Disc Edges : 4:(O)xy±z±w
-------------------------
x = (cos(t))/((-1)*sin(d) + (-1)*cos(d) + c)
y = (sin(t))/((-1)*sin(d) + (-1)*cos(d) + c)
z = ((-1)*cos(d) - (-1)*sin(d))/((-1)*sin(d) + (-1)*cos(d) + c)
----------
x = (cos(t))/((-1)*sin(d) + (1)*cos(d) + c)
y = (sin(t))/((-1)*sin(d) + (1)*cos(d) + c)
z = ((-1)*cos(d) - (1)*sin(d))/((-1)*sin(d) + (1)*cos(d) + c)
----------
x = (cos(t))/((1)*sin(d) + (-1)*cos(d) + c)
y = (sin(t))/((1)*sin(d) + (-1)*cos(d) + c)
z = ((1)*cos(d) - (-1)*sin(d))/((1)*sin(d) + (-1)*cos(d) + c)
----------
x = (cos(t))/((1)*sin(d) + (1)*cos(d) + c)
y = (sin(t))/((1)*sin(d) + (1)*cos(d) + c)
z = ((1)*cos(d) - (1)*sin(d))/((1)*sin(d) + (1)*cos(d) + c)
----------
0 < t < 2π
------------------------------------------------------------------
Circles : 4:IOxy,±z±w
---------------------
x = (u*cos(v))/((-1)*sin(d) + (-1)*cos(d) + c)
y = (u*sin(v))/((-1)*sin(d) + (-1)*cos(d) + c)
z = ((-1)*cos(d) - (-1)*sin(d))/((-1)*sin(d) + (-1)*cos(d) + c)
----------
x = (u*cos(v))/((-1)*sin(d) + (1)*cos(d) + c)
y = (u*sin(v))/((-1)*sin(d) + (1)*cos(d) + c)
z = ((-1)*cos(d) - (1)*sin(d))/((-1)*sin(d) + (1)*cos(d) + c)
----------
x = (u*cos(v))/((1)*sin(d) + (-1)*cos(d) + c)
y = (u*sin(v))/((1)*sin(d) + (-1)*cos(d) + c)
z = ((1)*cos(d) - (-1)*sin(d))/((1)*sin(d) + (-1)*cos(d) + c)
----------
x = (u*cos(v))/((1)*sin(d) + (1)*cos(d) + c)
y = (u*sin(v))/((1)*sin(d) + (1)*cos(d) + c)
z = ((1)*cos(d) - (1)*sin(d))/((1)*sin(d) + (1)*cos(d) + c)
-----------------------
-1 < u < 1 // 20 steps
0 < v < π // 20 steps
-------------------------------------------------------------------
Hollow Tubes : 2:Iz(O)xy,±w
---------------------------
x = (cos(v))/((u)*sin(d) + (-1)*cos(d) + c)
y = (sin(v))/((u)*sin(d) + (-1)*cos(d) + c)
z = ((u)*cos(d) - (-1)*sin(d))/((u)*sin(d) + (-1)*cos(d) + c)
----------
x = (cos(v))/((u)*sin(d) + (1)*cos(d) + c)
y = (sin(v))/((u)*sin(d) + (1)*cos(d) + c)
z = ((u)*cos(d) - (1)*sin(d))/((u)*sin(d) + (1)*cos(d) + c)
----------------------
-1 < u < 1 // 20 steps
0 < v < 2π // 40 steps
Hollow Tubes : 2:Iw(O)xy,±z
---------------------------
x = (cos(v))/((-1)*sin(d) + (u)*cos(d) + c)
y = (sin(v))/((-1)*sin(d) + (u)*cos(d) + c)
z = ((-1)*cos(d) - (u)*sin(d))/((-1)*sin(d) + (u)*cos(d) + c)
----------
x = (cos(v))/((1)*sin(d) + (u)*cos(d) + c)
y = (sin(v))/((1)*sin(d) + (u)*cos(d) + c)
z = ((1)*cos(d) - (u)*sin(d))/((1)*sin(d) + (u)*cos(d) + c)
----------------------
-1 < u < 1 // 20 steps
0 < v < 2π // 40 steps
Building triangular diprism in 1D elements:
• 4 pairs of 3 triangle edges
1. { sqrt(3)(t-1) , 3t+1 , -2*sqrt(3) , -2*sqrt(3) }
x = ((sqrt(3)(t-1))*cos(b) - (3t+1)*sin(b))/((((sqrt(3)(t-1))*sin(b) + (3t+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
y = ((((sqrt(3)(t-1))*sin(b) + (3t+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*sin(d) + (-2*sqrt(3))*cos(d))/((((sqrt(3)(t-1))*sin(b) + (3t+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
z = (((sqrt(3)(t-1))*sin(b) + (3t+1)*cos(b))*sin(c) + (-2*sqrt(3))*cos(c))/((((sqrt(3)(t-1))*sin(b) + (3t+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
2. { sqrt(3)(t-1) , 3t+1 , -2*sqrt(3) , 2*sqrt(3) }
x = ((sqrt(3)(t-1))*cos(b) - (3t+1)*sin(b))/((((sqrt(3)(t-1))*sin(b) + (3t+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
y = ((((sqrt(3)(t-1))*sin(b) + (3t+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*sin(d) + (2*sqrt(3))*cos(d))/((((sqrt(3)(t-1))*sin(b) + (3t+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
z = (((sqrt(3)(t-1))*sin(b) + (3t+1)*cos(b))*sin(c) + (-2*sqrt(3))*cos(c))/((((sqrt(3)(t-1))*sin(b) + (3t+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
3. { sqrt(3)(t-1) , 3t+1 , 2*sqrt(3) , -2*sqrt(3) }
x = ((sqrt(3)(t-1))*cos(b) - (3t+1)*sin(b))/((((sqrt(3)(t-1))*sin(b) + (3t+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
y = ((((sqrt(3)(t-1))*sin(b) + (3t+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*sin(d) + (-2*sqrt(3))*cos(d))/((((sqrt(3)(t-1))*sin(b) + (3t+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
z = (((sqrt(3)(t-1))*sin(b) + (3t+1)*cos(b))*sin(c) + (2*sqrt(3))*cos(c))/((((sqrt(3)(t-1))*sin(b) + (3t+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
4. { sqrt(3)(t-1) , 3t+1 , 2*sqrt(3) , 2*sqrt(3) }
x = ((sqrt(3)(t-1))*cos(b) - (3t+1)*sin(b))/((((sqrt(3)(t-1))*sin(b) + (3t+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
y = ((((sqrt(3)(t-1))*sin(b) + (3t+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*sin(d) + (2*sqrt(3))*cos(d))/((((sqrt(3)(t-1))*sin(b) + (3t+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
z = (((sqrt(3)(t-1))*sin(b) + (3t+1)*cos(b))*sin(c) + (2*sqrt(3))*cos(c))/((((sqrt(3)(t-1))*sin(b) + (3t+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
----
5. { sqrt(3)(-t+1) , 3t+1 , -2*sqrt(3) , -2*sqrt(3) }
x = ((sqrt(3)(-t+1))*cos(b) - (3t+1)*sin(b))/((((sqrt(3)(-t+1))*sin(b) + (3t+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
y = ((((sqrt(3)(-t+1))*sin(b) + (3t+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*sin(d) + (-2*sqrt(3))*cos(d))/((((sqrt(3)(-t+1))*sin(b) + (3t+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
z = (((sqrt(3)(-t+1))*sin(b) + (3t+1)*cos(b))*sin(c) + (-2*sqrt(3))*cos(c))/((((sqrt(3)(-t+1))*sin(b) + (3t+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
6. { sqrt(3)(-t+1) , 3t+1 , -2*sqrt(3) , 2*sqrt(3) }
x = ((sqrt(3)(-t+1))*cos(b) - (3t+1)*sin(b))/((((sqrt(3)(-t+1))*sin(b) + (3t+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
y = ((((sqrt(3)(-t+1))*sin(b) + (3t+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*sin(d) + (2*sqrt(3))*cos(d))/((((sqrt(3)(-t+1))*sin(b) + (3t+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
z = (((sqrt(3)(-t+1))*sin(b) + (3t+1)*cos(b))*sin(c) + (-2*sqrt(3))*cos(c))/((((sqrt(3)(-t+1))*sin(b) + (3t+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
7. { sqrt(3)(-t+1) , 3t+1 , 2*sqrt(3) , -2*sqrt(3) }
x = ((sqrt(3)(-t+1))*cos(b) - (3t+1)*sin(b))/((((sqrt(3)(-t+1))*sin(b) + (3t+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
y = ((((sqrt(3)(-t+1))*sin(b) + (3t+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*sin(d) + (-2*sqrt(3))*cos(d))/((((sqrt(3)(-t+1))*sin(b) + (3t+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
z = (((sqrt(3)(-t+1))*sin(b) + (3t+1)*cos(b))*sin(c) + (2*sqrt(3))*cos(c))/((((sqrt(3)(-t+1))*sin(b) + (3t+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
8. { sqrt(3)(-t+1) , 3t+1 , 2*sqrt(3) , 2*sqrt(3) }
x = ((sqrt(3)(-t+1))*cos(b) - (3t+1)*sin(b))/((((sqrt(3)(-t+1))*sin(b) + (3t+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
y = ((((sqrt(3)(-t+1))*sin(b) + (3t+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*sin(d) + (2*sqrt(3))*cos(d))/((((sqrt(3)(-t+1))*sin(b) + (3t+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
z = (((sqrt(3)(-t+1))*sin(b) + (3t+1)*cos(b))*sin(c) + (2*sqrt(3))*cos(c))/((((sqrt(3)(-t+1))*sin(b) + (3t+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
----
9. { 2*sqrt(3)t , -2 , -2*sqrt(3) , -2*sqrt(3) }
x = ((2*sqrt(3)t)*cos(b) - (-2)*sin(b))/((((2*sqrt(3)t)*sin(b) + (-2)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
y = ((((2*sqrt(3)t)*sin(b) + (-2)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*sin(d) + (-2*sqrt(3))*cos(d))/((((2*sqrt(3)t)*sin(b) + (-2)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
z = (((2*sqrt(3)t)*sin(b) + (-2)*cos(b))*sin(c) + (-2*sqrt(3))*cos(c))/((((2*sqrt(3)t)*sin(b) + (-2)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
10. { 2*sqrt(3)t , -2 , -2*sqrt(3) , 2*sqrt(3) }
x = ((2*sqrt(3)t)*cos(b) - (-2)*sin(b))/((((2*sqrt(3)t)*sin(b) + (-2)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
y = ((((2*sqrt(3)t)*sin(b) + (-2)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*sin(d) + (2*sqrt(3))*cos(d))/((((2*sqrt(3)t)*sin(b) + (-2)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
z = (((2*sqrt(3)t)*sin(b) + (-2)*cos(b))*sin(c) + (-2*sqrt(3))*cos(c))/((((2*sqrt(3)t)*sin(b) + (-2)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
11. { 2*sqrt(3)t , -2 , 2*sqrt(3) , -2*sqrt(3) }
x = ((2*sqrt(3)t)*cos(b) - (-2)*sin(b))/((((2*sqrt(3)t)*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
y = ((((2*sqrt(3)t)*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*sin(d) + (-2*sqrt(3))*cos(d))/((((2*sqrt(3)t)*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
z = (((2*sqrt(3)t)*sin(b) + (-2)*cos(b))*sin(c) + (2*sqrt(3))*cos(c))/((((2*sqrt(3)t)*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
12. { 2*sqrt(3)t , -2 , 2*sqrt(3) , 2*sqrt(3) }
x = ((2*sqrt(3)t)*cos(b) - (-2)*sin(b))/((((2*sqrt(3)t)*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
y = ((((2*sqrt(3)t)*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*sin(d) + (2*sqrt(3))*cos(d))/((((2*sqrt(3)t)*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
z = (((2*sqrt(3)t)*sin(b) + (-2)*cos(b))*sin(c) + (2*sqrt(3))*cos(c))/((((2*sqrt(3)t)*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
