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Intersection Area and Partial Volume in 2D

Intersection Area and Partial Volume in 2D
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2019-01-04 23:19
Coordinate GeometryCalculusIntegral
This example demonstrates the intersection area and partial volume in a 2D space. Key mathematical concepts include intersection, area, and volume. Using GeoGebra commands such as Segment, Intersect, and Polygon, complex geometric shapes can be constructed. Suitable for high school mathematics education and learning, particularly in coordinate geometry and calculus.

GeoGebra Commands & Parameters

  1. Segment(V_{--}, V_{+-}) → E_{b}
  2. Segment(V_{+-}, V_{++}) → E_{r}
  3. Segment(V_{++}, V_{-+}) → E_t
  4. Segment(V_{-+}, V_{--}) → E_l
  5. Segment(P_1, P_0) → E_{P01}
  6. Segment(C, P) → t_{<0}
  7. Segment(C, P) → t_{>0}
  8. Angle(A, C, If[t > 0, P, (3 * C) - (2 * P)]) → α
  9. Angle(If[t > 0, P, (3 * C) - (2 * P)], C, A) → β
  10. Intersect(E_{P01}, E_{b}) → I_{b}
  11. Intersect(E_{P01}, E_{r}) → I_{r}
  12. Intersect(E_{P01}, E_t) → I_{t}
  13. Intersect(E_{P01}, E_l) → I_{l}
  14. Segment(I_{b}, I_{t}) → l
  15. Segment(I_{l}, I_{r}) → m
  16. Segment(I_{b}, I_{r}) → n
  17. Segment(I_{r}, I_{t}) → p
  18. Segment(I_{t}, I_{l}) → q
  19. Segment(I_{l}, I_{b}) → r
  20. Polygon(V_{--}, I_{b}, I_{t}, V_{-+}) → q1, v_{--}, i_{b}, i_{t}, v_{-+}
  21. Polygon(V_{--}, V_{+-}, I_{r}, I_{t}, V_{-+}) → poly2, v_1, v_{+-}, i_{r}, i_1, v_2
  22. Polygon(V_{--}, V_{+-}, I_{r}, I_{l}) → q2, v_3, v_4, i_2, i_{l}
  23. Polygon(V_{--}, V_{+-}, V_{++}, I_{t}, I_{l}) → poly1, v_5, v_6, v_{++}, i_3, i_4
  24. Polygon(I_{b}, V_{+-}, V_{++}, I_{t}) → q3, i_5, v_7, v_8, i_6
  25. Polygon(I_{b}, V_{+-}, V_{++}, V_{-+}, I_{l}) → poly3, i_7, v_9, v_{10}, v_{11}, i_8
  26. Polygon(V_{--}, I_{b}, I_{r}, V_{++}, V_{-+}) → poly4, v_{12}, i_9, i_{10}, v_{13}, v_{14}
  27. Polygon(I_{r}, V_{++}, V_{-+}, I_{l}) → q4, i_{11}, v_{15}, v_{16}, i_{12}
  28. Polygon(I_{b}, V_{+-}, I_{r}) → t1, i_{13}, i_{14}, v_{17}
  29. Polygon(I_{r}, V_{++}, I_{t}) → t2, i_{15}, i_{16}, v_{18}
  30. Polygon(I_{t}, V_{-+}, I_{l}) → t3, i_{17}, i_{18}, v_{19}
  31. Polygon(V_{--}, I_{b}, I_{l}) → t4, i_{19}, v_{20}, i_{20}
  32. Polygon(V_{--}, V_{+-}, V_{++}, V_{-+}) → q5, v_{21}, v_{22}, v_{23}, v_{24}
  33. LaTeX(Term_{t,0}, false, true) → TextTerm_{0}
  34. LaTeX(Term_{t,1}, false, true) → TextTerm_{1}
  35. LaTeX(Term_{t,2}, false, true) → TextTerm_{2}
  36. LaTeX(Term_{t,3}, false, true) → TextTerm_{3}
  37. LaTeX(Term_{t,4}, false, true) → TextTerm_{4}
  38. LaTeX(Term_{t,5}, false, true) → TextTerm_{5}
  39. Segment(P_{--}, P_{+-}) → E_{x-}
  40. Segment(P_{+-}, P_{++}) → E_{+x}
  41. Segment(P_{++}, P_{-+}) → E_{x+}
  42. Segment(P_{-+}, P_{--}) → E_{-x}
  43. Segment((φ, 0), (φ, (2 * t_{max}))) → s_φ
  44. Segment(Q_{--}, Q_{+-}) → F_{x-}
  45. Segment(Q_{+-}, Q_{++}) → F_{+x}
  46. Segment(Q_{++}, Q_{-+}) → F_{x+}
  47. Segment(Q_{-+}, Q_{--}) → F_{-x}
  48. Segment(O_{g} + (φ, 0), Result_{g}) → f
  49. If(Defined[l], Length[l], If[Defined[m], Length[m], If[Defined[n], Length[n], If[Defined[p], Length[p], If[Defined[q], Length[q], If[Defined[r], Length[r], "Error"]]]]]) → LengthCheck
  50. Segment(O_{f} + (φ, 0), Result_{f}) → s

AI Drawing Prompt Reference

Draw a 2D geometric shape with four edges E_b, E_r, E_t, and E_l, and their intersection points I_b, I_r, I_t, and I_l. Construct multiple quadrilaterals such as q1, poly2, q2, etc., using these intersection points. Finally, connect these quadrilaterals to form a complex geometric area. Pay attention to the definitions of angles α and β, and how to construct line segments l, m, n, p, and q using the intersection points.
The prompt above was compiled from the original drawing commands and is for reference only.