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Modifiable Solid: Parabolic Cross Sections Parallel to yAxis

Modifiable Solid: Parabolic Cross Sections Parallel to yAxis
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2019-11-15 21:33
Solid GeometryCalculus
In collections Cross Sectional Specials (Calculus)
This example demonstrates parabolic cross sections parallel to the yAxis. Key mathematical concepts include parabolas and their properties. Using GeoGebra commands such as Intersect, Segment, and Midpoint, users can dynamically adjust the upper and lower functions to observe changes in the parabolic cross sections. Suitable for high school and above students studying calculus and solid geometry.

GeoGebra Commands & Parameters

  1. Intersect(lower, h, (0, -8)) → C
  2. Intersect(upper, h, (0, 9)) → B
  3. Segment(C, B) → q
  4. Intersect(upper, q, (0, 9)) → G
  5. Midpoint(C, G) → T
  6. Line(T, T + (1, 0, 0)) → l_1
  7. Segment(H, I) → s_1
  8. Midpoint(s_1) → E
  9. Line(E, E + (1, 0, 0)) → r_1
  10. Segment(J, K) → q_1
  11. Midpoint(q_1) → F
  12. Line(F, F + (1, 0, 0)) → c
  13. Line(B, D) → i
  14. Sphere(B, d) → e
  15. Segment(C, B) → p
  16. Textfield(upper) → InputBox1
  17. Textfield(lower) → InputBox2
  18. Textfield(a) → InputBox3
  19. Textfield(b) → InputBox4
  20. Segment(C, G) → k_1
  21. Line(C_1 + (0, 0, k), C_1 + (0, 1, k)) → g_2
  22. Parabola(T, g_2) → c_2
  23. CurveCartesian(x(C), y(C_1) + (t * (y(B) - y(C))), k - ((t * (y(B) - y(C))))^(2) / ((4 * k)), t, -1 / 2, 1 / 2) → o
  24. Surface(x(C), y(C_1) + ((v * t) * (y(B) - y(C))), k - (v * ((t * (y(B) - y(C))))^(2) / (((4 * k) * v))), v, 0, 1, t, -1 / 2, 1 / 2) → u
  25. CurveCartesian(x(E), y(E) + (t * (y(H) - y(I))), f - ((t * (y(H) - y(I))))^(2) / ((4 * f)), t, -1 / 2, 1 / 2) → g
  26. CurveCartesian(x(K), y(F) + (t * (y(J) - y(K))), l - ((t * (y(J) - y(K))))^(2) / ((4 * l)), t, -1 / 2, 1 / 2) → m
  27. Surface(b, y(F) + ((v * t) * (y(J) - y(K))), l - (v * ((t * (y(J) - y(K))))^(2) / (((4 * l) * v))), v, 0, 1, t, -1 / 2, 1 / 2) → n
  28. Surface(a, y(E) + ((v * t) * (y(H) - y(I))), f - (v * ((t * (y(H) - y(I))))^(2) / (((4 * f) * v))), v, 0, 1, t, -1 / 2, 1 / 2) → r
  29. Surface(u, (0.5 * (upper(u) + lower(u))) + (v * (upper(u) - lower(u))), (0.25 * (upper(u) - lower(u))) - ((v * (upper(u) - lower(u))))^(2) / (upper(u) - lower(u)), u, a, b, v, -1 / 2, 1 / 2) → t
  30. CurveCartesian(t, (0.5 * (upper(t) + lower(t))), (0.25 * (upper(t) - lower(t))), t, a, b) → s

AI Drawing Prompt Reference

Draw a modifiable parabolic solid with parabolic cross sections parallel to the y-axis, where the upper and lower functions can be adjusted via input boxes. First, draw the upper and lower parabolic functions, then construct parallel line segments to the y-axis using intersection and midpoint points, and finally generate the parabolic solid using these segments and parabolas.
The prompt above was compiled from the original drawing commands and is for reference only.