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Relative Arrangement of Two Parametric Surfaces

Relative Arrangement of Two Parametric Surfaces
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2013-11-11 20:18
VectorTrigonometrySolid GeometryParametric EquationGeometric Construction
This example demonstrates the relative arrangements and trace plots of two parametric surfaces, involving core concepts like parametric equations, trigonometric functions, vector operations, and surface geometry. It utilizes GeoGebra commands such as If, Join, Corner, Sequence, Zip, Max, Min, Polygon, Werkzeug2, Vector, Textfield, OrthogonalLine, and Element for dynamic surface construction and locus visualization. Suitable for high school and advanced mathematics education in understanding parametric surfaces, spatial geometry, and dynamic geometric modeling.

GeoGebra Commands & Parameters

  1. If(f0, 10, N1) → N01
  2. Join(Sequence[Sequence[(u1_{min} + i * dx1, v1_{min} + j * dy1), i, 0, N01 - 1], j, 0, N01 - 1]) → uva
  3. If(abs(x) < 0.2, 0.8, x) → h
  4. Corner(1) → Eckpunkt1
  5. Corner(2) → Eckpunkt2
  6. If(d, α1, α1_1) → α01
  7. Corner(3) → Eckpunkt3
  8. If(d, β1, β1_1) → β01
  9. Sequence(If[i ≟ ceil(i / N01) * N01, 0, i], i, 1, läX1') → ll
  10. Sequence(a31a * Element[X1a, i] + a32a * Element[Y1a, i] + a33a * Element[Z1aa, i], i, 1, läX1) → Z1aaaa'
  11. Sequence(a11a * Element[X1a, i] + a12a * Element[Y1a, i] + a13a * Element[Z1aa, i], i, 1, läX1) → X1aaa'
  12. Sequence(a21a * Element[X1a, i] + a22a * Element[Y1a, i] + a23a * Element[Z1aa, i], i, 1, läX1) → Y1aaa'
  13. Sequence(a31a * Element[X1a, i] + a32a * Element[Y1a, i] + a33a * Element[Z1aa, i], i, 1, läX1) → Z1aaa'
  14. If(d, β2, β2_1) → β02
  15. If(d, α2, α2_1) → α02
  16. Sequence(Polygon[Element[yPunktena, Element[ll, ii]], Element[yPunktena, Element[ll, ii] + N01], Element[yPunktena, Element[ll, ii] + N01 + 1], Element[yPunktena, Element[ll, ii] + 1]], ii, 1, läX1') → ylPolya
  17. Join(Sequence[Sequence[(u2_{min} + i * dx2, v2_{min} + j * dy2), i, 0, N01 - 1], j, 0, N01 - 1]) → uvb
  18. Zip(a31b * x1 + a32b * y1 + a33b * z1, x1, X2b, y1, Y2b, z1, Z2b) → Z2b'
  19. Zip(a21b * x1 + a22b * y1 + a23b * z1, x1, X2b, y1, Y2b, z1, Z2b) → Y2b'
  20. Zip(a11b * x1 + a12b * y1 + a13b * z1, x1, X2b, y1, Y2b, z1, Z2b) → X2b'
  21. Zip(Polygon[Element[yPunktenb, ii], Element[yPunktenb, ii + N01], Element[yPunktenb, ii + N01 + 1], Element[yPunktenb, ii + 1]], ii, ll) → ylPolyb
  22. Max(RemoveUndefined[Z1aa]) → z1max
  23. Min(RemoveUndefined[Z1aa]) → z1min
  24. Max(RemoveUndefined[Z2b]) → z2max
  25. Min(RemoveUndefined[Z2b]) → z2min
  26. Min(RemoveUndefined[Y2b]) → y2min
  27. Max(RemoveUndefined[Y2b]) → y2max
  28. Max(RemoveUndefined[Y1a]) → y1max
  29. Min(RemoveUndefined[Y1a]) → y1min
  30. Max(RemoveUndefined[X1a]) → x1max
  31. Min(RemoveUndefined[X1a]) → x1min
  32. Max(RemoveUndefined[X2b]) → x2max
  33. Min(RemoveUndefined[X2b]) → x2min
  34. Sequence(If[i ≤ läX1', Element[X1aaa', i], Element[X2b', i - läX1']], i, 1, läX'ab) → X'ab
  35. Sequence(If[i ≤ läX1', Element[ylPolya, i], Element[ylPolyb, i - läX1']], i, 1, läX'ab) → ypolyab