--------
• 2 pairs of 3 extruding edges
13. { -2*sqrt(3) , -2 , 2*sqrt(3)t , -2*sqrt(3) }
x = ((-2*sqrt(3))*cos(b) - (-2)*sin(b))/((((-2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
y = ((((-2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*sin(d) + (-2*sqrt(3))*cos(d))/((((-2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
z = (((-2*sqrt(3))*sin(b) + (-2)*cos(b))*sin(c) + (2*sqrt(3)t)*cos(c))/((((-2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
14. { -2*sqrt(3) , -2 , 2*sqrt(3)t , 2*sqrt(3) }
x = ((-2*sqrt(3))*cos(b) - (-2)*sin(b))/((((-2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
y = ((((-2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*sin(d) + (2*sqrt(3))*cos(d))/((((-2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
z = (((-2*sqrt(3))*sin(b) + (-2)*cos(b))*sin(c) + (2*sqrt(3)t)*cos(c))/((((-2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
15. { 2*sqrt(3) , -2 , 2*sqrt(3)t , -2*sqrt(3) }
x = ((2*sqrt(3))*cos(b) - (-2)*sin(b))/((((2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
y = ((((2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*sin(d) + (-2*sqrt(3))*cos(d))/((((2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
z = (((2*sqrt(3))*sin(b) + (-2)*cos(b))*sin(c) + (2*sqrt(3)t)*cos(c))/((((2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
16. { 2*sqrt(3) , -2 , 2*sqrt(3)t , 2*sqrt(3) }
x = ((2*sqrt(3))*cos(b) - (-2)*sin(b))/((((2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
y = ((((2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*sin(d) + (2*sqrt(3))*cos(d))/((((2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
z = (((2*sqrt(3))*sin(b) + (-2)*cos(b))*sin(c) + (2*sqrt(3)t)*cos(c))/((((2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
----
17. { 0 , 4 , 2*sqrt(3)t , -2*sqrt(3) }
x = ((0)*cos(b) - (4)*sin(b))/((((0)*sin(b) + (4)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
y = ((((0)*sin(b) + (4)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*sin(d) + (-2*sqrt(3))*cos(d))/((((0)*sin(b) + (4)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
z = (((0)*sin(b) + (4)*cos(b))*sin(c) + (2*sqrt(3)t)*cos(c))/((((0)*sin(b) + (4)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
18. { 0 , 4 , 2*sqrt(3)t , 2*sqrt(3) }
x = ((0)*cos(b) - (4)*sin(b))/((((0)*sin(b) + (4)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
y = ((((0)*sin(b) + (4)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*sin(d) + (2*sqrt(3))*cos(d))/((((0)*sin(b) + (4)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
z = (((0)*sin(b) + (4)*cos(b))*sin(c) + (2*sqrt(3)t)*cos(c))/((((0)*sin(b) + (4)*cos(b))*cos(c) - (2*sqrt(3)t)*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
--------
• 6 w-aligned edges
19. { -2*sqrt(3) , -2 , -2*sqrt(3) , 2*sqrt(3)t }
x = ((-2*sqrt(3))*cos(b) - (-2)*sin(b))/((((-2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)t)*sin(d) +a)
y = ((((-2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*sin(d) + (2*sqrt(3)t)*cos(d))/((((-2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)t)*sin(d) +a)
z = (((-2*sqrt(3))*sin(b) + (-2)*cos(b))*sin(c) + (-2*sqrt(3))*cos(c))/((((-2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)t)*sin(d) +a)
20. { -2*sqrt(3) , -2 , 2*sqrt(3) , 2*sqrt(3)t }
x = ((-2*sqrt(3))*cos(b) - (-2)*sin(b))/((((-2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)t)*sin(d) +a)
y = ((((-2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*sin(d) + (2*sqrt(3)t)*cos(d))/((((-2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)t)*sin(d) +a)
z = (((-2*sqrt(3))*sin(b) + (-2)*cos(b))*sin(c) + (2*sqrt(3))*cos(c))/((((-2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)t)*sin(d) +a)
21. { 2*sqrt(3) , -2 , -2*sqrt(3) , 2*sqrt(3)t }
x = ((2*sqrt(3))*cos(b) - (-2)*sin(b))/((((2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)t)*sin(d) +a)
y = ((((2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*sin(d) + (2*sqrt(3)t)*cos(d))/((((2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)t)*sin(d) +a)
z = (((2*sqrt(3))*sin(b) + (-2)*cos(b))*sin(c) + (-2*sqrt(3))*cos(c))/((((2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)t)*sin(d) +a)
22. { 2*sqrt(3) , -2 , 2*sqrt(3) , 2*sqrt(3)t }
x = ((2*sqrt(3))*cos(b) - (-2)*sin(b))/((((2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)t)*sin(d) +a)
y = ((((2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*sin(d) + (2*sqrt(3)t)*cos(d))/((((2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)t)*sin(d) +a)
z = (((2*sqrt(3))*sin(b) + (-2)*cos(b))*sin(c) + (2*sqrt(3))*cos(c))/((((2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)t)*sin(d) +a)
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23. { 0 , 4 , -2*sqrt(3) , 2*sqrt(3)t }
x = ((0)*cos(b) - (4)*sin(b))/((((0)*sin(b) + (4)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)t)*sin(d) +a)
y = ((((0)*sin(b) + (4)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*sin(d) + (2*sqrt(3)t)*cos(d))/((((0)*sin(b) + (4)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)t)*sin(d) +a)
z = (((0)*sin(b) + (4)*cos(b))*sin(c) + (-2*sqrt(3))*cos(c))/((((0)*sin(b) + (4)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)t)*sin(d) +a)
24. { 0 , 4 , 2*sqrt(3) , 2*sqrt(3)t }
x = ((0)*cos(b) - (4)*sin(b))/((((0)*sin(b) + (4)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)t)*sin(d) +a)
y = ((((0)*sin(b) + (4)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*sin(d) + (2*sqrt(3)t)*cos(d))/((((0)*sin(b) + (4)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)t)*sin(d) +a)
z = (((0)*sin(b) + (4)*cos(b))*sin(c) + (2*sqrt(3))*cos(c))/((((0)*sin(b) + (4)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)t)*sin(d) +a)
Building triangular diprism in 2D elements:
rotate xy , yz , yw // project xzw
x = ((X)*cos(b) - (Y)*sin(b))/((((X)*sin(b) + (Y)*cos(b))*cos(c) - (Z)*sin(c))*cos(d) - (W)*sin(d) +a)
y = ((((X)*sin(b) + (Y)*cos(b))*cos(c) - (Z)*sin(c))*sin(d) + (W)*cos(d))/((((X)*sin(b) + (Y)*cos(b))*cos(c) - (Z)*sin(c))*cos(d) - (W)*sin(d) +a)
z = (((X)*sin(b) + (Y)*cos(b))*sin(c) + (Z)*cos(c))/((((X)*sin(b) + (Y)*cos(b))*cos(c) - (Z)*sin(c))*cos(d) - (W)*sin(d) +a)
• 4 triangles
1. { sqrt(3)(v-1)u , 3v+1 , -2*sqrt(3) , -2*sqrt(3) }
x = ((sqrt(3)(v-1)u)*cos(b) - (3v+1)*sin(b))/((((sqrt(3)(v-1)u)*sin(b) + (3v+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
y = ((((sqrt(3)(v-1)u)*sin(b) + (3v+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*sin(d) + (-2*sqrt(3))*cos(d))/((((sqrt(3)(v-1)u)*sin(b) + (3v+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
z = (((sqrt(3)(v-1)u)*sin(b) + (3v+1)*cos(b))*sin(c) + (-2*sqrt(3))*cos(c))/((((sqrt(3)(v-1)u)*sin(b) + (3v+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
2. { sqrt(3)(v-1)u , 3v+1 , -2*sqrt(3) , 2*sqrt(3) }
x = ((sqrt(3)(v-1)u)*cos(b) - (3v+1)*sin(b))/((((sqrt(3)(v-1)u)*sin(b) + (3v+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
y = ((((sqrt(3)(v-1)u)*sin(b) + (3v+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*sin(d) + (2*sqrt(3))*cos(d))/((((sqrt(3)(v-1)u)*sin(b) + (3v+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
z = (((sqrt(3)(v-1)u)*sin(b) + (3v+1)*cos(b))*sin(c) + (-2*sqrt(3))*cos(c))/((((sqrt(3)(v-1)u)*sin(b) + (3v+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
3. { sqrt(3)(v-1)u , 3v+1 , 2*sqrt(3) , -2*sqrt(3) }
x = ((sqrt(3)(v-1)u)*cos(b) - (3v+1)*sin(b))/((((sqrt(3)(v-1)u)*sin(b) + (3v+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
y = ((((sqrt(3)(v-1)u)*sin(b) + (3v+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*sin(d) + (-2*sqrt(3))*cos(d))/((((sqrt(3)(v-1)u)*sin(b) + (3v+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
z = (((sqrt(3)(v-1)u)*sin(b) + (3v+1)*cos(b))*sin(c) + (2*sqrt(3))*cos(c))/((((sqrt(3)(v-1)u)*sin(b) + (3v+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
4. { sqrt(3)(v-1)u , 3v+1 , 2*sqrt(3) , 2*sqrt(3) }
x = ((sqrt(3)(v-1)u)*cos(b) - (3v+1)*sin(b))/((((sqrt(3)(v-1)u)*sin(b) + (3v+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
y = ((((sqrt(3)(v-1)u)*sin(b) + (3v+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*sin(d) + (2*sqrt(3))*cos(d))/((((sqrt(3)(v-1)u)*sin(b) + (3v+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
z = (((sqrt(3)(v-1)u)*sin(b) + (3v+1)*cos(b))*sin(c) + (2*sqrt(3))*cos(c))/((((sqrt(3)(v-1)u)*sin(b) + (3v+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
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• 6 squares
5. { sqrt(3)(u-1) , 3u+1 , 2*sqrt(3)v , -2*sqrt(3) }
x = ((sqrt(3)(u-1))*cos(b) - (3u+1)*sin(b))/((((sqrt(3)(u-1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
y = ((((sqrt(3)(u-1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*sin(d) + (-2*sqrt(3))*cos(d))/((((sqrt(3)(u-1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
z = (((sqrt(3)(u-1))*sin(b) + (3u+1)*cos(b))*sin(c) + (2*sqrt(3)v)*cos(c))/((((sqrt(3)(u-1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
