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圆锥曲线性质与几何作图演示

圆锥曲线性质与几何作图演示
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2016-03-10 10:14
解析几何立体几何几何作图轨迹
本范例演示圆锥与平面相交所生成的圆锥曲线及其几何性质，重点展示椭圆、抛物线与双曲线的焦点、准线、对称轴及截面形态。核心数学概念涵盖圆锥曲线、解析几何、立体几何（圆锥、球面、正交平面）、角平分线、刚体变换（旋转与平移）以及三角函数辅助定位。作图要点基于构造摘要，使用了 GeoGebra 命令 Circle、Intersect、Cone、OrthogonalPlane、IntersectPath、AngularBisector、Line、OrthogonalLine、Sphere、Point、Segment、Rotate、Translate，并涉及圆锥、切平面、球体、焦点（focus1、focus2）、圆心（center1、center2）、交点、线段及旋转平移后的图元等对象。该范例适合高中阶段解析几何与立体几何课堂教学，也可作为大学微积分前圆锥曲线的预习与自主探究资源，支持教师演示与学生动手构建。

GeoGebra 命令及参数

  1. Circle((0, 0, 3), 3, Vector[(0, 0, 1)]) → c
  2. Circle((0, 0, -3), 3, Vector[(0, 0, 1)]) → d
  3. Intersect(zAxis, xAxis) → A
  4. Cone(c, -3) → a, e, b
  5. Cone(d, 3) → f, g, h
  6. OrthogonalPlane((s, 0, 0), Vector[(sin((t * π)), 0, (-cos((t * π))))]) → cutplane
  7. IntersectPath(cutplane, l) → m
  8. Intersect(m, j) → B
  9. IntersectPath(cutplane, n) → q
  10. Intersect(q, j) → C
  11. IntersectPath(cutplane, j) → r
  12. IntersectPath(j, l) → a_1
  13. IntersectPath(j, n) → b_1
  14. AngularBisector(r, a_1) → cc, d_1
  15. AngularBisector(r, b_1) → ee, f_1
  16. Line((0, 0, 0), Vector[(0, 0, 1)]) → yy
  17. Intersect(cc, yy) → center1
  18. Intersect(ee, yy) → center2
  19. OrthogonalLine(center1, r, space) → c_1
  20. Intersect(c_1, r) → focus1
  21. OrthogonalLine(center2, r, space) → e_1
  22. Intersect(r, e_1) → focus2
  23. Sphere(center1, focus1) → sphere1
  24. Sphere(center2, focus2) → sphere2
  25. OrthogonalLine(center1, a_1, space) → g_1
  26. OrthogonalLine(center2, a_1, space) → h_1
  27. Intersect(h_1, a_1) → D
  28. Intersect(a_1, g_1) → E
  29. OrthogonalPlane(D, Vector[(0, 0, 1)]) → plane2
  30. OrthogonalPlane(E, Vector[(0, 0, 1)]) → plane1
  31. IntersectPath(cutplane, plane1) → directrix1
  32. IntersectPath(cutplane, plane2) → directrix2
  33. IntersectPath(plane1, sphere1) → circle1
  34. IntersectPath(plane2, sphere2) → circle2
  35. IntersectPath(cutplane, b) → k
  36. IntersectPath(cutplane, h) → p
  37. Point(k) → P
  38. Segment(P, focus1) → i
  39. Segment(P, focus2) → i_1
  40. Line(P, (0, 0, 0)) → j_1
  41. Intersect(j_1, circle2) → F,
  42. Segment(P, F) → k_1
  43. Intersect(j_1, circle1) → G,
  44. IntersectPath(o, cutplane) → l_1
  45. Rotate(k, (t * π), l_1) → k'
  46. Translate(k', Vector[((-s), 0, 0)]) → k''
  47. Segment(G, P) → m_1