## TENSEGRITY |

Here a description for calculating the lengths of struts and strings of tensegrities like arm chair and ellipse.

First a lot of definitions. See the picture on the right.

- n = number of struts
- k = strut number (no 1 is on the X-axes)
- r
_{a}= radius of circle a - w = distance between the heart of the circle and the Y-axes
- k+v = number of the strut that also holds string c from strut k
- g = radius of the ellipse in the X direction
- h = radius of the ellipse in de Y direction

_{ka} = - w - r_{a} * cos (360 * (k-1) / n) |
(1) |

_{ka} = r_{a} * sin (360 * (k-1) / n) |
(2) |

Strut k climbes up from the bottom (a red spot on the circle) to a blue point on the ellipse at the top. A string runs back from this top point to the bottom (another red point on the circle). This other red point is the bottom of strut k+v. The formula for the bottom of strut k+v is:

_{(k+v)a} = - w - r_{a} * cos (360 * (k+v-1) / n) |
(3) |

_{(k+v)a} = r_{a} * sin (360 * (k+v-1) / n) |
(4) |

From the bottom of strut k to the bottom of strut k+v, one can draw a line with direction f:

_{(k+v)a} - Y_{k}) / (X_{(k+v)a} - X_{k}) |
(5) |

The end of strut k lies on the intersection between the line from the heart of the circle with direction f (see the green line) and the ellipse

The equation that describes the green line is:

(6) |

The following equation applies to each ellipse:

^{2} / g^{2}) + (Y^{2} / h^{2}) = 1 |
(7) |

Equation (6) and (7) give the following result:

_{b}^{2} * ( f^{2} + (h/g)^{2}) + X_{b} * 2*f^{2}*w + f^{2}*w^{2} = h^{2} |
(8) |

Equation (8) and the abc-formula describe the X-coordinate at the top for each strut k, and with equation (6) als the Y-coordinate.

The length of the strut is:

_{k}^{2} = (X_{kb} - X_{ka})^{2} + (Y_{kb} - Y_{ka})^{2} + Z_{k}^{2} |
(9) |

with Z being a freely selectable height of the strut.

The length of string c (from the top of strut k to the bottom of strut k+v) is described as:

_{k}^{2} = (X_{kb} - X_{(k+v)a})^{2} + (Y_{kb} - Y_{(k+v)a})^{2} + Z_{k}^{2} |
(10) |

The length of string b (from the top op strut k to the top of strut k + 1):

_{k}^{2} = (X_{kb} - X_{(k+1)b})^{2} + (Y_{kb} - Y_{(k+1)b})^{2} + Z_{k}^{2} |
(11) |

And just for completeness (but this is nothing new): The length of string a (at the bottom):

_{a} * sin (180/n) |
(12) |

Schematic top view of tensegrity in which the bottom forms a circle and the top forms an ellipse.

Marcelo Pars