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Fish Curve: Interactive Parametric and Geometric Construction

Fish Curve: Interactive Parametric and Geometric Construction
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2020-06-28 05:12
FunctionParametric EquationGeometric ConstructionSlider AnimationGraphing
This GeoGebra example dynamically demonstrates a fish-shaped planar curve whose outline is driven by parametric equations, enriched with extensive geometric construction elements. Core mathematical concepts include parametric curves, power functions, quadratic equations, difference and slope, and multiplication, alongside basic geometric objects such as points, lines, circles, polygons, prisms, cylinders, and cones. The construction integrates commands like Line, Circle, Cone, Point, Midpoint, Mirror, Cylinder, CurveCartesian, Polygon, Prism, PointIn, Plane, and OrthogonalLine, building a complex scene with auxiliary points (e.g., P, Q, Q'), segments, planes, and 3D solids. Sliders (Phase, m, r, R) control parameters to dynamically alter the fish curve's shape and related geometric relationships. This example suits high school and above, helping students understand how parametric equations generate intricate curves and how geometric constructions interrelate with algebraic expressions. It also serves as a vivid case for mathematical modeling or computer-assisted geometry instruction.

GeoGebra Commands & Parameters

  1. Line(F, G) → n_1
  2. Circle(F, c, n_1) → r_1
  3. Cone(r_1, 0.925) → o_1, o_2, o_3
  4. Point(o_2) → P
  5. Midpoint(F, P) → Q
  6. Mirror(Q, P) → Q'
  7. Line(P, zAxis) → s_1
  8. Cylinder(A, B, c) → d, e, g, h
  9. Cylinder(B, C, c) → f, k, p, i
  10. Cylinder(C, D, c) → j, q, s, l
  11. Cylinder(D, E, c) → n, c_1, d_1, o
  12. Cylinder(E, F, c) → u, e_1, f_1, v
  13. Cylinder(F, G, c) → w, g_1, h_1, a_1
  14. Cylinder(G, H, c) → b_1, b_2, b_3, b_4
  15. CurveCartesian(x(O) + (R * cos(u)), y(O) + (R * sin(u)), z(O) + ((r * cos(((m * Phase) * u))) * cos(u)), u, 0, (2 * π)) → a
  16. Polygon(AA, BB, CC, DD) → q1, aa, bb, cc, dd
  17. Prism(q1, 4) → i_1, I, arêteAAI, arêteBBJ, arêteCCK, arêteDDL, arêteIJ, arêteJK, arêteKL, arêteIL, J, K, L, faceAABBJI, faceBBCCKJ, faceCCDDLK, faceAADDLI, faceIJKL
  18. Line(B, A) → j_1
  19. Circle(A, c, j_1) → k_1
  20. Cone(k_1, 0.35) → l_1, l_2, l_3
  21. PointIn(l_3) → M
  22. PointIn(l_3) → N
  23. Plane(Q', s_1) → q_1
  24. OrthogonalLine(Q', q_1) → m_1
  25. Point(m_1) → T
  26. Midpoint(Q', P) → S
  27. Midpoint(S, Q') → U
  28. Mirror(T, Q') → T'
  29. Polygon(P, T', U, T) → q2, p_2, t', u_1, t_1
  30. Line(B, s_1) → f_2
  31. Line(E, s_1) → g_2
  32. Point(f_2) → V
  33. Point(g_2) → W
  34. Polygon(E, B, V, W) → q3, e_2, b_5, v_1, w_1
  35. OrthogonalLine(B, q_1) → h_2
  36. OrthogonalLine(D, q_1) → i_2

AI Drawing Prompt Reference

Draw a fish-shaped planar curve defined by parametric equations, with sliders controlling phase and scale. Surround the curve with a 3D geometric scene: multiple cylinders arranged along a polyline, cones and frustums stacked, and auxiliary structures of points, segments, polygons, and prisms. Include key auxiliary points such as a moving point on the curve, its midpoint, and its mirror image, plus a plane through the mirror point and a perpendicular line. Generate quadrilateral and triangular regions in that plane, and place random points inside a cone. The overall composition highlights the dynamic interplay between parametric curves and solid geometry, emphasizing mathematical relationships and spatial construction.
The prompt above was compiled from the original drawing commands and is for reference only.