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Parametric Equations of 3-3 Duoprism

Philip_P
Nov 18th, 2017
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  1. Parametric Equations of a 3-3 Duoprism, with yw, zw rotation and xyz projection
  2.  
  3.  
  4.  
  5. 1D Elements
  6.  
  7. • 18 edges:
  8.  
  9. 1. { -sqrt(3)(t-1) , 3t+1 , -2*sqrt(3) , -2 }
  10.  
  11. x = (-sqrt(3)(t-1))/((-2*sqrt(3))*sin(c) + ((3t+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
  12.  
  13. y = ((3t+1)*cos(b) - (-2)*sin(b))/((-2*sqrt(3))*sin(c) + ((3t+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
  14.  
  15. 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)
  16.  
  17. ---
  18.  
  19. 2. { -sqrt(3)(t-1) , 3t+1 , 2*sqrt(3) , -2 }
  20.  
  21. x = (-sqrt(3)(t-1))/((2*sqrt(3))*sin(c) + ((3t+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
  22.  
  23. y = ((3t+1)*cos(b) - (-2)*sin(b))/((2*sqrt(3))*sin(c) + ((3t+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
  24.  
  25. 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)
  26.  
  27. ---
  28.  
  29. 3. { -sqrt(3)(t-1) , 3t+1 , 0 , 4 }
  30.  
  31. x = (-sqrt(3)(t-1))/((0)*sin(c) + ((3t+1)*sin(b) + (4)*cos(b))*cos(c) + a)
  32.  
  33. y = ((3t+1)*cos(b) - (4)*sin(b))/((0)*sin(c) + ((3t+1)*sin(b) + (4)*cos(b))*cos(c) + a)
  34.  
  35. 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)
  36.  
  37. ---
  38. -------
  39.  
  40. 4. { sqrt(3)(t-1) , 3t+1 , -2*sqrt(3) , -2 }
  41.  
  42. x = (sqrt(3)(t-1))/((-2*sqrt(3))*sin(c) + ((3t+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
  43.  
  44. y = ((3t+1)*cos(b) - (-2)*sin(b))/((-2*sqrt(3))*sin(c) + ((3t+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
  45.  
  46. 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)
  47.  
  48. ---
  49.  
  50. 5. { sqrt(3)(t-1) , 3t+1 , 2*sqrt(3) , -2 }
  51.  
  52. x = (sqrt(3)(t-1))/((2*sqrt(3))*sin(c) + ((3t+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
  53.  
  54. y = ((3t+1)*cos(b) - (-2)*sin(b))/((2*sqrt(3))*sin(c) + ((3t+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
  55.  
  56. 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)
  57.  
  58. ---
  59.  
  60. 6. { sqrt(3)(t-1) , 3t+1 , 0 , 4 }
  61.  
  62. x = (sqrt(3)(t-1))/((0)*sin(c) + ((3t+1)*sin(b) + (4)*cos(b))*cos(c) + a)
  63.  
  64. y = ((3t+1)*cos(b) - (4)*sin(b))/((0)*sin(c) + ((3t+1)*sin(b) + (4)*cos(b))*cos(c) + a)
  65.  
  66. 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)
  67.  
  68. ---
  69.  
  70. ------
  71.  
  72. 7. { 2*sqrt(3)t , -2 , -2*sqrt(3) , -2 }
  73.  
  74. x = (2*sqrt(3)t)/((-2*sqrt(3))*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
  75.  
  76. y = ((-2)*cos(b) - (-2)*sin(b))/((-2*sqrt(3))*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
  77.  
  78. 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)
  79.  
  80. ---
  81.  
  82. 8. { 2*sqrt(3)t , -2 , 2*sqrt(3) , -2 }
  83.  
  84. x = (2*sqrt(3)t)/((2*sqrt(3))*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
  85.  
  86. y = ((-2)*cos(b) - (-2)*sin(b))/((2*sqrt(3))*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
  87.  