6. { sqrt(3)(u-1) , 3u+1 , 2*sqrt(3)v , 2*sqrt(3) }
x = ((sqrt(3)(u-1))*cos(b) - (3u+1)*sin(b))/((((sqrt(3)(u-1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
y = ((((sqrt(3)(u-1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*sin(d) + (2*sqrt(3))*cos(d))/((((sqrt(3)(u-1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
z = (((sqrt(3)(u-1))*sin(b) + (3u+1)*cos(b))*sin(c) + (2*sqrt(3)v)*cos(c))/((((sqrt(3)(u-1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
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7. { sqrt(3)(-u+1) , 3u+1 , 2*sqrt(3)v , -2*sqrt(3) }
x = ((sqrt(3)(-u+1))*cos(b) - (3u+1)*sin(b))/((((sqrt(3)(-u+1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
y = ((((sqrt(3)(-u+1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*sin(d) + (-2*sqrt(3))*cos(d))/((((sqrt(3)(-u+1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
z = (((sqrt(3)(-u+1))*sin(b) + (3u+1)*cos(b))*sin(c) + (2*sqrt(3)v)*cos(c))/((((sqrt(3)(-u+1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
8. { sqrt(3)(-u+1) , 3u+1 , 2*sqrt(3)v , 2*sqrt(3) }
x = ((sqrt(3)(-u+1))*cos(b) - (3u+1)*sin(b))/((((sqrt(3)(-u+1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
y = ((((sqrt(3)(-u+1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*sin(d) + (2*sqrt(3))*cos(d))/((((sqrt(3)(-u+1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
z = (((sqrt(3)(-u+1))*sin(b) + (3u+1)*cos(b))*sin(c) + (2*sqrt(3)v)*cos(c))/((((sqrt(3)(-u+1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
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9. { 2*sqrt(3)u , -2 , 2*sqrt(3)v , -2*sqrt(3) }
x = ((2*sqrt(3)u)*cos(b) - (-2)*sin(b))/((((2*sqrt(3)u)*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
y = ((((2*sqrt(3)u)*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*sin(d) + (-2*sqrt(3))*cos(d))/((((2*sqrt(3)u)*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
z = (((2*sqrt(3)u)*sin(b) + (-2)*cos(b))*sin(c) + (2*sqrt(3)v)*cos(c))/((((2*sqrt(3)u)*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*cos(d) - (-2*sqrt(3))*sin(d) +a)
10. { 2*sqrt(3)u , -2 , 2*sqrt(3)v , 2*sqrt(3) }
x = ((2*sqrt(3)u)*cos(b) - (-2)*sin(b))/((((2*sqrt(3)u)*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
y = ((((2*sqrt(3)u)*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*sin(d) + (2*sqrt(3))*cos(d))/((((2*sqrt(3)u)*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
z = (((2*sqrt(3)u)*sin(b) + (-2)*cos(b))*sin(c) + (2*sqrt(3)v)*cos(c))/((((2*sqrt(3)u)*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)v)*sin(c))*cos(d) - (2*sqrt(3))*sin(d) +a)
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• 2 pairs of 3 squares
11. { sqrt(3)(u-1) , 3u+1 , -2*sqrt(3) , 2*sqrt(3)v }
x = ((sqrt(3)(u-1))*cos(b) - (3u+1)*sin(b))/((((sqrt(3)(u-1))*sin(b) + (3u+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
y = ((((sqrt(3)(u-1))*sin(b) + (3u+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*sin(d) + (2*sqrt(3)v)*cos(d))/((((sqrt(3)(u-1))*sin(b) + (3u+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
z = (((sqrt(3)(u-1))*sin(b) + (3u+1)*cos(b))*sin(c) + (-2*sqrt(3))*cos(c))/((((sqrt(3)(u-1))*sin(b) + (3u+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
12. { sqrt(3)(u-1) , 3u+1 , 2*sqrt(3) , 2*sqrt(3)v }
x = ((sqrt(3)(u-1))*cos(b) - (3u+1)*sin(b))/((((sqrt(3)(u-1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
y = ((((sqrt(3)(u-1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*sin(d) + (2*sqrt(3)v)*cos(d))/((((sqrt(3)(u-1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
z = (((sqrt(3)(u-1))*sin(b) + (3u+1)*cos(b))*sin(c) + (2*sqrt(3))*cos(c))/((((sqrt(3)(u-1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
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13. { sqrt(3)(-u+1) , 3u+1 , -2*sqrt(3) , 2*sqrt(3)v }
x = ((sqrt(3)(-u+1))*cos(b) - (3u+1)*sin(b))/((((sqrt(3)(-u+1))*sin(b) + (3u+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
y = ((((sqrt(3)(-u+1))*sin(b) + (3u+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*sin(d) + (2*sqrt(3)v)*cos(d))/((((sqrt(3)(-u+1))*sin(b) + (3u+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
z = (((sqrt(3)(-u+1))*sin(b) + (3u+1)*cos(b))*sin(c) + (-2*sqrt(3))*cos(c))/((((sqrt(3)(-u+1))*sin(b) + (3u+1)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
14. { sqrt(3)(-u+1) , 3u+1 , 2*sqrt(3) , 2*sqrt(3)v }
x = ((sqrt(3)(-u+1))*cos(b) - (3u+1)*sin(b))/((((sqrt(3)(-u+1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
y = ((((sqrt(3)(-u+1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*sin(d) + (2*sqrt(3)v)*cos(d))/((((sqrt(3)(-u+1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
z = (((sqrt(3)(-u+1))*sin(b) + (3u+1)*cos(b))*sin(c) + (2*sqrt(3))*cos(c))/((((sqrt(3)(-u+1))*sin(b) + (3u+1)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
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15. { 2*sqrt(3)u , -2 , -2*sqrt(3) , 2*sqrt(3)v }
x = ((2*sqrt(3)u)*cos(b) - (-2)*sin(b))/((((2*sqrt(3)u)*sin(b) + (-2)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
y = ((((2*sqrt(3)u)*sin(b) + (-2)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*sin(d) + (2*sqrt(3)v)*cos(d))/((((2*sqrt(3)u)*sin(b) + (-2)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
z = (((2*sqrt(3)u)*sin(b) + (-2)*cos(b))*sin(c) + (-2*sqrt(3))*cos(c))/((((2*sqrt(3)u)*sin(b) + (-2)*cos(b))*cos(c) - (-2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
16. { 2*sqrt(3)u , -2 , 2*sqrt(3) , 2*sqrt(3)v }
x = ((2*sqrt(3)u)*cos(b) - (-2)*sin(b))/((((2*sqrt(3)u)*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
y = ((((2*sqrt(3)u)*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*sin(d) + (2*sqrt(3)v)*cos(d))/((((2*sqrt(3)u)*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
z = (((2*sqrt(3)u)*sin(b) + (-2)*cos(b))*sin(c) + (2*sqrt(3))*cos(c))/((((2*sqrt(3)u)*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3))*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
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• 3 extruding squares
17. { -2*sqrt(3) , -2 , 2*sqrt(3)u , 2*sqrt(3)v }
x = ((-2*sqrt(3))*cos(b) - (-2)*sin(b))/((((-2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)u)*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
y = ((((-2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)u)*sin(c))*sin(d) + (2*sqrt(3)v)*cos(d))/((((-2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)u)*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
z = (((-2*sqrt(3))*sin(b) + (-2)*cos(b))*sin(c) + (2*sqrt(3)u)*cos(c))/((((-2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)u)*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
18. { 2*sqrt(3) , -2 , 2*sqrt(3)u , 2*sqrt(3)v }
x = ((2*sqrt(3))*cos(b) - (-2)*sin(b))/((((2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)u)*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
y = ((((2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)u)*sin(c))*sin(d) + (2*sqrt(3)v)*cos(d))/((((2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)u)*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
z = (((2*sqrt(3))*sin(b) + (-2)*cos(b))*sin(c) + (2*sqrt(3)u)*cos(c))/((((2*sqrt(3))*sin(b) + (-2)*cos(b))*cos(c) - (2*sqrt(3)u)*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
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19. { 0 , 4 , 2*sqrt(3)u , 2*sqrt(3)v }
x = ((0)*cos(b) - (4)*sin(b))/((((0)*sin(b) + (4)*cos(b))*cos(c) - (2*sqrt(3)u)*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
y = ((((0)*sin(b) + (4)*cos(b))*cos(c) - (2*sqrt(3)u)*sin(c))*sin(d) + (2*sqrt(3)v)*cos(d))/((((0)*sin(b) + (4)*cos(b))*cos(c) - (2*sqrt(3)u)*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
z = (((0)*sin(b) + (4)*cos(b))*sin(c) + (2*sqrt(3)u)*cos(c))/((((0)*sin(b) + (4)*cos(b))*cos(c) - (2*sqrt(3)u)*sin(c))*cos(d) - (2*sqrt(3)v)*sin(d) +a)
1D Elements
• 2 Circles
1. { 2*sqrt(3)*cos(t) , 2*sqrt(3)*sin(t) , -2 , -2*sqrt(3) }
{ (2*sqrt(3)*cos(t))/((-2)*sin(b) + (-2*sqrt(3))*cos(b)+a) , (2*sqrt(3)*sin(t))/((-2)*sin(b) + (-2*sqrt(3))*cos(b)+a) , ((-2)*cos(b) - (-2*sqrt(3))*sin(b))/((-2)*sin(b) + (-2*sqrt(3))*cos(b)+a) }
2. { 2*sqrt(3)*cos(t) , 2*sqrt(3)*sin(t) , -2 , 2*sqrt(3) }
{ (2*sqrt(3)*cos(t))/((-2)*sin(b) + (2*sqrt(3))*cos(b)+a) , (2*sqrt(3)*sin(t))/((-2)*sin(b) + (2*sqrt(3))*cos(b)+a) , ((-2)*cos(b) - (2*sqrt(3))*sin(b))/((-2)*sin(b) + (2*sqrt(3))*cos(b)+a) }
0 < t < 2π
-------
• Line Segment
3. { 0 , 0 , 4 , 2*sqrt(3)t}
{ (0)/((4)*sin(b) + (2*sqrt(3)t)*cos(b)+a) , (0)/((4)*sin(b) + (2*sqrt(3)t)*cos(b)+a) , ((4)*cos(b) - (2*sqrt(3)t)*sin(b))/((4)*sin(b) + (2*sqrt(3)t)*cos(b)+a) }
-1 < t < 1
--------------------
2D Elements
• 2 Curved Cone 2-Surfaces
1. { sqrt(3)(v-1)*cos(u) , sqrt(3)(v-1)*sin(u) , 3v+1 , -2*sqrt(3) }
{ (sqrt(3)(v-1)*cos(u))/((3v+1)*sin(b) + (-2*sqrt(3))*cos(b)+a) , (sqrt(3)(v-1)*sin(u))/((3v+1)*sin(b) + (-2*sqrt(3))*cos(b)+a) , ((3v+1)*cos(b) - (-2*sqrt(3))*sin(b))/((3v+1)*sin(b) + (-2*sqrt(3))*cos(b)+a) }