  36. Sequence(Element[ypolyab, Element[zupab, i]], i, 1, läX'ab) → ylPoly00_1
  37. Sequence(Element[ylPoly00_1, i], i, k, k + 179) → ylPoly00
  38. Polygon(Eckpunkt1, Eck11, Eck12, (x(Eckpunkt1), y(Eck12))) → Vieleck2, eckpunkt1, eck11_1, eck12_1, c
  39. Polygon(Eck11, Eckpunkt2, Eckpunkt3, Eck12) → Vieleck1, eck11, eckpunkt2, eckpunkt3, eck12
  40. Werkzeug2(Ba, {a_A, 0, 0}, {0, a_A, 0}, {0, 0, a_A}, α01, β01, γ1) → AchsenA_1, AchsenA_2
  41. Vector(Ba, Element[AchsenA_1, 1]) → Avx1
  42. Vector(Ba, Element[AchsenA_1, 2]) → Avy1
  43. Vector(Ba, Element[AchsenA_1, 3]) → Avz1
  44. Sequence(If[i ≤ läX1', (Element[X1a, i] - x1min) / (x1max - x1min) * b1, (Element[X2b, i - läX1'] - x2min) / (x2max - x2min) * b2], i, 1, läX'ab) → Xab
  45. Sequence(If[i ≤ läX1', (Element[Z1aa, i] - z1min) / (z1max - z1min) * r1, (Element[Z2b, i - läX1'] - z2min) / (z2max - z2min) * r2], i, 1, läX'ab) → Zab
  46. Werkzeug2(Bb, {a_A, 0, 0}, {0, a_A, 0}, {0, 0, a_A}, α02, β02, γ2) → AchsenB_1, AchsenB_2
  47. Vector(Bb, Element[AchsenB_1, 1]) → Avx2
  48. Vector(Bb, Element[AchsenB_1, 2]) → Avy2
  49. Vector(Bb, Element[AchsenB_1, 3]) → Avz2
  50. Polygon(Eck11, Eckpunkt2, Eckpunkt3, Eck12) → Vieleck10, eck11_2, eckpunkt2_1, eckpunkt3_1, eck12_2
  51. Textfield(u1_{min}) → Textfeld1
  52. Textfield(u1_{max}) → Textfeld2
  53. Textfield(v1_{min}) → Textfeld3
  54. Textfield(v1_{max}) → Textfeld4
  55. Textfield(v2_{min}) → Textfeld5
  56. Textfield(v2_{max}) → Textfeld6
  57. Textfield(u2_{min}) → Textfeld7
  58. Textfield(k) → inputBox1
  59. Textfield(v_k) → inputBox4
  60. Textfield(f1x) → inputBox5
  61. Textfield(f1y) → inputBox6
  62. Textfield(f2x) → inputBox8
  63. Textfield(f2y) → inputBox9
  64. Textfield(f2z) → inputBox10
  65. Textfield(N1) → inputBox2
  66. Textfield(f1z) → Textfeld9
  67. OrthogonalLine(LL0, xAxis) → LLy
  68. OrthogonalLine(LL0, yAxis) → LLx
  69. Textfield(u2_{min}) → Textfeld8
  70. If(f0, "⇒ N = 10", "⇒ N = N1 = " + N1 + "") → Text19
  71. Sequence(Element[Xab, Element[zupab, i]], i, 1, läX'ab) → Xabs
  72. Sequence(If[i ≤ läX1', (Element[Y1a, i] - y1min) / (y1max - y1min) * g1, (Element[Y2b, i - läX1'] - y2min) / (y2max - y2min) * g2], i, 1, läX'ab) → Yab
  73. Sequence(Element[Yab, Element[zupab, i]], i, 1, läX'ab) → Yabs
  74. Sequence(Element[Zab, Element[zupab, i]], i, 1, läX'ab) → Zabs
  75. Element(ylPoly00, 1) → M_{1}
  76. Element(ylPoly00, 2) → M_{2}
  77. Element(ylPoly00, 3) → M_{3}
  78. Element(ylPoly00, 4) → M_{4}
  79. Element(ylPoly00, 5) → M_{5}
  80. Element(ylPoly00, 6) → M_{6}
  81. Element(ylPoly00, 7) → M_{7}
  82. Element(ylPoly00, 8) → M_{8}
  83. Element(ylPoly00, 9) → M_{9}
  84. Element(ylPoly00, 10) → M_{10}
  85. Element(ylPoly00, 11) → M_{11}
  86. Element(ylPoly00, 12) → M_{12}
  87. Element(ylPoly00, 13) → M_{13}
  88. Element(ylPoly00, 14) → M_{14}
  89. Element(ylPoly00, 15) → M_{15}
  90. Element(ylPoly00, 16) → M_{16}
  91. Element(ylPoly00, 17) → M_{17}
  92. Element(ylPoly00, 18) → M_{18}
  93. Element(ylPoly00, 19) → M_{19}
  94. Element(ylPoly00, 20) → M_{20}
  95. Element(ylPoly00, 21) → M_{21}