  88. 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)
  89.  
  90. ---
  91.  
  92. 9. { 2*sqrt(3)t , -2 , 0 , 4 }
  93.  
  94. x = (2*sqrt(3)t)/((0)*sin(c) + ((-2)*sin(b) + (4)*cos(b))*cos(c) + a)
  95.  
  96. y = ((-2)*cos(b) - (4)*sin(b))/((0)*sin(c) + ((-2)*sin(b) + (4)*cos(b))*cos(c) + a)
  97.  
  98. z = ((0)*cos(c) - ((-2)*sin(b) + (4)*cos(b))*sin(c))/((0)*sin(c) + ((-2)*sin(b) + (4)*cos(b))*cos(c) + a)
  99.  
  100. ---
  101.  
  102.  
  103. ------
  104.  
  105. 10. { -2*sqrt(3) , -2 , -sqrt(3)(t-1) , 3t+1 }
  106.  
  107. x = (-2*sqrt(3))/((-sqrt(3)(t-1))*sin(c) + ((-2)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
  108.  
  109. y = ((-2)*cos(b) - (3t+1)*sin(b))/((-sqrt(3)(t-1))*sin(c) + ((-2)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
  110.  
  111. 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)
  112.  
  113. ---
  114.  
  115. 11. { -2*sqrt(3) , -2 , sqrt(3)(t-1) , 3t+1 }
  116.  
  117. x = (-2*sqrt(3))/((sqrt(3)(t-1))*sin(c) + ((-2)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
  118.  
  119. y = ((-2)*cos(b) - (3t+1)*sin(b))/((sqrt(3)(t-1))*sin(c) + ((-2)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
  120.  
  121. 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)
  122.  
  123. ---
  124.  
  125. 12. { -2*sqrt(3) , -2 , 2*sqrt(3)t , -2 }
  126.  
  127. x = (-2*sqrt(3))/((2*sqrt(3)t)*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
  128.  
  129. y = ((-2)*cos(b) - (-2)*sin(b))/((2*sqrt(3)t)*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
  130.  
  131. 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)
  132.  
  133. ---
  134.  
  135. ------
  136.  
  137. 13. { 2*sqrt(3) , -2 , -sqrt(3)(t-1) , 3t+1 }
  138.  
  139. x = (2*sqrt(3))/((-sqrt(3)(t-1))*sin(c) + ((-2)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
  140.  
  141. y = ((-2)*cos(b) - (3t+1)*sin(b))/((-sqrt(3)(t-1))*sin(c) + ((-2)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
  142.  
  143. 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)
  144.  
  145. ---
  146.  
  147. 14. { 2*sqrt(3) , -2 , sqrt(3)(t-1) , 3t+1 }
  148.  
  149. x = (2*sqrt(3))/((sqrt(3)(t-1))*sin(c) + ((-2)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
  150.  
  151. y = ((-2)*cos(b) - (3t+1)*sin(b))/((sqrt(3)(t-1))*sin(c) + ((-2)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
  152.  
  153. 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)
  154.  
  155. ---
  156.  
  157. 15. { 2*sqrt(3) , -2 , 2*sqrt(3)t , -2 }
  158.  
  159. x = (2*sqrt(3))/((2*sqrt(3)t)*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
  160.  
  161. y = ((-2)*cos(b) - (-2)*sin(b))/((2*sqrt(3)t)*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
  162.  
  163. 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)
  164.  
  165. ---
  166.  
  167. ------
  168.  
  169. 16. { 0 , 4 , -sqrt(3)(t-1) , 3t+1 }
  170.  
  171. x = (0)/((-sqrt(3)(t-1))*sin(c) + ((4)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
  172.  
  173. y = ((4)*cos(b) - (3t+1)*sin(b))/((-sqrt(3)(t-1))*sin(c) + ((4)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
  174.  
  175. 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)
  176.  
  177. ---
  178.  
