniform float tubeRadius; slider[0,0.7,3.]

//Radius of the goup of tubes
uniform float groupRadius; slider[0,0.13,0.4]

//Radius of the whole object (because it looks like a torus (-:)
uniform float objectRadius; slider[0,.7,1.7]

uniform float ggg; slider[0.,1.6,4.]
//uniform float hhh; slider[0.,3.,12.]
float Tor(vec3 p, float R, float tR, float Tube){
      float j;
      float a = p.x,b = p.y,c = p.z;
	j =sqrt(a*a+b*b)-R ;
	j =sqrt(j*j+c*c);
	return abs(j-tR)-Tube;
}

float ETor(vec3 p, float R, float tRmin, float tRmax, float Tube ){
      float j;
      float a = p.x,b = p.y,c = p.z;
	j =sqrt(a*a+b*b);
      a = (atan(b,a)/PI-ggg+2);
      a = clamp(a,tRmin,tRmax);
       j = j-R;
	j =sqrt(j*j+c*c);
	return abs(j-a)-Tube;
}

float YinYang(vec3 p, float R, float tR, float Tube){
      float j;
      float a = p.x,b = p.y,c = p.z;
      if (a>0) {b+=R;} else {b-=R;}
	j =sqrt(a*a+b*b)-R ;
	j =sqrt(j*j+c*c);
	return abs(j-tR)-Tube;
}

float DE(vec3 p) {
	float d1,d2,d3;
       d1=1000;//Tor(p,1.,.3,.1);
       if (p.z<0.0 &&0) {
           	return .1-p.z;
		} 
       if (p.x>0) {
           	d1=ETor(p,2.*1.,tubeRadius-2*groupRadius,tubeRadius,.02);
		} else {
             p.y +=2.;
		d1=Tor(p.yzx,tubeRadius-groupRadius,groupRadius,.02);
             p.y -= 2.;
		}
       d3=YinYang(p,1.,tubeRadius,.02);
       d1=min(d1,d3);
	return d1;
}
#preset default
FOV = 0.62536
Eye = -0.684574,2.97917,-0.56351
Target = 1.28819,-5.52322,0.968231
Up = -0.474302,-0.261493,-0.84063
AntiAlias = 1
Detail = -3
DetailAO = -1.57143
FudgeFactor = 0.66265
MaxRaySteps = 64
BoundingSphere = 4 Locked
Dither = 0 Locked
NormalBackStep = 1
AO = 0,0,0,0.90123
Specular = 4.4304
SpecularExp = 16
SpotLight = 0.435294,0.737255,1,0.5
SpotLightDir = 0.65626,0.5
CamLight = 1,0.941176,0.898039,1.0566
CamLightMin = 0
Glow = 1,1,1,0
GlowMax = 20
Fog = 0
HardShadow = 0
ShadowSoft = 12.9032
Reflection = 0
BaseColor = 0.701961,0.701961,0.701961
OrbitStrength = 0
X = 0.411765,0.6,0.560784,0.41748
Y = 0.666667,0.666667,0.498039,-0.16504
Z = 1,0.258824,0.207843,1
R = 0.0823529,0.278431,1,0.82352
BackgroundColor = 0.607843,0.866667,0.560784
GradientBackground = 0.3261
CycleColors = true
Cycles = 4.04901
EnableFloor = false
FloorNormal = 0,0,0
FloorHeight = 0
FloorColor = 1,1,1
tubeRadius = 0.07534
groupRadius = 0.29851
objectRadius = 0.43077
Rotations = 3
RotDenominator = 3
RotNumeratorX = -2
RotNumeratorY = -4
#endpreset

[fractower]

It seems that everyone with a Shapeways account has done a Klein bottle. This is my attempt

http://www.shapeways.com/model/242965/

The surface is is made from a 3d spirograph texture.

[DarkBeam]

From my second favorite math related sight http://paulbourke.net/geometry/klein/

Paul rules!!

The problem is that you can't (easily) extract a DE from a parametric equation. If that was possible easily I certainly had no need to ask

[kram1032]

Well, here you have an implicit formulation of the symmetric projection of the Klein bottle
http://www.wolframalpha.com/input/?i=klein+bottle

If you explicitly want the bottle-like one, I'm pretty sure, that requires yet another Polynomial of degree 12. Most likely one that, due to lack of symmetry, is even more complicated than the one given there...

[DarkBeam]

I have further improved my routine removing the "intersected" zone, so now I have a "real" bottle that I can explore going inside it. Sorry, a Klein bottle has no inside
Kram; I tried all parametrizations but none looked as pleasing as this; the symm one is like a "fat" bottle, very easy to implement as it is a single expression but a lot less elegant to my eye

[kram1032]

Well, I guess that's a pure opinion thing, but I personally am a sucker for symmetry (any kind of, also fractal symmetry, obviously), so the symmetric version of the Klein bottle seems nicer to me.
The "bottle" one seems very much like giving it a real life meaning which it really hasn't.

The symmetry also helps to understand the shape better without your mind constantly interfering associations...

[DarkBeam]

Images

[cKleinhuis]

ok, can we create drinking glasses from it and sell it as ff merchandise ?!
or is it open at the other end

[DarkBeam]

ok, can we create drinking glasses from it and sell it as ff merchandise ?!
or is it open at the other end

Oh yeah, why don't you affiliate with those guys! http://www.kleinbottle.com/

 

[kram1032]

Lawl...

Well, since the 3D projection of such a Klein Bottle will always intersect with itself, you have a part where liquid can be kept in... However, that part will not allow the liquid to get out again, so it's kinda pointless...

In an unprojected, "flat" version (by means of curvature and other things, the Klein Bottle actually behaves like the euclidean plane despite being limited in surface), you obviously couldn't keep any liquid. Especially once you start arguing what side is inside and after going in a circle once you suddenly realize, EVERY side is inside.
Or outside for that matter.