  96. Element(ylPoly00, 22) → M_{22}
  97. Element(ylPoly00, 23) → M_{23}
  98. Element(ylPoly00, 24) → M_{24}
  99. Element(ylPoly00, 25) → M_{25}
  100. Element(ylPoly00, 26) → M_{26}
  101. Element(ylPoly00, 27) → M_{27}
  102. Element(ylPoly00, 28) → M_{28}
  103. Element(ylPoly00, 29) → M_{29}
  104. Element(ylPoly00, 30) → M_{30}
  105. Element(ylPoly00, 31) → M_{31}
  106. Element(ylPoly00, 32) → M_{32}
  107. Element(ylPoly00, 33) → M_{33}
  108. Element(ylPoly00, 34) → M_{34}
  109. Element(ylPoly00, 35) → M_{35}
  110. Element(ylPoly00, 36) → M_{36}
  111. Element(ylPoly00, 37) → M_{37}
  112. Element(ylPoly00, 38) → M_{38}
  113. Element(ylPoly00, 39) → M_{39}
  114. Element(ylPoly00, 40) → M_{40}
  115. Element(ylPoly00, 41) → M_{41}
  116. Element(ylPoly00, 42) → M_{42}
  117. Element(ylPoly00, 43) → M_{43}
  118. Element(ylPoly00, 44) → M_{44}
  119. Element(ylPoly00, 45) → M_{45}
  120. Element(ylPoly00, 46) → M_{46}
  121. Element(ylPoly00, 47) → M_{47}
  122. Element(ylPoly00, 48) → M_{48}
  123. Element(ylPoly00, 49) → M_{49}
  124. Element(ylPoly00, 50) → M_{50}
  125. Element(ylPoly00, 51) → M_{51}
  126. Element(ylPoly00, 52) → M_{52}
  127. Element(ylPoly00, 53) → M_{53}
  128. Element(ylPoly00, 54) → M_{54}
  129. Element(ylPoly00, 55) → M_{55}
  130. Element(ylPoly00, 56) → M_{56}
  131. Element(ylPoly00, 57) → M_{57}
  132. Element(ylPoly00, 58) → M_{58}
  133. Element(ylPoly00, 59) → M_{59}
  134. Element(ylPoly00, 60) → M_{60}
  135. Element(ylPoly00, 61) → M_{61}
  136. Element(ylPoly00, 62) → M_{62}
  137. Element(ylPoly00, 63) → M_{63}
  138. Element(ylPoly00, 64) → M_{64}
  139. Element(ylPoly00, 65) → M_{65}
  140. Element(ylPoly00, 66) → M_{66}
  141. Element(ylPoly00, 67) → M_{67}
  142. Element(ylPoly00, 68) → M_{68}
  143. Element(ylPoly00, 69) → M_{69}
  144. Element(ylPoly00, 70) → M_{70}
  145. Element(ylPoly00, 71) → M_{71}
  146. Element(ylPoly00, 72) → M_{72}
  147. Element(ylPoly00, 73) → M_{73}
  148. Element(ylPoly00, 74) → M_{74}
  149. Element(ylPoly00, 75) → M_{75}
  150. Element(ylPoly00, 76) → M_{76}
  151. Element(ylPoly00, 77) → M_{77}
  152. Element(ylPoly00, 78) → M_{78}
  153. Element(ylPoly00, 79) → M_{79}
  154. Element(ylPoly00, 80) → M_{80}
  155. Element(ylPoly00, 81) → M_{81}
  156. Element(ylPoly00, 82) → M_{82}
  157. Element(ylPoly00, 83) → M_{83}
  158. Element(ylPoly00, 84) → M_{84}
  159. Element(ylPoly00, 85) → M_{85}
  160. Element(ylPoly00, 86) → M_{86}
  161. Element(ylPoly00, 87) → M_{87}
  162. Element(ylPoly00, 88) → M_{88}
  163. Element(ylPoly00, 89) → M_{89}
  164. Element(ylPoly00, 90) → M_{90}
  165. Element(ylPoly00, 91) → M_{91}
  166. Element(ylPoly00, 92) → M_{92}
  167. Element(ylPoly00, 93) → M_{93}
  168. Element(ylPoly00, 94) → M_{94}
  169. Element(ylPoly00, 95) → M_{95}
  170. Element(ylPoly00, 96) → M_{96}
  171. Element(ylPoly00, 97) → M_{97}
  172. Element(ylPoly00, 98) → M_{98}
  173. Element(ylPoly00, 99) → M_{99}
  174. Element(ylPoly00, 100) → M_{100}
  175. Element(ylPoly00, 101) → M_{101}
  176. Element(ylPoly00, 102) → M_{102}
  177. Element(ylPoly00, 103) → M_{103}