  179. 17. { 0 , 4 , sqrt(3)(t-1) , 3t+1 }
  180.  
  181. x = (0)/((sqrt(3)(t-1))*sin(c) + ((4)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
  182.  
  183. y = ((4)*cos(b) - (3t+1)*sin(b))/((sqrt(3)(t-1))*sin(c) + ((4)*sin(b) + (3t+1)*cos(b))*cos(c) + a)
  184.  
  185. 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)
  186.  
  187. ---
  188.  
  189. 18. { 0 , 4 , 2*sqrt(3)t , -2 }
  190.  
  191. x = (0)/((2*sqrt(3)t)*sin(c) + ((4)*sin(b) + (-2)*cos(b))*cos(c) + a)
  192.  
  193. y = ((4)*cos(b) - (-2)*sin(b))/((2*sqrt(3)t)*sin(c) + ((4)*sin(b) + (-2)*cos(b))*cos(c) + a)
  194.  
  195. 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)
  196.  
  197.  
  198.  
  199.  
  200.  
  201. ---------------
  202.  
  203.  
  204.  
  205.  
  206. 2D Elements:
  207.  
  208.  
  209.  
  210. • 6 triangles: [ 2D elem A x 0D elem B ] + [ 0D elem A x 2D elem B ]
  211.  
  212. 1. { sqrt(3)(v-1)u , 3v+1 , -2*sqrt(3) , -2 }
  213.  
  214. x = (sqrt(3)(v-1)u)/((-2*sqrt(3))*sin(c) + ((3v+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
  215.  
  216. y = ((3v+1)*cos(b) - (-2)*sin(b))/((-2*sqrt(3))*sin(c) + ((3v+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
  217.  
  218. 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)
  219.  
  220. ---
  221.  
  222. 2. { sqrt(3)(v-1)u , 3v+1 , 2*sqrt(3) , -2 }
  223.  
  224. x = (sqrt(3)(v-1)u)/((2*sqrt(3))*sin(c) + ((3v+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
  225.  
  226. y = ((3v+1)*cos(b) - (-2)*sin(b))/((2*sqrt(3))*sin(c) + ((3v+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
  227.  
  228. 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)
  229.  
  230. ---
  231.  
  232. 3. { sqrt(3)(v-1)u , 3v+1 , 0 , 4 }
  233.  
  234. x = (sqrt(3)(v-1)u)/((0)*sin(c) + ((3v+1)*sin(b) + (4)*cos(b))*cos(c) + a)
  235.  
  236. y = ((3v+1)*cos(b) - (4)*sin(b))/((0)*sin(c) + ((3v+1)*sin(b) + (4)*cos(b))*cos(c) + a)
  237.  
  238. 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)
  239.  
  240. ---
  241. ------
  242.  
  243. 4. { -2*sqrt(3) , -2 , sqrt(3)(v-1)u , 3v+1 }
  244.  
  245. x = (-2*sqrt(3))/((sqrt(3)(v-1)u)*sin(c) + ((-2)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
  246.  
  247. 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)
  248.  
  249. 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)
  250.  
  251. ---
  252.  
  253. 5. { 2*sqrt(3) , -2 , sqrt(3)(v-1)u , 3v+1 }
  254.  
  255. x = (2*sqrt(3))/((sqrt(3)(v-1)u)*sin(c) + ((-2)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
  256.  
  257. 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)
  258.  
  259. 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)
  260.  
  261. ---
  262.  
  263. 6. { 0 , 4 , sqrt(3)(v-1)u , 3v+1 }
  264.  
  265. x = (0)/((sqrt(3)(v-1)u)*sin(c) + ((4)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
  266.  
  267. 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)
  268.  
  269. 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)
  270.  
  271. ---
  272.  
  273.  
  274.  
  275.  
  276.  
  277.  
  278.  
  279.  
  280. • 9 squares: 1D elem A x 1D elem B
  281.  