2. { sqrt(3)(v-1)*cos(u) , sqrt(3)(v-1)*sin(u) , 3v+1 , 2*sqrt(3) }
{ (sqrt(3)(v-1)*cos(u))/((3v+1)*sin(b) + (2*sqrt(3))*cos(b)+a) , (sqrt(3)(v-1)*sin(u))/((3v+1)*sin(b) + (2*sqrt(3))*cos(b)+a) , ((3v+1)*cos(b) - (2*sqrt(3))*sin(b))/((3v+1)*sin(b) + (2*sqrt(3))*cos(b)+a) }
0 < u < 2π , 40 steps // -1 < v < 1 , 20 steps
-----
• 2 Solid discs
3. { 2*sqrt(3)v*cos(u) , 2*sqrt(3)v*sin(u) , -2 , -2*sqrt(3) }
{ (2*sqrt(3)v*cos(u))/((-2)*sin(b) + (-2*sqrt(3))*cos(b)+a) , (2*sqrt(3)v*sin(u))/((-2)*sin(b) + (-2*sqrt(3))*cos(b)+a) , ((-2)*cos(b) - (-2*sqrt(3))*sin(b))/((-2)*sin(b) + (-2*sqrt(3))*cos(b)+a) }
4. { 2*sqrt(3)v*cos(u) , 2*sqrt(3)v*sin(u) , -2 , 2*sqrt(3) }
{ (2*sqrt(3)v*cos(u))/((-2)*sin(b) + (2*sqrt(3))*cos(b)+a) , (2*sqrt(3)v*sin(u))/((-2)*sin(b) + (2*sqrt(3))*cos(b)+a) , ((-2)*cos(b) - (2*sqrt(3))*sin(b))/((-2)*sin(b) + (2*sqrt(3))*cos(b)+a) }
0 < u < π // -1 < v < 1
-----
• Hollow Tube
5. { 2*sqrt(3)*cos(u) , 2*sqrt(3)*sin(u) , -2 , 2*sqrt(3)v}
{ (2*sqrt(3)*cos(u))/((-2)*sin(b) + (2*sqrt(3)v)*cos(b)+a) , (2*sqrt(3)*sin(u))/((-2)*sin(b) + (2*sqrt(3)v)*cos(b)+a) , ((-2)*cos(b) - (2*sqrt(3)v)*sin(b))/((-2)*sin(b) + (2*sqrt(3)v)*cos(b)+a) }
0 < u < 2π // -1 < v < 1
1D Elements
• 18 edges:
1. { -sqrt(3)(t-1) , 3t+1 , -2*sqrt(3) , -2 }
x = (-sqrt(3)(t-1))/((-2*sqrt(3))*sin(c) + ((3t+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
y = ((3t+1)*cos(b) - (-2)*sin(b))/((-2*sqrt(3))*sin(c) + ((3t+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
z = ((-2*sqrt(3))*cos(c) - ((3t+1)*sin(b) + (-2)*cos(b))*sin(c))/((-2*sqrt(3))*sin(c) + ((3t+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
---
2. { -sqrt(3)(t-1) , 3t+1 , 2*sqrt(3) , -2 }
x = (-sqrt(3)(t-1))/((2*sqrt(3))*sin(c) + ((3t+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
y = ((3t+1)*cos(b) - (-2)*sin(b))/((2*sqrt(3))*sin(c) + ((3t+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
z = ((2*sqrt(3))*cos(c) - ((3t+1)*sin(b) + (-2)*cos(b))*sin(c))/((2*sqrt(3))*sin(c) + ((3t+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
---
3. { -sqrt(3)(t-1) , 3t+1 , 0 , 4 }
x = (-sqrt(3)(t-1))/((0)*sin(c) + ((3t+1)*sin(b) + (4)*cos(b))*cos(c) + a)
y = ((3t+1)*cos(b) - (4)*sin(b))/((0)*sin(c) + ((3t+1)*sin(b) + (4)*cos(b))*cos(c) + a)
z = ((0)*cos(c) - ((3t+1)*sin(b) + (4)*cos(b))*sin(c))/((0)*sin(c) + ((3t+1)*sin(b) + (4)*cos(b))*cos(c) + a)
---
-------
4. { sqrt(3)(t-1) , 3t+1 , -2*sqrt(3) , -2 }
x = (sqrt(3)(t-1))/((-2*sqrt(3))*sin(c) + ((3t+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
y = ((3t+1)*cos(b) - (-2)*sin(b))/((-2*sqrt(3))*sin(c) + ((3t+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
z = ((-2*sqrt(3))*cos(c) - ((3t+1)*sin(b) + (-2)*cos(b))*sin(c))/((-2*sqrt(3))*sin(c) + ((3t+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
---
5. { sqrt(3)(t-1) , 3t+1 , 2*sqrt(3) , -2 }
x = (sqrt(3)(t-1))/((2*sqrt(3))*sin(c) + ((3t+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
y = ((3t+1)*cos(b) - (-2)*sin(b))/((2*sqrt(3))*sin(c) + ((3t+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
z = ((2*sqrt(3))*cos(c) - ((3t+1)*sin(b) + (-2)*cos(b))*sin(c))/((2*sqrt(3))*sin(c) + ((3t+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
---
6. { sqrt(3)(t-1) , 3t+1 , 0 , 4 }
x = (sqrt(3)(t-1))/((0)*sin(c) + ((3t+1)*sin(b) + (4)*cos(b))*cos(c) + a)
y = ((3t+1)*cos(b) - (4)*sin(b))/((0)*sin(c) + ((3t+1)*sin(b) + (4)*cos(b))*cos(c) + a)
z = ((0)*cos(c) - ((3t+1)*sin(b) + (4)*cos(b))*sin(c))/((0)*sin(c) + ((3t+1)*sin(b) + (4)*cos(b))*cos(c) + a)
---
------
7. { 2*sqrt(3)t , -2 , -2*sqrt(3) , -2 }
x = (2*sqrt(3)t)/((-2*sqrt(3))*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
y = ((-2)*cos(b) - (-2)*sin(b))/((-2*sqrt(3))*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
z = ((-2*sqrt(3))*cos(c) - ((-2)*sin(b) + (-2)*cos(b))*sin(c))/((-2*sqrt(3))*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
---
8. { 2*sqrt(3)t , -2 , 2*sqrt(3) , -2 }
x = (2*sqrt(3)t)/((2*sqrt(3))*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
y = ((-2)*cos(b) - (-2)*sin(b))/((2*sqrt(3))*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
z = ((2*sqrt(3))*cos(c) - ((-2)*sin(b) + (-2)*cos(b))*sin(c))/((2*sqrt(3))*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
---
9. { 2*sqrt(3)t , -2 , 0 , 4 }
x = (2*sqrt(3)t)/((0)*sin(c) + ((-2)*sin(b) + (4)*cos(b))*cos(c) + a)
y = ((-2)*cos(b) - (4)*sin(b))/((0)*sin(c) + ((-2)*sin(b) + (4)*cos(b))*cos(c) + a)
z = ((0)*cos(c) - ((-2)*sin(b) + (4)*cos(b))*sin(c))/((0)*sin(c) + ((-2)*sin(b) + (4)*cos(b))*cos(c) + a)
---
------
10. { -2*sqrt(3) , -2 , -sqrt(3)(t-1) , 3t+1 }
x = (-2*sqrt(3))/((-sqrt(3)(t-1))*sin(c) + ((-2)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
y = ((-2)*cos(b) - (3t+1)*sin(b))/((-sqrt(3)(t-1))*sin(c) + ((-2)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
z = ((-sqrt(3)(t-1))*cos(c) - ((-2)*sin(b) + (3t+1)*cos(b))*sin(c))/((-sqrt(3)(t-1))*sin(c) + ((-2)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
---
11. { -2*sqrt(3) , -2 , sqrt(3)(t-1) , 3t+1 }
x = (-2*sqrt(3))/((sqrt(3)(t-1))*sin(c) + ((-2)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
y = ((-2)*cos(b) - (3t+1)*sin(b))/((sqrt(3)(t-1))*sin(c) + ((-2)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
z = ((sqrt(3)(t-1))*cos(c) - ((-2)*sin(b) + (3t+1)*cos(b))*sin(c))/((sqrt(3)(t-1))*sin(c) + ((-2)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
---
12. { -2*sqrt(3) , -2 , 2*sqrt(3)t , -2 }
x = (-2*sqrt(3))/((2*sqrt(3)t)*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
y = ((-2)*cos(b) - (-2)*sin(b))/((2*sqrt(3)t)*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
z = ((2*sqrt(3)t)*cos(c) - ((-2)*sin(b) + (-2)*cos(b))*sin(c))/((2*sqrt(3)t)*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
---
------
13. { 2*sqrt(3) , -2 , -sqrt(3)(t-1) , 3t+1 }
x = (2*sqrt(3))/((-sqrt(3)(t-1))*sin(c) + ((-2)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
y = ((-2)*cos(b) - (3t+1)*sin(b))/((-sqrt(3)(t-1))*sin(c) + ((-2)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
z = ((-sqrt(3)(t-1))*cos(c) - ((-2)*sin(b) + (3t+1)*cos(b))*sin(c))/((-sqrt(3)(t-1))*sin(c) + ((-2)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
---
14. { 2*sqrt(3) , -2 , sqrt(3)(t-1) , 3t+1 }
x = (2*sqrt(3))/((sqrt(3)(t-1))*sin(c) + ((-2)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
y = ((-2)*cos(b) - (3t+1)*sin(b))/((sqrt(3)(t-1))*sin(c) + ((-2)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
z = ((sqrt(3)(t-1))*cos(c) - ((-2)*sin(b) + (3t+1)*cos(b))*sin(c))/((sqrt(3)(t-1))*sin(c) + ((-2)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
---
15. { 2*sqrt(3) , -2 , 2*sqrt(3)t , -2 }
x = (2*sqrt(3))/((2*sqrt(3)t)*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
y = ((-2)*cos(b) - (-2)*sin(b))/((2*sqrt(3)t)*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
z = ((2*sqrt(3)t)*cos(c) - ((-2)*sin(b) + (-2)*cos(b))*sin(c))/((2*sqrt(3)t)*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
---
------
16. { 0 , 4 , -sqrt(3)(t-1) , 3t+1 }
x = (0)/((-sqrt(3)(t-1))*sin(c) + ((4)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
y = ((4)*cos(b) - (3t+1)*sin(b))/((-sqrt(3)(t-1))*sin(c) + ((4)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
z = ((-sqrt(3)(t-1))*cos(c) - ((4)*sin(b) + (3t+1)*cos(b))*sin(c))/((-sqrt(3)(t-1))*sin(c) + ((4)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
---
17. { 0 , 4 , sqrt(3)(t-1) , 3t+1 }
x = (0)/((sqrt(3)(t-1))*sin(c) + ((4)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
y = ((4)*cos(b) - (3t+1)*sin(b))/((sqrt(3)(t-1))*sin(c) + ((4)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
z = ((sqrt(3)(t-1))*cos(c) - ((4)*sin(b) + (3t+1)*cos(b))*sin(c))/((sqrt(3)(t-1))*sin(c) + ((4)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
---
18. { 0 , 4 , 2*sqrt(3)t , -2 }
x = (0)/((2*sqrt(3)t)*sin(c) + ((4)*sin(b) + (-2)*cos(b))*cos(c) + a)
y = ((4)*cos(b) - (-2)*sin(b))/((2*sqrt(3)t)*sin(c) + ((4)*sin(b) + (-2)*cos(b))*cos(c) + a)
z = ((2*sqrt(3)t)*cos(c) - ((4)*sin(b) + (-2)*cos(b))*sin(c))/((2*sqrt(3)t)*sin(c) + ((4)*sin(b) + (-2)*cos(b))*cos(c) + a)
---------------
2D Elements:
• 6 triangles: [ 2D elem A x 0D elem B ] + [ 0D elem A x 2D elem B ]
1. { sqrt(3)(v-1)u , 3v+1 , -2*sqrt(3) , -2 }
x = (sqrt(3)(v-1)u)/((-2*sqrt(3))*sin(c) + ((3v+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
y = ((3v+1)*cos(b) - (-2)*sin(b))/((-2*sqrt(3))*sin(c) + ((3v+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
z = ((-2*sqrt(3))*cos(c) - ((3v+1)*sin(b) + (-2)*cos(b))*sin(c))/((-2*sqrt(3))*sin(c) + ((3v+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
---
2. { sqrt(3)(v-1)u , 3v+1 , 2*sqrt(3) , -2 }
x = (sqrt(3)(v-1)u)/((2*sqrt(3))*sin(c) + ((3v+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
y = ((3v+1)*cos(b) - (-2)*sin(b))/((2*sqrt(3))*sin(c) + ((3v+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
