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三维曲面绘制与曲线层积

三维曲面绘制与曲线层积
几何画板及绘图内容加载中，请稍等……
2013-11-05 19:47
函数立体几何微积分函数图像
所属专题 surface 3D /3D-Oberflächen 1 surface 3D /3D-Oberflächen 1
①本范例演示了三维空间中 z = f(x,y) 函数的曲面绘制及曲线层积动画效果；②涉及的核心数学概念包括函数映射、三维坐标系统、参数化曲面、曲线层积等；③作图要点提炼自构造摘要，使用了 If, Corner, Mod, Sequence, Element, Polygon, Vector, Segment, Ray, Textfield, Point, Locus, LaTeX 等GeoGebra命令与对象；④适合高中及以上数学学习者，用于理解函数图像在三维空间的表现及动态变化过程。

GeoGebra 命令及参数

  1. If(x ≟ 0, 1, sin(x) / x) → sinc
  2. Corner(1) → Eckpunkt1
  3. Corner(2) → Eckpunkt2
  4. Corner(3) → Eckpunkt3
  5. If(c, α, x(C - Eckpunkt1) / dEx * c_{Move}) → α1
  6. If(c, β, y(C - Eckpunkt2) / dEy * c_{Move}) → β1
  7. If(jg, t, x_{min} + x_{max} - t) → tj
  8. If(a ∧ bP, tj) → tt2
  9. If(a ∧ (¬bP), tj) → tt1
  10. Mod(β1, 6.283) → β2
  11. Mod(α1, 6.283) → α2
  12. Sequence(a11 * Element[zWX1, i] + a12 * Element[zWY1, i] + a13 * Element[zWZ1, i], i, 1, 4) → zWX1'
  13. Element(zWW_0, 3) → zW_2
  14. Element(zWW_0, 4) → zW_1
  15. Element(zWW_0, 2) → zW_3
  16. Polygon(Eckpunkt1, Eck11, Eck12, (x(Eckpunkt1), y(Eck12))) → Vieleck2, eckpunkt1, eck11, eck12, a_1
  17. Sequence(Segment[Element[zl10, i], Element[zl10, i + 1]], i, 1, 7) → BoxStr
  18. Vector(Element[zl10, 4], Element[zl10, 1]) → zVectx
  19. Vector(Element[zl10, 4], Element[zl10, 3]) → zVecty
  20. Vector(Element[zl10, 4], Element[zl10, 5]) → zVectz
  21. Polygon(Eck11, Eckpunkt2, Eckpunkt3, Eck12) → Vieleck1, eck11_1, eckpunkt2, eckpunkt3, eck12_1
  22. Element(zTT, zW_1) → zY1
  23. Element(zTT, zW_2) → zY2
  24. Element(zTT, zW_3) → zY3
  25. Polygon(zTT) → zTTPoly, , , ,
  26. Element(zl10, zW_1) → zlN11
  27. Element(zl10, 9 - zW_1) → zlN111
  28. Element(zl10, zW_2) → zlN22
  29. Element(zl10, 9 - zW_2) → zlN222
  30. Element(zl10, zW_3) → zlN33
  31. Element(zWW_0, 1) → zW_4
  32. Element(zl10, zW_4) → zlN44
  33. Polygon({zlN11, zlN22, zlN222, zlN111}) → Vgrund01, , , ,
  34. Polygon(Sequence[Element[zl10, i], i, 1, 4]) → Vgrund000, , , ,
  35. Element(zl10, 7) → P_R
  36. Element(zl10, 8) → P_L
  37. Polygon({Element[zl10, 1], Element[zl10, 2], Element[zl10, 7], Element[zl10, 8]}) → Vvrd0, , , ,
  38. Segment(P_R, P_L) → StrRL
  39. Polygon(Sequence[Element[zl10, i], i, 9, 12]) → zlPoly, n, r, s, c_1
  40. Ray(B, Element[zl10, 13]) → Strx
  41. Ray(B, Element[zl10, 14]) → Stry
  42. Ray(B, Element[zl10, 15]) → Strz
  43. Textfield(x_{min}) → inputBox1
  44. Textfield(x_{max}) → inputBox4
  45. Textfield(v_{Animation}) → inputBox8
  46. Textfield(Step) → inputBox7
  47. Textfield(N) → inputBox9
  48. If(b, t, t0) → Rot
  49. If(b, exp(t), sin(π * t0)) → Grün
  50. If(b, ℯ^(-t), 1 - (t0 - 0.5)^(2)) → Blau
  51. Textfield(t) → inputBox10
