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双参数曲面相对排列演示

双参数曲面相对排列演示
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2013-11-11 20:18
向量三角函数立体几何参数方程几何作图
①本范例演示了两个参数曲面的相对排列关系及轨迹生成；②涉及核心数学概念包括参数方程、三角函数、向量运算及曲面几何；③构造中运用了If、Join、Corner、Sequence、Zip、Max、Min、Polygon、Werkzeug2、Vector、Textfield、OrthogonalLine、Element等GeoGebra命令与对象，实现了动态曲面构建与轨迹展示；④适合高中及以上数学教学，用于理解参数曲面、空间几何及动态几何建模。

GeoGebra 命令及参数

  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