z = ((2*sqrt(3))*cos(c) - ((3v+1)*sin(b) + (-2)*cos(b))*sin(c))/((2*sqrt(3))*sin(c) + ((3v+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
---
3. { sqrt(3)(v-1)u , 3v+1 , 0 , 4 }
x = (sqrt(3)(v-1)u)/((0)*sin(c) + ((3v+1)*sin(b) + (4)*cos(b))*cos(c) + a)
y = ((3v+1)*cos(b) - (4)*sin(b))/((0)*sin(c) + ((3v+1)*sin(b) + (4)*cos(b))*cos(c) + a)
z = ((0)*cos(c) - ((3v+1)*sin(b) + (4)*cos(b))*sin(c))/((0)*sin(c) + ((3v+1)*sin(b) + (4)*cos(b))*cos(c) + a)
---
------
4. { -2*sqrt(3) , -2 , sqrt(3)(v-1)u , 3v+1 }
x = (-2*sqrt(3))/((sqrt(3)(v-1)u)*sin(c) + ((-2)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
y = ((-2)*cos(b) - (3v+1)*sin(b))/((sqrt(3)(v-1)u)*sin(c) + ((-2)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
z = ((sqrt(3)(v-1)u)*cos(c) - ((-2)*sin(b) + (3v+1)*cos(b))*sin(c))/((sqrt(3)(v-1)u)*sin(c) + ((-2)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
---
5. { 2*sqrt(3) , -2 , sqrt(3)(v-1)u , 3v+1 }
x = (2*sqrt(3))/((sqrt(3)(v-1)u)*sin(c) + ((-2)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
y = ((-2)*cos(b) - (3v+1)*sin(b))/((sqrt(3)(v-1)u)*sin(c) + ((-2)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
z = ((sqrt(3)(v-1)u)*cos(c) - ((-2)*sin(b) + (3v+1)*cos(b))*sin(c))/((sqrt(3)(v-1)u)*sin(c) + ((-2)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
---
6. { 0 , 4 , sqrt(3)(v-1)u , 3v+1 }
x = (0)/((sqrt(3)(v-1)u)*sin(c) + ((4)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
y = ((4)*cos(b) - (3v+1)*sin(b))/((sqrt(3)(v-1)u)*sin(c) + ((4)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
z = ((sqrt(3)(v-1)u)*cos(c) - ((4)*sin(b) + (3v+1)*cos(b))*sin(c))/((sqrt(3)(v-1)u)*sin(c) + ((4)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
---
• 9 squares: 1D elem A x 1D elem B
7. { -sqrt(3)(u-1) , 3u+1 , -sqrt(3)(v-1) , 3v+1 }
x = (-sqrt(3)(u-1))/((-sqrt(3)(v-1))*sin(c) + ((3u+1)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
y = ((3u+1)*cos(b) - (3v+1)*sin(b))/((-sqrt(3)(v-1))*sin(c) + ((3u+1)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
z = ((-sqrt(3)(v-1))*cos(c) - ((3u+1)*sin(b) + (3v+1)*cos(b))*sin(c))/((-sqrt(3)(v-1))*sin(c) + ((3u+1)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
---
8. { -sqrt(3)(u-1) , 3u+1 , sqrt(3)(v-1) , 3v+1 }
x = (-sqrt(3)(u-1))/((sqrt(3)(v-1))*sin(c) + ((3u+1)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
y = ((3u+1)*cos(b) - (3v+1)*sin(b))/((sqrt(3)(v-1))*sin(c) + ((3u+1)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
z = ((sqrt(3)(v-1))*cos(c) - ((3u+1)*sin(b) + (3v+1)*cos(b))*sin(c))/((sqrt(3)(v-1))*sin(c) + ((3u+1)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
---
9. { -sqrt(3)(u-1) , 3u+1 , 2*sqrt(3)v , -2 }
x = (-sqrt(3)(u-1))/((2*sqrt(3)v)*sin(c) + ((3u+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
y = ((3u+1)*cos(b) - (-2)*sin(b))/((2*sqrt(3)v)*sin(c) + ((3u+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
z = ((2*sqrt(3)v)*cos(c) - ((3u+1)*sin(b) + (-2)*cos(b))*sin(c))/((2*sqrt(3)v)*sin(c) + ((3u+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
---
------
10. { sqrt(3)(u-1) , 3u+1 , -sqrt(3)(v-1) , 3v+1 }
x = (sqrt(3)(u-1))/((-sqrt(3)(v-1))*sin(c) + ((3u+1)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
y = ((3u+1)*cos(b) - (3v+1)*sin(b))/((-sqrt(3)(v-1))*sin(c) + ((3u+1)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
z = ((-sqrt(3)(v-1))*cos(c) - ((3u+1)*sin(b) + (3v+1)*cos(b))*sin(c))/((-sqrt(3)(v-1))*sin(c) + ((3u+1)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
---
11. { sqrt(3)(u-1) , 3u+1 , sqrt(3)(v-1) , 3v+1 }
x = (sqrt(3)(u-1))/((sqrt(3)(v-1))*sin(c) + ((3u+1)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
y = ((3u+1)*cos(b) - (3v+1)*sin(b))/((sqrt(3)(v-1))*sin(c) + ((3u+1)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
z = ((sqrt(3)(v-1))*cos(c) - ((3u+1)*sin(b) + (3v+1)*cos(b))*sin(c))/((sqrt(3)(v-1))*sin(c) + ((3u+1)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
---
12. { sqrt(3)(u-1) , 3u+1 , 2*sqrt(3)v , -2 }
x = (sqrt(3)(u-1))/((2*sqrt(3)v)*sin(c) + ((3u+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
y = ((3u+1)*cos(b) - (-2)*sin(b))/((2*sqrt(3)v)*sin(c) + ((3u+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
z = ((2*sqrt(3)v)*cos(c) - ((3u+1)*sin(b) + (-2)*cos(b))*sin(c))/((2*sqrt(3)v)*sin(c) + ((3u+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
---
------
13. { 2*sqrt(3)u , -2 , -sqrt(3)(v-1) , 3v+1 }
x = (2*sqrt(3)u)/((-sqrt(3)(v-1))*sin(c) + ((-2)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
y = ((-2)*cos(b) - (3v+1)*sin(b))/((-sqrt(3)(v-1))*sin(c) + ((-2)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
z = ((-sqrt(3)(v-1))*cos(c) - ((-2)*sin(b) + (3v+1)*cos(b))*sin(c))/((-sqrt(3)(v-1))*sin(c) + ((-2)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
---
14. { 2*sqrt(3)u , -2 , sqrt(3)(v-1) , 3v+1 }
x = (2*sqrt(3)u)/((sqrt(3)(v-1))*sin(c) + ((-2)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
y = ((-2)*cos(b) - (3v+1)*sin(b))/((sqrt(3)(v-1))*sin(c) + ((-2)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
z = ((sqrt(3)(v-1))*cos(c) - ((-2)*sin(b) + (3v+1)*cos(b))*sin(c))/((sqrt(3)(v-1))*sin(c) + ((-2)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
---
15. { 2*sqrt(3)u , -2 , 2*sqrt(3)v , -2 }
x = (2*sqrt(3)u)/((2*sqrt(3)v)*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
y = ((-2)*cos(b) - (-2)*sin(b))/((2*sqrt(3)v)*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
z = ((2*sqrt(3)v)*cos(c) - ((-2)*sin(b) + (-2)*cos(b))*sin(c))/((2*sqrt(3)v)*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
---
Implement Rotate + Project Function with the 1D,2D elements for final equations:
1D Elements
------------
1. { 2*sqrt(3)(t-1) , 6t+2 , -2*sqrt(2) , -4*sqrt(3) }
x = (2*sqrt(3)(t-1))/(((6t+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
y = ((6t+2)*cos(b) - (-2*sqrt(2))*sin(b))/(((6t+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
z = (((6t+2)*sin(b) + (-2*sqrt(2))*cos(b))*sin(c) + (-4*sqrt(3))*cos(c))/(((6t+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
2. { -2*sqrt(3)(t-1) , 6t+2 , -2*sqrt(2) , -4*sqrt(3) }
x = (-2*sqrt(3)(t-1))/(((6t+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
y = ((6t+2)*cos(b) - (-2*sqrt(2))*sin(b))/(((6t+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
z = (((6t+2)*sin(b) + (-2*sqrt(2))*cos(b))*sin(c) + (-4*sqrt(3))*cos(c))/(((6t+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
3. { 4*sqrt(3)t , -4 , -2*sqrt(2) , -4*sqrt(3) }
x = (4*sqrt(3)t)/(((-4)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
y = ((-4)*cos(b) - (-2*sqrt(2))*sin(b))/(((-4)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
z = (((-4)*sin(b) + (-2*sqrt(2))*cos(b))*sin(c) + (-4*sqrt(3))*cos(c))/(((-4)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
4. { 0 , -4(t-1) , 2*sqrt(2)(2t+1) , -4*sqrt(3) }
x = (0)/(((-4(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
y = ((-4(t-1))*cos(b) - (2*sqrt(2)(2t+1))*sin(b))/(((-4(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
z = (((-4(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*sin(c) + (-4*sqrt(3))*cos(c))/(((-4(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
5. { -2*sqrt(3)(t-1) , 2(t-1) , 2*sqrt(2)(2t+1) , -4*sqrt(3) }
x = (-2*sqrt(3)(t-1))/(((2(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
y = ((2(t-1))*cos(b) - (2*sqrt(2)(2t+1))*sin(b))/(((2(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
z = (((2(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*sin(c) + (-4*sqrt(3))*cos(c))/(((2(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
6. { 2*sqrt(3)(t-1) , 2(t-1) , 2*sqrt(2)(2t+1) , -4*sqrt(3) }
x = (2*sqrt(3)(t-1))/(((2(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
y = ((2(t-1))*cos(b) - (2*sqrt(2)(2t+1))*sin(b))/(((2(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
z = (((2(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*sin(c) + (-4*sqrt(3))*cos(c))/(((2(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
---
7. { 2*sqrt(3)(t-1) , 6t+2 , -2*sqrt(2) , 4*sqrt(3) }
x = (2*sqrt(3)(t-1))/(((6t+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
y = ((6t+2)*cos(b) - (-2*sqrt(2))*sin(b))/(((6t+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
z = (((6t+2)*sin(b) + (-2*sqrt(2))*cos(b))*sin(c) + (4*sqrt(3))*cos(c))/(((6t+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
8. { -2*sqrt(3)(t-1) , 6t+2 , -2*sqrt(2) , 4*sqrt(3) }
x = (-2*sqrt(3)(t-1))/(((6t+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
y = ((6t+2)*cos(b) - (-2*sqrt(2))*sin(b))/(((6t+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
z = (((6t+2)*sin(b) + (-2*sqrt(2))*cos(b))*sin(c) + (4*sqrt(3))*cos(c))/(((6t+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
9. { 4*sqrt(3)t , -4 , -2*sqrt(2) , 4*sqrt(3) }
x = (4*sqrt(3)t)/(((-4)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
y = ((-4)*cos(b) - (-2*sqrt(2))*sin(b))/(((-4)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
