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	From: 	  mike@econym.demon.co.uk
	Subject: 	Paramnetric equations of seashells
	Date: 	January 19, 2005 12:35:17 AM PST
	To: 	  xah@xahlee.org

I hear that you're interested in parametric equations that approximate
spiral seashells. Here are a few that I've made.

They're all based on this parameterisation of the torus

  Fx = R*cos(u)*(R2+cos(v))
  Fy = R*sin(u)*(R2+cos(v))
  Fz = R*sin(v)

I then replaced R with R*u/(2*pi) so that the radius of the tube
increases linearly as u increases, and replaced sin(u) with sin(N*u) so
that the tube spirals round the origin N times. Then I added an offset
factor H*(u/2*pi)^P so that the turns of the spiral are offset in the z
direction  by a distance that varies from 0 to H.


This gives a periwinkle:

R=1;    // radius of tube
N=4.6;  // number of turns
H=2;    // height
P=2;    // power

W(u) = u/(2*pi)*R
Fx = W(u)*cos(N*u)*(1+cos(v))
Fy = W(u)*sin(N*u)*(1+cos(v))
Fz = W(u)*sin(v) + H*(u/(2*pi))^P


And a top

R=1;    // radius of tube
N=7.6;  // number of turns
H=2.5;  // height
P=1.3;  // power

W = u/(2*pi)*R
Fx = W(u)*cos(N*u)*(1+cos(v))
Fy = W(u)*sin(N*u)*(1+cos(v))
Fz = W(u)*sin(v) + H*(u/(2*pi))^P


A cone shell is something like this:

R=1;    // radius of tube
N=4.6;  // number of turns
H=0.5;  // height
P=2;    // power

W = u/(2*pi)*R
Fx = W(u)*cos(N*u)*(1+cos(v))
Fy = W(u)*sin(N*u)*(1+cos(v))
Fz = W(u)*sin(v)*1.25 + H*(u/(2*pi))^P +W(u)*cos(v)*1.25


And here's a wrinkled periwinkle

R=1;    // radius of tube
N=4.6;  // number of turns
H=2.5;  // height
F=80;   // wave frequency
A=0.2;  // wave amplitude
P=1.9;  // power

W = u/(2*pi)*R
Fx = W(u)*cos(N*u)*(1+cos(v)+cos(F*u)*A)
Fy = W(u)*sin(N*u)*(1+cos(v)+cos(F*u)*A)
Fz = W(u)*sin(v) + H*(u/(2*pi))^P

--
Mike Williams
Gentleman of Leisure

ssss---------------------------------------------------

	From: 	  private@econym.demon.co.uk
	Subject: 	Parametric equations of seashells
	Date: 	January 20, 2005 9:41:46 PM PST
	To: 	  xah@xahlee.org

You wrote

> hi Mike,
> 
> Great work!
> 
> of your results, i find the wrinkled version of greatest interest. Now  
> it would be great if one can add to the equation periodic spikes, as to  
> make one of those horny ones,

I reckon that this general approach would only be able to place spikes
that point in the +-x and +-y directions. This might make something that
at first glance resembles a Venus Comb, but the rows of spikes should
really point in directions separated by about 120 degrees rather than
90.

For spikes that point in other directions I think you'd need an entirely
different approach.


> I think the ideal "top" one should have a flat surface, very angular  
> triangular opening, and with a cone shaped hole at the bottom. See
> * http://xahlee.org/xamsi_calku/top/top.html

We only have one species of "top" in my country (Britain), the "Toothed
Topshell" (Monodonta lineata) and it is quite rounded. A flat sided
topshell is more like this:

R=1;    // radius of tube
N=7.6;  // number of turns
H=2.5;  // height
P=1.3;  // power
T=0.8;  // Triangleness of cross section
A=-0.3;  // Angle of tilt of cross section (radians)

W = u/(2*pi)*R}

Fx = W(u)*cos(N*u)*(1+cos(v+A)+sin(2*v+A)*T/4)}
Fy = W(u)*sin(N*u)*(1+cos(v+A)+sin(2*v+A)*T/4)}
Fz = W(u)*(sin(v+A)+cos(2*v+A)*T/4)  + H*(u/(2*pi))^P}


Which also gives the possibility of creating something like a turret
shell (oddly, my book of British fauna doesn't include these, but
there's loads of them on the local beach). I can't quite seem to get the
coils to touch each other correctly at the tip.

R=1;    // radius of tube
N=9.6;  // number of turns
H=5.0;  // height
P=1.5;  // power
P1=1.1; // another power
T=0.8;  // Triangleness of cross section
A=0.1;  // Angle of tilt of cross section (radians)
S=1.5;  // Stretch

W = (u/(2*pi)*R)^P1

Fx = W(u)*cos(N*u)*(1+cos(v+A)+sin(2*v+A)*T/4)}
Fy = W(u)*sin(N*u)*(1+cos(v+A)+sin(2*v+A)*T/4)}
Fz = S*W(u)*(sin(v+A)+cos(2*v+A)*T/4)  + S*H*(u/(2*pi))^P}


Whilst trying to get the turret working, I happened to create this by
mistake. 

R=1;    // radius of tube
N=7.6;  // number of turns
H=2.5;  // height
P=1.3;  // power
T=0.8;  // Triangleness of cross section
A=-0.3; // Angle of tilt of cross section (radians)
S=3.0;  // Stretch

W = u/(2*pi)*R}

Fx = W(u)*cos(N*u)*(1+cos(v+A)+sin(2*v+A)*T/4)}
Fy = W(u)*sin(N*u)*(1+cos(v+A)+sin(2*v+A)*T/4)}
Fz = S*W(u)*(sin(v+A)+cos(2*v+A)*T/4)  + S*H*(u/(2*pi))^P}


I should have mentioned (but I guess you've worked it out for yourself)
that all these surfaces are designed to be generated with u and v both
being in the range 0 to 2*pi.

> PS may i forward your results to the curves-surfaces mailing list?

Yes.

Is there an actual mailing list? All I could find at the URL you gave me
was one of those horrible Google Groups. 

-- 
Mike Williams
Gentleman of Leisure

ssss---------------------------------------------------

spindle shell

R=1;    // radius of tube
N=5.6;  // number of turns
H=4.5;  // height
P=1.4;  // power
L=4;    // Controls spike length
K=9;    // Controls spike sharpness

W = (u/(2*pi)*R)^0.9

Fx = W(u)*cos(N*u)*(1+cos(v))
Fy = W(u)*sin(N*u)*(1+cos(v))
Fz = W(u)*(sin(v)+L*(sin(v/2))^K)  + H*(u/(2*pi))^P}

ssss---------------------------------------------------
top

R=1;    // radius of tube
N=7.6;  // number of turns
H=2.5;  // height
P=1.3;  // power
T=0.8;  // Triangleness of cross section
A=-0.3;  // Angle of tilt of cross section (radians)

W = u/(2*pi)*R}

Fx = W(u)*cos(N*u)*(1+cos(v+A)+sin(2*v+A)*T/4)}
Fy = W(u)*sin(N*u)*(1+cos(v+A)+sin(2*v+A)*T/4)}
Fz = W(u)*(sin(v+A)+cos(2*v+A)*T/4)  + H*(u/(2*pi))^P}