MathPlotAI Logo MathPlotAI

Intersection Area and Partial Volume in 2D by Ron van den Burg (2)

Intersection Area and Partial Volume in 2D by Ron van den Burg (2)
Loading the math board and drawing, please wait…
2018-05-24 03:44
Coordinate GeometryGeometric Construction
This example demonstrates the calculation of the intersection area and partial volume of two shapes in a 2D plane. Key math concepts include area calculation, intersection, volume concepts, curve sketching, tangent lines, general quadrilaterals, ellipses, and matrix operations. The construction uses GeoGebra commands like Segment, Intersect, Polygon, If, and Angle, along with objects such as φ_{max}, w_{0}, w_{1}, w, ω, C, V_{--}, V_{+-}, V_{-+}, V_{++}, E_{b}, E_{r}, E_t, E_l, t_{max}, t, φ_{min}, φ, φ_{left}, φ_{right}, P, P_0, P_1, E_{P01}, t_{<0}, t_{>0}, A, P_{--}, P_{+-}, P_{++}, P_{-+}, E_{x-}, E_{+x}, E_{x+}, E_{-x}, f_{--}, f_{+-}, f_{-+}, f_{++}, f_{t}. It is suitable for high school and above students learning geometric intersections, volume calculations, and parametric graphing with GeoGebra.

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. Segment(P_{--}, P_{+-}) → E_{x-}
  9. Segment(P_{+-}, P_{++}) → E_{+x}
  10. Segment(P_{++}, P_{-+}) → E_{x+}
  11. Segment(P_{-+}, P_{--}) → E_{-x}
  12. Intersect(f_{--}, k, (0.3138055098076551, 10.214635073645745)) → I_{--}
  13. Intersect(f_{+-}, k, (0.3138055098076551, -3.411484981784172)) → I_{+-}
  14. Intersect(f_{-+}, k, (0.3138055098076551, -5.7990009979994435)) → I_{-+}
  15. Intersect(f_{++}, k, (0.3138055098076551, 0)) → I_{++}
  16. Intersect(f_{t}, k, (0.3138055098076551, 1.0041490938621296)) → Result
  17. Segment((φ, 0), (φ, (2 * t_{max}))) → s_φ
  18. Intersect(E_{P01}, E_{b}) → I_{b}
  19. Intersect(E_{P01}, E_{r}) → I_{r}
  20. Intersect(E_{P01}, E_t) → I_{t}
  21. Intersect(E_{P01}, E_l) → I_{l}
  22. Segment(I_{b}, I_{t}) → l
  23. Segment(I_{l}, I_{r}) → m
  24. Segment(I_{b}, I_{r}) → n
  25. Segment(I_{r}, I_{t}) → p
  26. Segment(I_{t}, I_{l}) → q
  27. Segment(I_{l}, I_{b}) → r
  28. Polygon(V_{--}, I_{b}, I_{t}, V_{-+}) → q1, v_{--}, i_{b}, i_{t}, v_{-+}
  29. Polygon(V_{--}, V_{+-}, I_{r}, I_{t}, V_{-+}) → poly2, v_1, v_{+-}, i_{r}, i_1, v_2
  30. Polygon(V_{--}, V_{+-}, I_{r}, I_{l}) → q2, v_3, v_4, i_2, i_{l}
  31. Polygon(V_{--}, V_{+-}, V_{++}, I_{t}, I_{l}) → poly1, v_5, v_6, v_{++}, i_3, i_4
  32. Polygon(I_{b}, V_{+-}, V_{++}, I_{t}) → q3, i_5, v_7, v_8, i_6
  33. Polygon(I_{b}, V_{+-}, V_{++}, V_{-+}, I_{l}) → poly3, i_7, v_9, v_{10}, v_{11}, i_8
  34. Polygon(V_{--}, I_{b}, I_{r}, V_{++}, V_{-+}) → poly4, v_{12}, i_9, i_{10}, v_{13}, v_{14}
  35. Polygon(I_{r}, V_{++}, V_{-+}, I_{l}) → q4, i_{11}, v_{15}, v_{16}, i_{12}
  36. Polygon(I_{b}, V_{+-}, I_{r}) → t1, i_{13}, i_{14}, v_{17}
  37. Polygon(I_{r}, V_{++}, I_{t}) → t2, i_{15}, i_{16}, v_{18}
  38. Polygon(I_{t}, V_{-+}, I_{l}) → t3, i_{17}, i_{18}, v_{19}
  39. Polygon(V_{--}, I_{b}, I_{l}) → t4, i_{19}, v_{20}, i_{20}
  40. Polygon(V_{--}, V_{+-}, V_{++}, V_{-+}) → q5, v_{21}, v_{22}, v_{23}, v_{24}
  41. 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
  42. Segment(B, Result) → s
  43. Angle(A, C, If[t > 0, P, (3 * C) - (2 * P)]) → α
  44. Angle(If[t > 0, P, (3 * C) - (2 * P)], C, A) → β

AI Drawing Prompt Reference

Draw the intersection area and partial volume of two shapes in a 2D plane. Start by sketching four sides and a line connecting two points, then find the intersections of five curves with a horizontal line and connect them to form multiple polygons. Calculate the area of these polygons and label key points and segments. Finally, draw the outline and shaded regions of the shapes to show their intersection and volume.
The prompt above was compiled from the original drawing commands and is for reference only.