MathPlotAI Logo MathPlotAI

Vector Representation of a Sphere

Vector Representation of a Sphere
Loading the math board and drawing, please wait…
2013-04-18 21:32
VectorCoordinate GeometrySolid Geometry
In collections surface 3D /3D-Oberflächen 1
This example demonstrates the vector representation of the spherical equation in 3D space. It involves core mathematical concepts such as vectors, 3D space, spherical equations, midpoints, and sequences. The construction utilizes GeoGebra commands/objects like Line, Segment, Polygon, PolyLine, Point, Vector, Midpoint, and Sequence. It is suitable for high school and above students to learn and understand vectors and 3D geometry.

GeoGebra Commands & Parameters

  1. Line(D6, D7) → i_1
  2. Line(A, i_1) → j_1
  3. Segment(D6, D7) → d_1
  4. Segment(D7, D4) → d_2
  5. Segment(D4, D5) → d_3
  6. Segment(D5, D6) → d_4
  7. Segment(D2, D3) → d_5
  8. Segment(D3, D0) → d_6
  9. Segment(D0, D1) → d_7
  10. Segment(D1, D2) → d_8
  11. Segment(D6, D2) → d_9
  12. Segment(D5, D1) → d_{12}
  13. Segment(D4, D0) → d_{11}
  14. Segment(D7, D3) → d_{10}
  15. Polygon(A - e / 5 * u - e / 5 * v, A + e / 5 * u - e / 5 * v, A + e / 5 * u + e / 5 * v, A - e / 5 * u + e / 5 * v) → Vieleck1, f, g, h, i
  16. Segment(P_x, P_{xy}) → s2_x
  17. Segment(P_y, P_{xy}) → s2_y
  18. Segment(P, P_{xy}) → s_p
  19. Segment(P_{yz}, P) → s3_x
  20. Segment(P_{xz}, P) → s3_y
  21. Segment(P_{xz}, P_y) → s2_z
  22. Segment(P_{yz}, P_x) → s3_z
  23. Segment(P_{yz}, P_z) → c
  24. Segment(P_{xz}, P_z) → d
  25. PolyLine(D0, P_y, P_{xy}, P) → szy
  26. Point(szy) → Q_2
  27. Vector(D0, Q_2) → vqy
  28. Vector(A, P) → vp
  29. Vector(A, P_{xy}) → e_1
  30. Segment(A, P_y) → s_x
  31. Segment(A, P_x) → s_y
  32. Segment(A, P_z) → s_z
  33. Polygon(A - e / 5 * u - e / 5 * w, A + e / 5 * u - e / 5 * w, A + e / 5 * u + e / 5 * w, A - e / 5 * u + e / 5 * w) → Vieleck2, j, k, l, m
  34. Polygon(A - e / 5 * w - e / 5 * v, A + e / 5 * w - e / 5 * v, A + e / 5 * w + e / 5 * v, A - e / 5 * w + e / 5 * v) → Vieleck3, n, p, q, s
  35. Midpoint(A, P) → Pos_1
  36. Vector(A, M) → v_M
  37. Vector(M, Q) → v_{MQ}
  38. Vector(A, Q) → v_Q
  39. Midpoint(M, Q) → Pos_2
  40. Midpoint(A, Q) → Pos_3
  41. Midpoint(A, M) → Pos_4
  42. Sequence(Segment[A - e / 5 * u - e / 5 * v + i / 5 * v, A + e / 5 * u - e / 5 * v + i / 5 * v], i, 0, 2 * e) → Liste1
  43. Sequence(Segment[A - e / 5 * v - e / 5 * u + i / 5 * u, A + e / 5 * v - e / 5 * u + i / 5 * u], i, 0, 2 * e) → Liste2
  44. Sequence(Segment[A - e / 5 * w - e / 5 * v + i / 5 * v, A + e / 5 * w - e / 5 * v + i / 5 * v], i, 0, 2 * e) → Liste3
  45. Sequence(Segment[A - e / 5 * v - e / 5 * w + i / 5 * w, A + e / 5 * v - e / 5 * w + i / 5 * w], i, 0, 2 * e) → Liste4
  46. Sequence(Segment[A + e / 5 * u - e / 5 * w + i / 5 * w, A - e / 5 * u - e / 5 * w + i / 5 * w], i, 0, 2 * e) → Liste5
  47. Sequence(Segment[A - e / 5 * w - e / 5 * u + i / 5 * u, A + e / 5 * w - e / 5 * u + i / 5 * u], i, 0, 2 * e) → Liste6
  48. Segment(B_1, B_6) → bb_1
  49. Segment(B_8, B_9) → bb_2
  50. Segment(B_{10}, B_{11}) → bb_3
  51. Segment(B_4, B_7) → bb_4
  52. Polygon(B, B_2, B_{13}, B_{12}) → Vieleck4, b_1, b_2, b_{13}, b_{12}
  53. Vector(A - e / 5 * v, Pos_y) → v_2
  54. Vector(A - e / 5 * u, Pos_x) → u_2
  55. Vector(A - e / 5 * w, Pos_z) → w_2

AI Drawing Prompt Reference

Visualize the vector representation of a spherical equation in 3D space, highlighting vectors, 3D geometry, and spherical equations. Start by constructing the basic geometric shape of a sphere using multiple line segments connecting key points on its surface. Then, draw vectors from the center of the sphere to various points on the surface, illustrating the relationship between vectors and the spherical equation. Finally, use midpoints and vector operations to visualize the vector field on the sphere, demonstrating the application of vectors in 3D geometry.
The prompt above was compiled from the original drawing commands and is for reference only.