z = (((-4)*sin(b) + (-2*sqrt(2))*cos(b))*sin(c) + (4*sqrt(3))*cos(c))/(((-4)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
10. { 0 , -4(t-1) , 2*sqrt(2)(2t+1) , 4*sqrt(3) }
x = (0)/(((-4(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
y = ((-4(t-1))*cos(b) - (2*sqrt(2)(2t+1))*sin(b))/(((-4(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
z = (((-4(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*sin(c) + (4*sqrt(3))*cos(c))/(((-4(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
11. { -2*sqrt(3)(t-1) , 2(t-1) , 2*sqrt(2)(2t+1) , 4*sqrt(3) }
x = (-2*sqrt(3)(t-1))/(((2(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
y = ((2(t-1))*cos(b) - (2*sqrt(2)(2t+1))*sin(b))/(((2(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
z = (((2(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*sin(c) + (4*sqrt(3))*cos(c))/(((2(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
12. { 2*sqrt(3)(t-1) , 2(t-1) , 2*sqrt(2)(2t+1) , 4*sqrt(3) }
x = (2*sqrt(3)(t-1))/(((2(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
y = ((2(t-1))*cos(b) - (2*sqrt(2)(2t+1))*sin(b))/(((2(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
z = (((2(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*sin(c) + (4*sqrt(3))*cos(c))/(((2(t-1))*sin(b) + (2*sqrt(2)(2t+1))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
---
13. { -4*sqrt(3) , -4 , -2*sqrt(2) , 4*sqrt(3)t }
x = (-4*sqrt(3))/(((-4)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3)t)*sin(c)+a)
y = ((-4)*cos(b) - (-2*sqrt(2))*sin(b))/(((-4)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3)t)*sin(c)+a)
z = (((-4)*sin(b) + (-2*sqrt(2))*cos(b))*sin(c) + (4*sqrt(3)t)*cos(c))/(((-4)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3)t)*sin(c)+a)
14. { 4*sqrt(3) , -4 , -2*sqrt(2) , 4*sqrt(3)t }
x = (4*sqrt(3))/(((-4)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3)t)*sin(c)+a)
y = ((-4)*cos(b) - (-2*sqrt(2))*sin(b))/(((-4)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3)t)*sin(c)+a)
z = (((-4)*sin(b) + (-2*sqrt(2))*cos(b))*sin(c) + (4*sqrt(3)t)*cos(c))/(((-4)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3)t)*sin(c)+a)
15. { 0 , 8 , -2*sqrt(2) , 4*sqrt(3)t }
x = (0)/(((8)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3)t)*sin(c)+a)
y = ((8)*cos(b) - (-2*sqrt(2))*sin(b))/(((8)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3)t)*sin(c)+a)
z = (((8)*sin(b) + (-2*sqrt(2))*cos(b))*sin(c) + (4*sqrt(3)t)*cos(c))/(((8)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3)t)*sin(c)+a)
16. { 0 , 0 , 6*sqrt(2) , 4*sqrt(3)t }
x = (0)/(((0)*sin(b) + (6*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3)t)*sin(c)+a)
y = ((0)*cos(b) - (6*sqrt(2))*sin(b))/(((0)*sin(b) + (6*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3)t)*sin(c)+a)
z = (((0)*sin(b) + (6*sqrt(2))*cos(b))*sin(c) + (4*sqrt(3)t)*cos(c))/(((0)*sin(b) + (6*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3)t)*sin(c)+a)
------------
2D Elements
------------
1.{ 2*sqrt(3)(v-1)u , 6v+2 , -2*sqrt(2) , -4*sqrt(3) }
x = (2*sqrt(3)(v-1)u)/(((6v+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
y = ((6v+2)*cos(b) - (-2*sqrt(2))*sin(b))/(((6v+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
z = (((6v+2)*sin(b) + (-2*sqrt(2))*cos(b))*sin(c) + (-4*sqrt(3))*cos(c))/(((6v+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
2.{ sqrt(3)(u-1)(v-1), -(3u+1)(v-1), 2*sqrt(2)(2v+1), 4*sqrt(3) }
x = (sqrt(3)(u-1)(v-1))/(((-(3u+1)(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
y = ((-(3u+1)(v-1))*cos(b) - (2*sqrt(2)(2v+1))*sin(b))/(((-(3u+1)(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
z = (((-(3u+1)(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*sin(c) + (4*sqrt(3))*cos(c))/(((-(3u+1)(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
3.{ -sqrt(3)(u-1)(v-1), -(3u+1)(v-1), 2*sqrt(2)(2v+1), -4*sqrt(3) }
x = (-sqrt(3)(u-1)(v-1))/(((-(3u+1)(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
y = ((-(3u+1)(v-1))*cos(b) - (2*sqrt(2)(2v+1))*sin(b))/(((-(3u+1)(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
z = (((-(3u+1)(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*sin(c) + (-4*sqrt(3))*cos(c))/(((-(3u+1)(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
4.{ 2*sqrt(3)(v-1)u , 2(v-1) , 2*sqrt(2)(2v+1) , -4*sqrt(3) }
x = (2*sqrt(3)(v-1)u)/(((2(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
y = ((2(v-1))*cos(b) - (2*sqrt(2)(2v+1))*sin(b))/(((2(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
z = (((2(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*sin(c) + (-4*sqrt(3))*cos(c))/(((2(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*cos(c) - (-4*sqrt(3))*sin(c)+a)
---
5. { 2*sqrt(3)(v-1)u , 6v+2 , -2*sqrt(2) , 4*sqrt(3) }
x = (2*sqrt(3)(v-1)u)/(((6v+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
y = ((6v+2)*cos(b) - (-2*sqrt(2))*sin(b))/(((6v+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
z = (((6v+2)*sin(b) + (-2*sqrt(2))*cos(b))*sin(c) + (4*sqrt(3))*cos(c))/(((6v+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
6. { sqrt(3)(u-1)(v-1) , -(3u+1)(v-1) , 2*sqrt(2)(2v+1), 4*sqrt(3) }
x = (sqrt(3)(u-1)(v-1))/(((-(3u+1)(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
y = ((-(3u+1)(v-1))*cos(b) - (2*sqrt(2)(2v+1))*sin(b))/(((-(3u+1)(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
z = (((-(3u+1)(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*sin(c) + (4*sqrt(3))*cos(c))/(((-(3u+1)(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
7. { -sqrt(3)(u-1)(v-1) , -(3u+1)(v-1) , 2*sqrt(2)(2v+1) , 4*sqrt(3) }
x = (-sqrt(3)(u-1)(v-1))/(((-(3u+1)(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
y = ((-(3u+1)(v-1))*cos(b) - (2*sqrt(2)(2v+1))*sin(b))/(((-(3u+1)(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
z = (((-(3u+1)(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*sin(c) + (4*sqrt(3))*cos(c))/(((-(3u+1)(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
8. { 2*sqrt(3)(v-1)u , 2(v-1) , 2*sqrt(2)(2v+1) , 4*sqrt(3) }
x = (2*sqrt(3)(v-1)u)/(((2(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
y = ((2(v-1))*cos(b) - (2*sqrt(2)(2v+1))*sin(b))/(((2(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
z = (((2(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*sin(c) + (4*sqrt(3))*cos(c))/(((2(v-1))*sin(b) + (2*sqrt(2)(2v+1))*cos(b))*cos(c) - (4*sqrt(3))*sin(c)+a)
---
9. { 2*sqrt(3)(u-1) , 6u+2 , -2*sqrt(2) , 4*sqrt(3)v }
x = (2*sqrt(3)(u-1))/(((6u+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3)v)*sin(c)+a)
y = ((6u+2)*cos(b) - (-2*sqrt(2))*sin(b))/(((6u+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3)v)*sin(c)+a)
z = (((6u+2)*sin(b) + (-2*sqrt(2))*cos(b))*sin(c) + (4*sqrt(3)v)*cos(c))/(((6u+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3)v)*sin(c)+a)
10. { -2*sqrt(3)(u-1) , 6u+2 , -2*sqrt(2) , 4*sqrt(3)v }
x = (-2*sqrt(3)(u-1))/(((6u+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3)v)*sin(c)+a)
y = ((6u+2)*cos(b) - (-2*sqrt(2))*sin(b))/(((6u+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3)v)*sin(c)+a)
z = (((6u+2)*sin(b) + (-2*sqrt(2))*cos(b))*sin(c) + (4*sqrt(3)v)*cos(c))/(((6u+2)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3)v)*sin(c)+a)
11. { 4*sqrt(3)u , -4 , -2*sqrt(2) , 4*sqrt(3)v }
x = (4*sqrt(3)u)/(((-4)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3)v)*sin(c)+a)
y = ((-4)*cos(b) - (-2*sqrt(2))*sin(b))/(((-4)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3)v)*sin(c)+a)
z = (((-4)*sin(b) + (-2*sqrt(2))*cos(b))*sin(c) + (4*sqrt(3)v)*cos(c))/(((-4)*sin(b) + (-2*sqrt(2))*cos(b))*cos(c) - (4*sqrt(3)v)*sin(c)+a)
12. { 0 , -4(u-1) , 2*sqrt(2)(2u+1) , 4*sqrt(3)v }
x = (0)/(((-4(u-1))*sin(b) + (2*sqrt(2)(2u+1))*cos(b))*cos(c) - (4*sqrt(3)v)*sin(c)+a)
y = ((-4(u-1))*cos(b) - (2*sqrt(2)(2u+1))*sin(b))/(((-4(u-1))*sin(b) + (2*sqrt(2)(2u+1))*cos(b))*cos(c) - (4*sqrt(3)v)*sin(c)+a)
z = (((-4(u-1))*sin(b) + (2*sqrt(2)(2u+1))*cos(b))*sin(c) + (4*sqrt(3)v)*cos(c))/(((-4(u-1))*sin(b) + (2*sqrt(2)(2u+1))*cos(b))*cos(c) - (4*sqrt(3)v)*sin(c)+a)
13. { -2*sqrt(3)(u-1) , 2(u-1) , 2*sqrt(2)(2u+1), 4*sqrt(3)v }
x = (-2*sqrt(3)(u-1))/(((2(u-1))*sin(b) + (2*sqrt(2)(2u+1))*cos(b))*cos(c) - (4*sqrt(3)v)*sin(c)+a)
y = ((2(u-1))*cos(b) - (2*sqrt(2)(2u+1))*sin(b))/(((2(u-1))*sin(b) + (2*sqrt(2)(2u+1))*cos(b))*cos(c) - (4*sqrt(3)v)*sin(c)+a)
z = (((2(u-1))*sin(b) + (2*sqrt(2)(2u+1))*cos(b))*sin(c) + (4*sqrt(3)v)*cos(c))/(((2(u-1))*sin(b) + (2*sqrt(2)(2u+1))*cos(b))*cos(c) - (4*sqrt(3)v)*sin(c)+a)
14. { 2*sqrt(3)(u-1) , 2(u-1) , 2*sqrt(2)(2u+1), 4*sqrt(3)v }
x = (2*sqrt(3)(u-1))/(((2(u-1))*sin(b) + (2*sqrt(2)(2u+1))*cos(b))*cos(c) - (4*sqrt(3)v)*sin(c)+a)
y = ((2(u-1))*cos(b) - (2*sqrt(2)(2u+1))*sin(b))/(((2(u-1))*sin(b) + (2*sqrt(2)(2u+1))*cos(b))*cos(c) - (4*sqrt(3)v)*sin(c)+a)
z = (((2(u-1))*sin(b) + (2*sqrt(2)(2u+1))*cos(b))*sin(c) + (4*sqrt(3)v)*cos(c))/(((2(u-1))*sin(b) + (2*sqrt(2)(2u+1))*cos(b))*cos(c) - (4*sqrt(3)v)*sin(c)+a)
Klitzing wrote:all these animated gifs are truely great visualizations!
