# TENSEGRITY TENSEGRITIES

## Formula for elliptical tensegrities

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)
• ra = 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
Formula for the X- and Y-coordinates of the bottom of strut k:
 Xka = - w - ra * cos (360 * (k-1) / n) (1)

 Yka = ra * 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:

 X(k+v)a = - w - ra * cos (360 * (k+v-1) / n) (3)

 Y(k+v)a = ra * 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:

 f = (Y(k+v)a - Yk) / (X(k+v)a - Xk) (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:

 Y = f * (X + w) (6)

The following equation applies to each ellipse:

 (X2 / g2) + (Y2 / h2) = 1 (7)

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

 Xb2 * ( f2 + (h/g)2) + Xb * 2*f2*w + f2*w2 = h2 (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:

 sk2 = (Xkb - Xka)2 + (Ykb - Yka)2 + Zk2 (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:

 ck2 = (Xkb - X(k+v)a)2 + (Ykb - Y(k+v)a)2 + Zk2 (10)

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

 bk2 = (Xkb - X(k+1)b)2 + (Ykb - Y(k+1)b)2 + Zk2 (11)

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

 a = 2 * ra * sin (180/n) (12) Schematic top view of tensegrity in which the bottom forms a circle and the top forms an ellipse.
Marcelo Pars