  178. Element(ylPoly00, 104) → M_{104}
  179. Element(ylPoly00, 105) → M_{105}
  180. Element(ylPoly00, 106) → M_{106}
  181. Element(ylPoly00, 107) → M_{107}
  182. Element(ylPoly00, 108) → M_{108}
  183. Element(ylPoly00, 109) → M_{109}
  184. Element(ylPoly00, 110) → M_{110}
  185. Element(ylPoly00, 111) → M_{111}
  186. Element(ylPoly00, 112) → M_{112}
  187. Element(ylPoly00, 113) → M_{113}
  188. Element(ylPoly00, 114) → M_{114}
  189. Element(ylPoly00, 115) → M_{115}
  190. Element(ylPoly00, 116) → M_{116}
  191. Element(ylPoly00, 117) → M_{117}
  192. Element(ylPoly00, 118) → M_{118}
  193. Element(ylPoly00, 119) → M_{119}
  194. Element(ylPoly00, 120) → M_{120}
  195. Element(ylPoly00, 121) → M_{121}
  196. Element(ylPoly00, 122) → M_{122}
  197. Element(ylPoly00, 123) → M_{123}
  198. Element(ylPoly00, 124) → M_{124}
  199. Element(ylPoly00, 125) → M_{125}
  200. Element(ylPoly00, 126) → M_{126}
  201. Element(ylPoly00, 127) → M_{127}
  202. Element(ylPoly00, 128) → M_{128}
  203. Element(ylPoly00, 129) → M_{129}
  204. Element(ylPoly00, 130) → M_{130}
  205. Element(ylPoly00, 131) → M_{131}
  206. Element(ylPoly00, 132) → M_{132}
  207. Element(ylPoly00, 133) → M_{133}
  208. Element(ylPoly00, 134) → M_{134}
  209. Element(ylPoly00, 135) → M_{135}
  210. Element(ylPoly00, 136) → M_{136}
  211. Element(ylPoly00, 137) → M_{137}
  212. Element(ylPoly00, 138) → M_{138}
  213. Element(ylPoly00, 139) → M_{139}
  214. Element(ylPoly00, 140) → M_{140}
  215. Element(ylPoly00, 141) → M_{141}
  216. Element(ylPoly00, 142) → M_{142}
  217. Element(ylPoly00, 143) → M_{143}
  218. Element(ylPoly00, 144) → M_{144}
  219. Element(ylPoly00, 145) → M_{145}
  220. Element(ylPoly00, 146) → M_{146}
  221. Element(ylPoly00, 147) → M_{147}
  222. Element(ylPoly00, 148) → M_{148}
  223. Element(ylPoly00, 149) → M_{149}
  224. Element(ylPoly00, 150) → M_{150}
  225. Element(ylPoly00, 151) → M_{151}
  226. Element(ylPoly00, 152) → M_{152}
  227. Element(ylPoly00, 153) → M_{153}
  228. Element(ylPoly00, 154) → M_{154}
  229. Element(ylPoly00, 155) → M_{155}
  230. Element(ylPoly00, 156) → M_{156}
  231. Element(ylPoly00, 157) → M_{157}
  232. Element(ylPoly00, 158) → M_{158}
  233. Element(ylPoly00, 159) → M_{159}
  234. Element(ylPoly00, 160) → M_{160}
  235. Element(ylPoly00, 161) → M_{161}
  236. Element(ylPoly00, 162) → M_{162}
  237. Element(ylPoly00, 163) → M_{163}
  238. Element(ylPoly00, 164) → M_{164}
  239. Element(ylPoly00, 165) → M_{165}
  240. Element(ylPoly00, 166) → M_{166}
  241. Element(ylPoly00, 167) → M_{167}
  242. Element(ylPoly00, 168) → M_{168}
  243. Element(ylPoly00, 169) → M_{169}
  244. Element(ylPoly00, 170) → M_{170}
  245. Element(ylPoly00, 171) → M_{171}
  246. Element(ylPoly00, 172) → M_{172}
  247. Element(ylPoly00, 173) → M_{173}
  248. Element(ylPoly00, 174) → M_{174}
  249. Element(ylPoly00, 175) → M_{175}
  250. Element(ylPoly00, 176) → M_{176}
  251. Element(ylPoly00, 177) → M_{177}
  252. Element(ylPoly00, 178) → M_{178}
  253. Element(ylPoly00, 179) → M_{179}
  254. Element(ylPoly00, 180) → M_{180}
  255. If(f0, If[f, true, false], If[a, true, false]) → fz0