  282. 7. { -sqrt(3)(u-1) , 3u+1 , -sqrt(3)(v-1) , 3v+1 }
  283.  
  284. x = (-sqrt(3)(u-1))/((-sqrt(3)(v-1))*sin(c) + ((3u+1)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
  285.  
  286. 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)
  287.  
  288. 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)
  289.  
  290. ---
  291.  
  292. 8. { -sqrt(3)(u-1) , 3u+1 , sqrt(3)(v-1) , 3v+1 }
  293.  
  294. x = (-sqrt(3)(u-1))/((sqrt(3)(v-1))*sin(c) + ((3u+1)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
  295.  
  296. 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)
  297.  
  298. 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)
  299.  
  300. ---
  301.  
  302. 9. { -sqrt(3)(u-1) , 3u+1 , 2*sqrt(3)v , -2 }
  303.  
  304. x = (-sqrt(3)(u-1))/((2*sqrt(3)v)*sin(c) + ((3u+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
  305.  
  306. y = ((3u+1)*cos(b) - (-2)*sin(b))/((2*sqrt(3)v)*sin(c) + ((3u+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
  307.  
  308. 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)
  309.  
  310. ---
  311. ------
  312.  
  313. 10. { sqrt(3)(u-1) , 3u+1 , -sqrt(3)(v-1) , 3v+1 }
  314.  
  315. x = (sqrt(3)(u-1))/((-sqrt(3)(v-1))*sin(c) + ((3u+1)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
  316.  
  317. 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)
  318.  
  319. 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)
  320.  
  321. ---
  322.  
  323. 11. { sqrt(3)(u-1) , 3u+1 , sqrt(3)(v-1) , 3v+1 }
  324.  
  325. x = (sqrt(3)(u-1))/((sqrt(3)(v-1))*sin(c) + ((3u+1)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
  326.  
  327. 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)
  328.  
  329. 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)
  330.  
  331. ---
  332.  
  333. 12. { sqrt(3)(u-1) , 3u+1 , 2*sqrt(3)v , -2 }
  334.  
  335. x = (sqrt(3)(u-1))/((2*sqrt(3)v)*sin(c) + ((3u+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
  336.  
  337. y = ((3u+1)*cos(b) - (-2)*sin(b))/((2*sqrt(3)v)*sin(c) + ((3u+1)*sin(b) + (-2)*cos(b))*cos(c) + a)
  338.  
  339. 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)
  340.  
  341. ---
  342. ------
  343.  
  344. 13. { 2*sqrt(3)u , -2 , -sqrt(3)(v-1) , 3v+1 }
  345.  
  346. x = (2*sqrt(3)u)/((-sqrt(3)(v-1))*sin(c) + ((-2)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
  347.  
  348. y = ((-2)*cos(b) - (3v+1)*sin(b))/((-sqrt(3)(v-1))*sin(c) + ((-2)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
  349.  
  350. 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)
  351.  
  352. ---
  353.  
  354. 14. { 2*sqrt(3)u , -2 , sqrt(3)(v-1) , 3v+1 }
  355.  
  356. x = (2*sqrt(3)u)/((sqrt(3)(v-1))*sin(c) + ((-2)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
  357.  
  358. y = ((-2)*cos(b) - (3v+1)*sin(b))/((sqrt(3)(v-1))*sin(c) + ((-2)*sin(b) + (3v+1)*cos(b))*cos(c) + a)
  359.  
  360. 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)
  361.  
  362. ---
  363.  
  364. 15. { 2*sqrt(3)u , -2 , 2*sqrt(3)v , -2 }
  365.  
  366. x = (2*sqrt(3)u)/((2*sqrt(3)v)*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
  367.  
  368. y = ((-2)*cos(b) - (-2)*sin(b))/((2*sqrt(3)v)*sin(c) + ((-2)*sin(b) + (-2)*cos(b))*cos(c) + a)
  369.  
  370. 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)
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