  52. Textfield(Th) → inputBox3
  53. Textfield(y_{min}) → inputBox5
  54. Textfield(z_{min}) → inputBox6
  55. Textfield(z_{max}) → inputBox11
  56. Textfield(y_{max}) → inputBox12
  57. Segment((x_{min}, 0), (x_{max}, 0)) → Sx
  58. Segment((0, y_{min}), (0, y_{max})) → Sy
  59. Point(Sx) → Ax
  60. Point(Sy) → Ay
  61. Locus(A', y0) → locu
  62. Locus(A, y0) → locy
  63. Locus(A', x0) → locv
  64. Textfield(f) → inputBox2
  65. LaTeX(f, true, true) → Text3_1
  66. LaTeX(f, true, true) → Text12_1
  67. Locus(M5_{1}, Ay) → M6_{1}
  68. Locus(M5_{2}, Ay) → M6_{2}
  69. Locus(M5_{3}, Ay) → M6_{3}
  70. Locus(M5_{4}, Ay) → M6_{4}
  71. Locus(M5_{5}, Ay) → M6_{5}
  72. Locus(M5_{6}, Ay) → M6_{6}
  73. Locus(M5_{7}, Ay) → M6_{7}
  74. Locus(M5_{8}, Ay) → M6_{8}
  75. Locus(M5_{9}, Ay) → M6_{9}
  76. Locus(M5_{10}, Ay) → M6_{10}
  77. Locus(M5_{11}, Ay) → M6_{11}
  78. Locus(M5_{12}, Ay) → M6_{12}
  79. Locus(M5_{13}, Ay) → M6_{13}
  80. Locus(M5_{14}, Ay) → M6_{14}
  81. Locus(M5_{15}, Ay) → M6_{15}
  82. Locus(M5_{16}, Ay) → M6_{16}
  83. Locus(M5_{17}, Ay) → M6_{17}
  84. Locus(M5_{18}, Ay) → M6_{18}
  85. Locus(M5_{19}, Ay) → M6_{19}
  86. Locus(M5_{20}, Ay) → M6_{20}
  87. Locus(M5_{21}, Ay) → M6_{21}
  88. Locus(M5_{22}, Ay) → M6_{22}
  89. Locus(M5_{23}, Ay) → M6_{23}
  90. Locus(M5_{24}, Ay) → M6_{24}
  91. Locus(M5_{25}, Ay) → M6_{25}
  92. Locus(M5_{26}, Ay) → M6_{26}
  93. Locus(M5_{27}, Ay) → M6_{27}
  94. Locus(M5_{28}, Ay) → M6_{28}
  95. Locus(M5_{29}, Ay) → M6_{29}
  96. Locus(M5_{30}, Ay) → M6_{30}
  97. Locus(M5_{31}, Ay) → M6_{31}
  98. Locus(M5_{32}, Ay) → M6_{32}
  99. Locus(M5_{33}, Ay) → M6_{33}
  100. Locus(M5_{34}, Ay) → M6_{34}
  101. Locus(M5_{35}, Ay) → M6_{35}
  102. Locus(M5_{36}, Ay) → M6_{36}
  103. Locus(M5_{37}, Ay) → M6_{37}
  104. Locus(M5_{38}, Ay) → M6_{38}
  105. Locus(M5_{39}, Ay) → M6_{39}
  106. Locus(M5_{40}, Ay) → M6_{40}
  107. Locus(M5_{41}, Ay) → M6_{41}
  108. Locus(M5_{42}, Ay) → M6_{42}
  109. Locus(M5_{43}, Ay) → M6_{43}
  110. Locus(M5_{44}, Ay) → M6_{44}
  111. Locus(M5_{45}, Ay) → M6_{45}
  112. Locus(M5_{46}, Ay) → M6_{46}
  113. Locus(M5_{47}, Ay) → M6_{47}
  114. Locus(M5_{48}, Ay) → M6_{48}
  115. Locus(M5_{49}, Ay) → M6_{49}
  116. Locus(M5_{50}, Ay) → M6_{50}
  117. Locus(M5_{51}, Ay) → M6_{51}
  118. Locus(M5_{52}, Ay) → M6_{52}
  119. Locus(M5_{53}, Ay) → M6_{53}
  120. Locus(M5_{54}, Ay) → M6_{54}
  121. Locus(M5_{55}, Ay) → M6_{55}
  122. Locus(M5_{56}, Ay) → M6_{56}
  123. Locus(M5_{57}, Ay) → M6_{57}
  124. Locus(M5_{58}, Ay) → M6_{58}
  125. Locus(M5_{59}, Ay) → M6_{59}
  126. Locus(M5_{60}, Ay) → M6_{60}
  127. Locus(M5_{61}, Ay) → M6_{61}
  128. Locus(M5_{62}, Ay) → M6_{62}
