# Math Parametric Equation for Seashell

By Xah Lee. Date:

The following are some math formulas for seashell, by Mike Willams. (slightly modified)

```	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

```(* a torus *)

R = 1;
R2 = 2;

ParametricPlot3D[ {
R* Cos[u]*(R2+Cos[v]),
R*Sin[u]*(R2+Cos[v]),
R*Sin[v]
},
{u, 0, 6},
{v, 0, 10}
]
```

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 seashell:

### periwinkle seashell

```R =1;   (* radius of tube *)
cN = 4.6; (* number of turns *)
H =2;   (* height *)
P =2;   (* power *)

W = Function[ {u}, u/(2*Pi)*R ];
(* W = (#/(2*Pi)*R &) *)

ParametricPlot3D[
{
W[u]*Cos[cN*u]*(1+Cos[v]),
W[u]*Sin[cN*u]*(1+Cos[v]),
W[u]*Sin[v] + H*(u/(2*Pi))^P
},
{u, 0, 6}, {v, 0, 6}, PlotPoints -> 60, PlotRange -> All,
ColorFunction -> "TemperatureMap"
]
```

### top shell

And a top

```R=1; (* radius of tube *)
cN=7.6; (* number of turns *)
H=2.5; (* height *)
P=1.3; (* power *)

W = Function[ {u}, u/(2*Pi)*R ];

ParametricPlot3D[ {W[u]*Cos[cN*u]*(1+Cos[v]),
W[u]*Sin[cN*u]*(1+Cos[v]),
W[u]*Sin[v] + H*(u/(2*Pi))^P
},
{u,0,7},
{v,0,6},
PlotPoints -> {160,  10}, PlotRange -> All, ColorFunction -> "TemperatureMap"
]
```

### cone shell

A cone shell is something like this:

```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
```
```--
Mike Williams
Gentleman of Leisure```

```	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 * Seashell Gallery: Top Shell

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```

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
```

top

```tubeRadius = 1;     "radius of tube";
turns = 7.6;   "number of turns";
height = 2.5;   "height";
powerP = 1.3;   "power";
triangleX = 0.8;   "Triangleness of cross section";
crossTilt = -0.3;   "crossTiltngle of tilt of cross section (radians)";

ff = ((#/(2*Pi)*tubeRadius) &);

ParametricPlot3D[  {
ff[u]*Cos[turns*u]*(1+Cos[v+crossTilt]+Sin[2*v+crossTilt]*triangleX/4),
ff[u]*Sin[turns*u]*(1+Cos[v+crossTilt]+Sin[2*v+crossTilt]*triangleX/4),
ff[u]*(Sin[v+crossTilt]+Cos[2*v+crossTilt]*triangleX/4)  + height*(u/(2*Pi))^powerP
},
{u,0,3},
{v,0,3}
]```

back to Math of Seashell Shapes