--- rk
Mercurial, the Spectre wrote:Yeah, would love to see the 4D tori and the 5D sphere-circle duoprism
• Implement ZW Rotation + XYZ Projection to Edge equations
x = (X)/((Z)*sin(a)+(W)*cos(a)+3)
y = (Y)/((Z)*sin(a)+(W)*cos(a)+3)
z = ((Z)*cos(a)-(W)*sin(a))/((Z)*sin(a)+(W)*cos(a)+3)
1. { b*t , -1/b , -c , -d }
x = (b*t)/((-c)*sin(a)+(-d)*cos(a)+3)
y = (-1/b)/((-c)*sin(a)+(-d)*cos(a)+3)
z = ((-c)*cos(a)-(-d)*sin(a))/((-c)*sin(a)+(-d)*cos(a)+3)
2. { b*t , -1/b , c , -d }
x = (b*t)/((c)*sin(a)+(-d)*cos(a)+3)
y = (-1/b)/((c)*sin(a)+(-d)*cos(a)+3)
z = ((c)*cos(a)-(-d)*sin(a))/((c)*sin(a)+(-d)*cos(a)+3)
3. { b*t , 1/b , -c , -d }
x = (b*t)/((-c)*sin(a)+(-d)*cos(a)+3)
y = (1/b)/((-c)*sin(a)+(-d)*cos(a)+3)
z = ((-c)*cos(a)-(-d)*sin(a))/((-c)*sin(a)+(-d)*cos(a)+3)
4. { b*t , 1/b , c , -d }
x = (b*t)/((c)*sin(a)+(-d)*cos(a)+3)
y = (1/b)/((c)*sin(a)+(-d)*cos(a)+3)
z = ((c)*cos(a)-(-d)*sin(a))/((c)*sin(a)+(-d)*cos(a)+3)
--
5. { b*t , -1/b , -c , d }
x = (b*t)/((-c)*sin(a)+(d)*cos(a)+3)
y = (-1/b)/((-c)*sin(a)+(d)*cos(a)+3)
z = ((-c)*cos(a)-(d)*sin(a))/((-c)*sin(a)+(d)*cos(a)+3)
6. { b*t , -1/b , c , d }
x = (b*t)/((c)*sin(a)+(d)*cos(a)+3)
y = (-1/b)/((c)*sin(a)+(d)*cos(a)+3)
z = ((c)*cos(a)-(d)*sin(a))/((c)*sin(a)+(d)*cos(a)+3)
7. { b*t , 1/b , -c , d }
x = (b*t)/((-c)*sin(a)+(d)*cos(a)+3)
y = (1/b)/((-c)*sin(a)+(d)*cos(a)+3)
z = ((-c)*cos(a)-(d)*sin(a))/((-c)*sin(a)+(d)*cos(a)+3)
8. { b*t , 1/b , c , d }
x = (b*t)/((c)*sin(a)+(d)*cos(a)+3)
y = (1/b)/((c)*sin(a)+(d)*cos(a)+3)
z = ((c)*cos(a)-(d)*sin(a))/((c)*sin(a)+(d)*cos(a)+3)
-------
9. { -b , t/b , -c , -d }
x = (-b)/((-c)*sin(a)+(-d)*cos(a)+3)
y = (t/b)/((-c)*sin(a)+(-d)*cos(a)+3)
z = ((-c)*cos(a)-(-d)*sin(a))/((-c)*sin(a)+(-d)*cos(a)+3)
10. { -b , t/b , c , -d }
x = (-b)/((c)*sin(a)+(-d)*cos(a)+3)
y = (t/b)/((c)*sin(a)+(-d)*cos(a)+3)
z = ((c)*cos(a)-(-d)*sin(a))/((c)*sin(a)+(-d)*cos(a)+3)
11. { b , t/b , -c , -d }
x = (b)/((-c)*sin(a)+(-d)*cos(a)+3)
y = (t/b)/((-c)*sin(a)+(-d)*cos(a)+3)
z = ((-c)*cos(a)-(-d)*sin(a))/((-c)*sin(a)+(-d)*cos(a)+3)
12. { b , t/b , c , -d }
x = (b)/((c)*sin(a)+(-d)*cos(a)+3)
y = (t/b)/((c)*sin(a)+(-d)*cos(a)+3)
z = ((c)*cos(a)-(-d)*sin(a))/((c)*sin(a)+(-d)*cos(a)+3)
---
13. { -b , t/b , -c , d }
x = (-b)/((-c)*sin(a)+(d)*cos(a)+3)
y = (t/b)/((-c)*sin(a)+(d)*cos(a)+3)
z = ((-c)*cos(a)-(d)*sin(a))/((-c)*sin(a)+(d)*cos(a)+3)
14. { -b , t/b , c , d }
x = (-b)/((c)*sin(a)+(d)*cos(a)+3)
y = (t/b)/((c)*sin(a)+(d)*cos(a)+3)
z = ((c)*cos(a)-(d)*sin(a))/((c)*sin(a)+(d)*cos(a)+3)
15. { b , t/b , -c , d }
x = (b)/((-c)*sin(a)+(d)*cos(a)+3)
y = (t/b)/((-c)*sin(a)+(d)*cos(a)+3)
z = ((-c)*cos(a)-(d)*sin(a))/((-c)*sin(a)+(d)*cos(a)+3)
16. { b , t/b , c , d }
x = (b)/((c)*sin(a)+(d)*cos(a)+3)
y = (t/b)/((c)*sin(a)+(d)*cos(a)+3)
z = ((c)*cos(a)-(d)*sin(a))/((c)*sin(a)+(d)*cos(a)+3)
-------
17. { -b , -1/b , c*t , -d }
x = (-b)/((c*t)*sin(a)+(-d)*cos(a)+3)
y = (-1/b)/((c*t)*sin(a)+(-d)*cos(a)+3)
z = ((c*t)*cos(a)-(-d)*sin(a))/((c*t)*sin(a)+(-d)*cos(a)+3)
18. { -b , 1/b , c*t , -d }
x = (-b)/((c*t)*sin(a)+(-d)*cos(a)+3)
y = (1/b)/((c*t)*sin(a)+(-d)*cos(a)+3)
z = ((c*t)*cos(a)-(-d)*sin(a))/((c*t)*sin(a)+(-d)*cos(a)+3)
19. { b , -1/b , c*t , -d }
x = (b)/((c*t)*sin(a)+(-d)*cos(a)+3)
y = (-1/b)/((c*t)*sin(a)+(-d)*cos(a)+3)
z = ((c*t)*cos(a)-(-d)*sin(a))/((c*t)*sin(a)+(-d)*cos(a)+3)
20. { b , 1/b , c*t , -d }
x = (b)/((c*t)*sin(a)+(-d)*cos(a)+3)
y = (1/b)/((c*t)*sin(a)+(-d)*cos(a)+3)
z = ((c*t)*cos(a)-(-d)*sin(a))/((c*t)*sin(a)+(-d)*cos(a)+3)
---
21. { -b , -1/b , c*t , d }
x = (-b)/((c*t)*sin(a)+(d)*cos(a)+3)
y = (-1/b)/((c*t)*sin(a)+(d)*cos(a)+3)
z = ((c*t)*cos(a)-(d)*sin(a))/((c*t)*sin(a)+(d)*cos(a)+3)
22. { -b , 1/b , c*t , d }
x = (-b)/((c*t)*sin(a)+(d)*cos(a)+3)
y = (1/b)/((c*t)*sin(a)+(d)*cos(a)+3)
z = ((c*t)*cos(a)-(d)*sin(a))/((c*t)*sin(a)+(d)*cos(a)+3)
23. { b , -1/b , c*t , d }
x = (b)/((c*t)*sin(a)+(d)*cos(a)+3)
y = (-1/b)/((c*t)*sin(a)+(d)*cos(a)+3)
z = ((c*t)*cos(a)-(d)*sin(a))/((c*t)*sin(a)+(d)*cos(a)+3)
24. { b , 1/b , c*t , d }
x = (b)/((c*t)*sin(a)+(d)*cos(a)+3)
y = (1/b)/((c*t)*sin(a)+(d)*cos(a)+3)
z = ((c*t)*cos(a)-(d)*sin(a))/((c*t)*sin(a)+(d)*cos(a)+3)
-------
25. { -b , -1/b , -c , d*t }
x = (-b)/((-c)*sin(a)+(d*t)*cos(a)+3)
y = (-1/b)/((-c)*sin(a)+(d*t)*cos(a)+3)
z = ((-c)*cos(a)-(d*t)*sin(a))/((-c)*sin(a)+(d*t)*cos(a)+3)
26. { -b , 1/b , -c , d*t }
x = (-b)/((-c)*sin(a)+(d*t)*cos(a)+3)
y = (1/b)/((-c)*sin(a)+(d*t)*cos(a)+3)
z = ((-c)*cos(a)-(d*t)*sin(a))/((-c)*sin(a)+(d*t)*cos(a)+3)
27. { b , -1/b , -c , d*t }
x = (b)/((-c)*sin(a)+(d*t)*cos(a)+3)
y = (-1/b)/((-c)*sin(a)+(d*t)*cos(a)+3)
z = ((-c)*cos(a)-(d*t)*sin(a))/((-c)*sin(a)+(d*t)*cos(a)+3)
28. { b , 1/b , -c , d*t }
x = (b)/((-c)*sin(a)+(d*t)*cos(a)+3)
y = (1/b)/((-c)*sin(a)+(d*t)*cos(a)+3)
z = ((-c)*cos(a)-(d*t)*sin(a))/((-c)*sin(a)+(d*t)*cos(a)+3)
---
29. { -b , -1/b , c , d*t }
x = (-b)/((c)*sin(a)+(d*t)*cos(a)+3)
y = (-1/b)/((c)*sin(a)+(d*t)*cos(a)+3)
z = ((c)*cos(a)-(d*t)*sin(a))/((c)*sin(a)+(d*t)*cos(a)+3)
30. { -b , 1/b , c , d*t }
x = (-b)/((c)*sin(a)+(d*t)*cos(a)+3)
y = (1/b)/((c)*sin(a)+(d*t)*cos(a)+3)
z = ((c)*cos(a)-(d*t)*sin(a))/((c)*sin(a)+(d*t)*cos(a)+3)
31. { b , -1/b , c , d*t }
x = (b)/((c)*sin(a)+(d*t)*cos(a)+3)
y = (-1/b)/((c)*sin(a)+(d*t)*cos(a)+3)
z = ((c)*cos(a)-(d*t)*sin(a))/((c)*sin(a)+(d*t)*cos(a)+3)
32. { b , 1/b , c , d*t }
x = (b)/((c)*sin(a)+(d*t)*cos(a)+3)
y = (1/b)/((c)*sin(a)+(d*t)*cos(a)+3)
z = ((c)*cos(a)-(d*t)*sin(a))/((c)*sin(a)+(d*t)*cos(a)+3)
=======================================
• 2D Faces
{ b*u , v/b , ±c , ±d }
{ b*u , ±1/b , c*v , ±d }
{ b*u , ±1/b , ±c , d*v }
{ ±b , u/b , c*v , ±d }
{ ±b , u/b , ±c , d*v }
{ ±b , ±1/b , c*u , d*v }
Expanded 2D Faces Equations
1. { b*u , v/b , -c , -d }
2. { b*u , v/b , -c , d }
3. { b*u , v/b , c , -d }
4. { b*u , v/b , c , d }
---
5. { b*u , -1/b , c*v , -d }
6. { b*u , -1/b , c*v , d }
7. { b*u , 1/b , c*v , -d }
