3D Pipes - Tutorial

• The variable t defines the time from 0 to 2*pi
• X, Y, Z define a 3D curve as a function of t.
• D defines the width in mm of the generating circle/polygon as a function of time.
• S defines the generating shape: circle, hexagon, pentagon, square, triangle, ribbon.
• If S is not a circle, TW defines the twist angle of the generating shape as a function of time

This formula will generate a torus:

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This formula will generate a cone and by modifying S a variety of pyramids:

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There is a single predefined constant: pi (3.14159....)

The variable t (time) ranges initially from 0 to 2*pi

Some auxiliary variables can be used, namely:   a, b, k, r, n, w

Auxiliary variable assignment is of the form:  variable = expr;
Several assignments can be done on the same line, like
a = (t-pi)/2; b=t/(2*pi); a
Note that the last term must be the value applied to X, Y or Z...

Variables u and v are used to generate surfaces, see here.

Variables x, y, z, d can be used once they have been computed.

Following arithmetic operators can be used: + - * / < > =  +=  -=  *=  /=  <=  >=  ==  != 
and the conditional operator ? vt : vf see here.

Following functions can be used: sin(), cos(), asin(), acos(), sinh(), cosh(), abs(), exp(), log(), pow(), random(), sqrt(), tan(), atan2(), round(), floor().
pow() needs two parameters, like pow(n, 5) n to the power of 5, atan2() needs 2 parameters: y and x, random() has no parameters and generates a random number between 0 an 1.

Examples:
 

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Using variables x, y, z, d once they have been computed

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Using hyperbolic function cosh.
sinh and cosh are available only on modern browsers.

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Modifying t range from [0, 2*pi] to [-1, +1]

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Examples:

Truncated pyramid with twisted handle

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Locking key

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• Pipe mode :
  16 values of v are used and for each v, 32 values of u are generated. The selected shape follows the X ,Y, Z 3D curve to generate a volume.
• Surface mode  :
  u, v are used to generate a surface. Note that in most cases, this does not generate a closed printable volume. Therefore, an additional control is displayed. By checking it, the surface is given a 2mm thickness and becomes printable.

Examples:

A very simple u, v pipe plane. 16 pipes are generated in the Y direction

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A very simple printable u, v surface with 2mm thickness

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Printable surface of revolution. The generating curve is Z. Note the v==0? condition to avoid 0/0 division.

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A non printable Mobius strip. Note that v range is modified from [0,2*pi] to [-1,1].
There's a much easier way to define a printable Mobius strip

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A non printable Klein bottle

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Partial 3D print showing inside of Klein bottle,
using above formula with

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• if (condition) {statements; value;} else {statements; value}
• switch(expression) {case value: statements; value; break; ... case... ; default:... }

The following example shows the use of "switch" to generate a tetrapod. It uses the fact that for each value of v, 32 values of u are generated. The first 4 "v" values out of 16 are used.


X Expression: 
u = u / pi / 2; // u range from 0 to 1
a = floor(v/pi*8); // a range from 0 to 15, a=0 top leg, a=1 left leg, a=2 right leg, a = 3 back leg, a>3 discard
b = 1.5 * u; // scaling
switch(a) {
  case 0: case 3: 0; break; // top leg, back leg
  case 1: -b*cos(pi/6); break; // left leg
  case 2: b*cos(pi/6); break; // right leg
  default: 0;}

Y Expression:
switch(a) {
   case 0: 0; break; // top leg
   case 1: case 2: -b*sin(pi/6); break; // left and right legs
   case 3: b; break; // back leg
   default: 0;}

Z Expression:
switch(a){
   case 0: b; break; // top leg
   case 1: case 2: case 3: -b*sin(pi/6); break; // left, right, back legs
   default: 0;}

D Expression:
a>3 ? 0 : 8*(2.5 - b); // truncated cone, use 1.5 instead of 2.5 for full cone

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On the same principle, an octopod, using ribbon and twist:

 

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• All angles are assumed to be in radians.
• X, Y, Z range from -16 to 16. If bigger they will be clipped.
• The actual print size in mm is obtained by multiplying X, Y, Z by 12.5.
• The build plane (the grey plane) X range and Y range are -2 to +2.
• The shape is always Z translated to be above the build plane.
• D is in mm, TW is in radians.

Issues

• Do not use both u and v as general purpose auxiliary variables. If you do so, the app will assume that you are using u, v surface coordinates and unpredictable results may occur. The safe way is to use a, b, k, r, n, w auxiliary variables.
• When drawing a printable surface, a tiny line may appear on the shape. This has no impact on the final 3D print result.