  129. Locus(M5_{63}, Ay) → M6_{63}
  130. Locus(M5_{64}, Ay) → M6_{64}
  131. Locus(M5_{65}, Ay) → M6_{65}
  132. Locus(M5_{66}, Ay) → M6_{66}
  133. Locus(M5_{67}, Ay) → M6_{67}
  134. Locus(M5_{68}, Ay) → M6_{68}
  135. Locus(M5_{69}, Ay) → M6_{69}
  136. Locus(M5_{70}, Ay) → M6_{70}
  137. Locus(M5_{71}, Ay) → M6_{71}
  138. Locus(M5_{72}, Ay) → M6_{72}
  139. Locus(M5_{73}, Ay) → M6_{73}
  140. Locus(M5_{74}, Ay) → M6_{74}
  141. Locus(M5_{75}, Ay) → M6_{75}
  142. Locus(M5_{76}, Ay) → M6_{76}
  143. Locus(M5_{77}, Ay) → M6_{77}
  144. Locus(M5_{78}, Ay) → M6_{78}
  145. Locus(M5_{79}, Ay) → M6_{79}
  146. Locus(M5_{80}, Ay) → M6_{80}
  147. Locus(M5_{81}, Ay) → M6_{81}
  148. Locus(M5_{82}, Ay) → M6_{82}
  149. Locus(M5_{83}, Ay) → M6_{83}
  150. Locus(M5_{84}, Ay) → M6_{84}
  151. Locus(M5_{85}, Ay) → M6_{85}
  152. Locus(M5_{86}, Ay) → M6_{86}
  153. Locus(M5_{87}, Ay) → M6_{87}
  154. Locus(M5_{88}, Ay) → M6_{88}
  155. Locus(M5_{89}, Ay) → M6_{89}
  156. Locus(M5_{90}, Ay) → M6_{90}
  157. Locus(M5_{91}, Ay) → M6_{91}
  158. Locus(M5_{92}, Ay) → M6_{92}
  159. Locus(M5_{93}, Ay) → M6_{93}
  160. Locus(M5_{94}, Ay) → M6_{94}
  161. Locus(M5_{95}, Ay) → M6_{95}
  162. Locus(M5_{96}, Ay) → M6_{96}
  163. Locus(M5_{97}, Ay) → M6_{97}
  164. Locus(M5_{98}, Ay) → M6_{98}
  165. Locus(M5_{99}, Ay) → M6_{99}
  166. Locus(M5_{100}, Ay) → M6_{100}
  167. Locus(M5_{101}, Ay) → M6_{101}
  168. Locus(M5_{102}, Ay) → M6_{102}
  169. Locus(M5_{103}, Ay) → M6_{103}
  170. Locus(M5_{104}, Ay) → M6_{104}
  171. Locus(M5_{105}, Ay) → M6_{105}
  172. Locus(M5_{106}, Ay) → M6_{106}
  173. Locus(M5_{107}, Ay) → M6_{107}
  174. Locus(M5_{108}, Ay) → M6_{108}
  175. Locus(M5_{109}, Ay) → M6_{109}
  176. Locus(M5_{110}, Ay) → M6_{110}
  177. Locus(M5_{111}, Ay) → M6_{111}
  178. Locus(M5_{112}, Ay) → M6_{112}
  179. Locus(M5_{113}, Ay) → M6_{113}
  180. Locus(M5_{114}, Ay) → M6_{114}
  181. Locus(M5_{115}, Ay) → M6_{115}
  182. Locus(M5_{116}, Ay) → M6_{116}
  183. Locus(M5_{117}, Ay) → M6_{117}
  184. Locus(M5_{118}, Ay) → M6_{118}
  185. Locus(M5_{119}, Ay) → M6_{119}
  186. Locus(M5_{120}, Ay) → M6_{120}
  187. Locus(M5_{121}, Ay) → M6_{121}
  188. Locus(M5_{122}, Ay) → M6_{122}
  189. Locus(M5_{123}, Ay) → M6_{123}
  190. Locus(M5_{124}, Ay) → M6_{124}
  191. Locus(M5_{125}, Ay) → M6_{125}
  192. Locus(M5_{126}, Ay) → M6_{126}
  193. Locus(M5_{127}, Ay) → M6_{127}
  194. Locus(M5_{128}, Ay) → M6_{128}
  195. Locus(M5_{129}, Ay) → M6_{129}
  196. Locus(M5_{130}, Ay) → M6_{130}
  197. Locus(M5_{131}, Ay) → M6_{131}
  198. Locus(M5_{132}, Ay) → M6_{132}
  199. Locus(M5_{133}, Ay) → M6_{133}
  200. Locus(M5_{134}, Ay) → M6_{134}
  201. Locus(M5_{135}, Ay) → M6_{135}