8. { b*u , 1/b , c*v , d }
---
9. { b*u , -1/b , -c , d*v }
10. { b*u , -1/b , c , d*v }
11. { b*u , 1/b , -c , d*v }
12. { b*u , 1/b , c , d*v }
---
13. { -b , u/b , c*v , -d }
14. { -b , u/b , c*v , d }
15. { b , u/b , c*v , -d }
16. { b , u/b , c*v , d }
---
17. { -b , u/b , -c , d*v }
18. { -b , u/b , c , d*v }
19. { b , u/b , -c , d*v }
20. { b , u/b , c , d*v }
---
21. { -b , -1/b , c*u , d*v }
22. { -b , 1/b , c*u , d*v }
23. { b , -1/b , c*u , d*v }
24. { b , 1/b , c*u , d*v }
• Implement ZW Rotation + XYZ Projection to Square equations
x = (X)/((Z)*sin(a)+(W)*cos(a)+3)
y = (Y)/((Z)*sin(a)+(W)*cos(a)+3)
z = ((Z)*cos(a)-(W)*sin(a))/((Z)*sin(a)+(W)*cos(a)+3)
1. { b*u , v/b , -c , -d }
x = (b*u)/((-c)*sin(a)+(-d)*cos(a)+3)
y = (v/b)/((-c)*sin(a)+(-d)*cos(a)+3)
z = ((-c)*cos(a)-(-d)*sin(a))/((-c)*sin(a)+(-d)*cos(a)+3)
2. { b*u , v/b , -c , d }
x = (b*u)/((-c)*sin(a)+(d)*cos(a)+3)
y = (v/b)/((-c)*sin(a)+(d)*cos(a)+3)
z = ((-c)*cos(a)-(d)*sin(a))/((-c)*sin(a)+(d)*cos(a)+3)
3. { b*u , v/b , c , -d }
x = (b*u)/((c)*sin(a)+(-d)*cos(a)+3)
y = (v/b)/((c)*sin(a)+(-d)*cos(a)+3)
z = ((c)*cos(a)-(-d)*sin(a))/((c)*sin(a)+(-d)*cos(a)+3)
4. { b*u , v/b , c , d }
x = (b*u)/((c)*sin(a)+(d)*cos(a)+3)
y = (v/b)/((c)*sin(a)+(d)*cos(a)+3)
z = ((c)*cos(a)-(d)*sin(a))/((c)*sin(a)+(d)*cos(a)+3)
---
5. { b*u , -1/b , c*v , -d }
x = (b*u)/((c*v)*sin(a)+(-d)*cos(a)+3)
y = (-1/b)/((c*v)*sin(a)+(-d)*cos(a)+3)
z = ((c*v)*cos(a)-(-d)*sin(a))/((c*v)*sin(a)+(-d)*cos(a)+3)
6. { b*u , -1/b , c*v , d }
x = (b*u)/((c*v)*sin(a)+(d)*cos(a)+3)
y = (-1/b)/((c*v)*sin(a)+(d)*cos(a)+3)
z = ((c*v)*cos(a)-(d)*sin(a))/((c*v)*sin(a)+(d)*cos(a)+3)
7. { b*u , 1/b , c*v , -d }
x = (b*u)/((c*v)*sin(a)+(-d)*cos(a)+3)
y = (1/b)/((c*v)*sin(a)+(-d)*cos(a)+3)
z = ((c*v)*cos(a)-(-d)*sin(a))/((c*v)*sin(a)+(-d)*cos(a)+3)
8. { b*u , 1/b , c*v , d }
x = (b*u)/((c*v)*sin(a)+(d)*cos(a)+3)
y = (1/b)/((c*v)*sin(a)+(d)*cos(a)+3)
z = ((c*v)*cos(a)-(d)*sin(a))/((c*v)*sin(a)+(d)*cos(a)+3)
---
9. { b*u , -1/b , -c , d*v }
x = (b*u)/((-c)*sin(a)+(d*v)*cos(a)+3)
y = (-1/b)/((-c)*sin(a)+(d*v)*cos(a)+3)
z = ((-c)*cos(a)-(d*v)*sin(a))/((-c)*sin(a)+(d*v)*cos(a)+3)
10. { b*u , -1/b , c , d*v }
x = (b*u)/((c)*sin(a)+(d*v)*cos(a)+3)
y = (-1/b)/((c)*sin(a)+(d*v)*cos(a)+3)
z = ((c)*cos(a)-(d*v)*sin(a))/((c)*sin(a)+(d*v)*cos(a)+3)
11. { b*u , 1/b , -c , d*v }
x = (b*u)/((-c)*sin(a)+(d*v)*cos(a)+3)
y = (1/b)/((-c)*sin(a)+(d*v)*cos(a)+3)
z = ((-c)*cos(a)-(d*v)*sin(a))/((-c)*sin(a)+(d*v)*cos(a)+3)
12. { b*u , 1/b , c , d*v }
x = (b*u)/((c)*sin(a)+(d*v)*cos(a)+3)
y = (1/b)/((c)*sin(a)+(d*v)*cos(a)+3)
z = ((c)*cos(a)-(d*v)*sin(a))/((c)*sin(a)+(d*v)*cos(a)+3)
---
13. { -b , u/b , c*v , -d }
x = (-b)/((c*v)*sin(a)+(-d)*cos(a)+3)
y = (u/b)/((c*v)*sin(a)+(-d)*cos(a)+3)
z = ((c*v)*cos(a)-(-d)*sin(a))/((c*v)*sin(a)+(-d)*cos(a)+3)
14. { -b , u/b , c*v , d }
x = (-b)/((c*v)*sin(a)+(d)*cos(a)+3)
y = (u/b)/((c*v)*sin(a)+(d)*cos(a)+3)
z = ((c*v)*cos(a)-(d)*sin(a))/((c*v)*sin(a)+(d)*cos(a)+3)
15. { b , u/b , c*v , -d }
x = (b)/((c*v)*sin(a)+(-d)*cos(a)+3)
y = (u/b)/((c*v)*sin(a)+(-d)*cos(a)+3)
z = ((c*v)*cos(a)-(-d)*sin(a))/((c*v)*sin(a)+(-d)*cos(a)+3)
16. { b , u/b , c*v , d }
x = (b)/((c*v)*sin(a)+(d)*cos(a)+3)
y = (u/b)/((c*v)*sin(a)+(d)*cos(a)+3)
z = ((c*v)*cos(a)-(d)*sin(a))/((c*v)*sin(a)+(d)*cos(a)+3)
---
17. { -b , u/b , -c , d*v }
x = (-b)/((-c)*sin(a)+(d*v)*cos(a)+3)
y = (u/b)/((-c)*sin(a)+(d*v)*cos(a)+3)
z = ((-c)*cos(a)-(d*v)*sin(a))/((-c)*sin(a)+(d*v)*cos(a)+3)
18. { -b , u/b , c , d*v }
x = (-b)/((c)*sin(a)+(d*v)*cos(a)+3)
y = (u/b)/((c)*sin(a)+(d*v)*cos(a)+3)
z = ((c)*cos(a)-(d*v)*sin(a))/((c)*sin(a)+(d*v)*cos(a)+3)
19. { b , u/b , -c , d*v }
x = (b)/((-c)*sin(a)+(d*v)*cos(a)+3)
y = (u/b)/((-c)*sin(a)+(d*v)*cos(a)+3)
z = ((-c)*cos(a)-(d*v)*sin(a))/((-c)*sin(a)+(d*v)*cos(a)+3)
20. { b , u/b , c , d*v }
x = (b)/((c)*sin(a)+(d*v)*cos(a)+3)
y = (u/b)/((c)*sin(a)+(d*v)*cos(a)+3)
z = ((c)*cos(a)-(d*v)*sin(a))/((c)*sin(a)+(d*v)*cos(a)+3)
---
21. { -b , -1/b , c*u , d*v }
x = (-b)/((c*u)*sin(a)+(d*v)*cos(a)+3)
y = (-1/b)/((c*u)*sin(a)+(d*v)*cos(a)+3)
z = ((c*u)*cos(a)-(d*v)*sin(a))/((c*u)*sin(a)+(d*v)*cos(a)+3)
22. { -b , 1/b , c*u , d*v }
x = (-b)/((c*u)*sin(a)+(d*v)*cos(a)+3)
y = (1/b)/((c*u)*sin(a)+(d*v)*cos(a)+3)
z = ((c*u)*cos(a)-(d*v)*sin(a))/((c*u)*sin(a)+(d*v)*cos(a)+3)
23. { b , -1/b , c*u , d*v }
x = (b)/((c*u)*sin(a)+(d*v)*cos(a)+3)
y = (-1/b)/((c*u)*sin(a)+(d*v)*cos(a)+3)
z = ((c*u)*cos(a)-(d*v)*sin(a))/((c*u)*sin(a)+(d*v)*cos(a)+3)
24. { b , 1/b , c*u , d*v }
x = (b)/((c*u)*sin(a)+(d*v)*cos(a)+3)
y = (1/b)/((c*u)*sin(a)+(d*v)*cos(a)+3)
z = ((c*u)*cos(a)-(d*v)*sin(a))/((c*u)*sin(a)+(d*v)*cos(a)+3)
quickfur wrote:Hmm. Are you by any chance referring to this format? If so, you're in luck, since exporting to that format should be trivial to implement.
16
-1 -1 -1 -1
-1 -1 -1 1
-1 -1 1 -1
-1 -1 1 1
-1 1 -1 -1
-1 1 -1 1
-1 1 1 -1
-1 1 1 1
1 -1 -1 -1
1 -1 -1 1
1 -1 1 -1
1 -1 1 1
1 1 -1 -1
1 1 -1 1
1 1 1 -1
1 1 1 1
32
0 2
2 3
3 1
1 0
0 4
4 5
5 1
4 6
6 2
0 8
8 9
9 1
8 10
10 2
8 12
12 4
6 7
7 5
7 3
10 11
11 3
9 11
9 13
13 5
12 13
12 14
14 10
14 6
14 15
15 13
15 11
15 7
24
4 0 2 3 1
4 0 4 5 1
4 0 4 6 2
4 0 8 9 1
4 0 8 10 2
4 0 8 12 4
4 4 6 7 5
4 2 6 7 3
4 1 5 7 3
4 2 10 11 3
4 1 9 11 3
4 1 9 13 5
4 8 10 11 9
4 8 12 13 9
4 8 12 14 10
4 4 12 13 5
4 4 12 14 6
4 2 10 14 6
4 12 14 15 13
4 10 14 15 11
4 9 13 15 11
4 6 14 15 7
4 5 13 15 7
4 3 11 15 7
quickfur wrote:Of course, you can approximate a curved surface by breaking it into many pieces, e.g., the cubinder can be approximated by a 4,n-duoprism for sufficiently large n. Just how large, depends on what you want to do with it.
quickfur wrote:And btw, the triangle-square product is just a 3,4-duoprism (I even have animations of it on that page). Similarly, the triangle-triangle product is just a 3,3-duoprism (also with animation ).
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