  202. Locus(M5_{136}, Ay) → M6_{136}
  203. Locus(M5_{137}, Ay) → M6_{137}
  204. Locus(M5_{138}, Ay) → M6_{138}
  205. Locus(M5_{139}, Ay) → M6_{139}
  206. Locus(M5_{140}, Ay) → M6_{140}
  207. Locus(M5_{141}, Ay) → M6_{141}
  208. Locus(M5_{142}, Ay) → M6_{142}
  209. Locus(M5_{143}, Ay) → M6_{143}
  210. Locus(M5_{144}, Ay) → M6_{144}
  211. Locus(M5_{145}, Ay) → M6_{145}
  212. Locus(M5_{146}, Ay) → M6_{146}
  213. Locus(M5_{147}, Ay) → M6_{147}
  214. Locus(M5_{148}, Ay) → M6_{148}
  215. Locus(M5_{149}, Ay) → M6_{149}
  216. Locus(M5_{150}, Ay) → M6_{150}
  217. Locus(M5_{151}, Ay) → M6_{151}
  218. Locus(M5_{152}, Ay) → M6_{152}
  219. Locus(M5_{153}, Ay) → M6_{153}
  220. Locus(M5_{154}, Ay) → M6_{154}
  221. Locus(M5_{155}, Ay) → M6_{155}
  222. Locus(M5_{156}, Ay) → M6_{156}
  223. Locus(M5_{157}, Ay) → M6_{157}
  224. Locus(M5_{158}, Ay) → M6_{158}
  225. Locus(M5_{159}, Ay) → M6_{159}
  226. Locus(M5_{160}, Ay) → M6_{160}
  227. Locus(M5_{161}, Ay) → M6_{161}
  228. Locus(M5_{162}, Ay) → M6_{162}
  229. Locus(M5_{163}, Ay) → M6_{163}
  230. Locus(M5_{164}, Ay) → M6_{164}
  231. Locus(M5_{165}, Ay) → M6_{165}
  232. Locus(M5_{166}, Ay) → M6_{166}
  233. Locus(M5_{167}, Ay) → M6_{167}
  234. Locus(M5_{168}, Ay) → M6_{168}
  235. Locus(M5_{169}, Ay) → M6_{169}
  236. Locus(M5_{170}, Ay) → M6_{170}
  237. Locus(M5_{171}, Ay) → M6_{171}
  238. Locus(M5_{172}, Ay) → M6_{172}
  239. Locus(M5_{173}, Ay) → M6_{173}
  240. Locus(M5_{174}, Ay) → M6_{174}
  241. Locus(M5_{175}, Ay) → M6_{175}
  242. Locus(M5_{176}, Ay) → M6_{176}
  243. Locus(M5_{177}, Ay) → M6_{177}
  244. Locus(M5_{178}, Ay) → M6_{178}
  245. Locus(M5_{179}, Ay) → M6_{179}
  246. Locus(M5_{180}, Ay) → M6_{180}
  247. Locus(M5_{181}, Ay) → M6_{181}
  248. Locus(M5_{182}, Ay) → M6_{182}
  249. Locus(M5_{183}, Ay) → M6_{183}
  250. Locus(M5_{184}, Ay) → M6_{184}
  251. Locus(M5_{185}, Ay) → M6_{185}
  252. Locus(M5_{186}, Ay) → M6_{186}
  253. Locus(M5_{187}, Ay) → M6_{187}
  254. Locus(M5_{188}, Ay) → M6_{188}
  255. Locus(M5_{189}, Ay) → M6_{189}
  256. Locus(M5_{190}, Ay) → M6_{190}
  257. Locus(M5_{191}, Ay) → M6_{191}
  258. Locus(M5_{192}, Ay) → M6_{192}
  259. Locus(M5_{193}, Ay) → M6_{193}
  260. Locus(M5_{194}, Ay) → M6_{194}
  261. Locus(M5_{195}, Ay) → M6_{195}
  262. Locus(M5_{196}, Ay) → M6_{196}
  263. Locus(M5_{197}, Ay) → M6_{197}
  264. Locus(M5_{198}, Ay) → M6_{198}
  265. Locus(M5_{199}, Ay) → M6_{199}
  266. Locus(M5_{200}, Ay) → M6_{200}
  267. Locus(M5_{201}, Ay) → M6_{201}

AI 绘图提示语参考

绘制一个三维曲面图，展示函数 z = f(x,y) 的图像，并动态显示其等高线。图中包含一个正方形底面和四个侧面，构成一个立方体框架。正方形底面中心连接三条射线，分别代表 x、y、z 轴方向。等高线围绕曲面动态变化，展示函数的梯度。曲面下方标注了 x_min 和 x_max 的范围，以及动画速度和步长参数。
以上提示语为根据原始绘图命令整理而来，仅供参考。