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Apple-Shaped Parametric Curves and Surfaces

Apple-Shaped Parametric Curves and Surfaces
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2016-01-16 20:38
Solid GeometryParametric EquationGeometric Construction
This example demonstrates how to draw an apple-shaped curve and surface using parametric equations. It involves core mathematical concepts such as parametric curves, surfaces, orthogonal lines, intersections, reflections, circular arcs, distances, and angles. The construction utilizes GeoGebra commands like CurveCartesian, Tangent, OrthogonalLine, Intersect, Mirror, CircleArc, Distance, Angle, and Surface. It is suitable for high school and above students to learn the application of parametric equations in geometric plotting.

GeoGebra Commands & Parameters

  1. CurveCartesian((-(p_1 + (3.8 * cos(v)))), (((cos(v) + sin(v) - 1) * (1 + sin(v))) * log(1 - (π * v / 10))) + (p_2 * sin(v)), v, b_1, b_2) → c_3
  2. Tangent(B, c_3) → tg
  3. OrthogonalLine(B, tg) → lot
  4. Intersect(lot, yAxis) → C
  5. Mirror(B, yAxis) → B'
  6. CircleArc(C, B', B) → c
  7. Distance(C, B) → r
  8. Angle(lot, yAxis) → wi
  9. Surface(((r * cos(u)) * sin(v)), ((r * sin(u)) * sin(v)), z(M) - (r * cos(v)), u, π, π + a, v, π, (2 * π) - wi) → e
  10. CurveCartesian((r * sin(v)), ((r * 0) * sin(v)), z(M) - (r * cos(v)), v, π, (2 * π) - wi) → k_1
  11. CurveCartesian(((r * (-1)) * sin(v)), ((r * 0) * sin(v)), z(M) - (r * cos(v)), v, π, (2 * π) - wi) → k_2
  12. Surface((cos(u) * (p_1 + (3.8 * cos(v)))), (sin(u) * (p_1 + (3.8 * cos(v)))), (((cos(v) + sin(v) - 1) * (1 + sin(v))) * log(1 - (π * v / 10))) + (p_2 * sin(v)), u, 0, a, v, b_1, b_2) → apple
  13. CurveCartesian((-(p_1 + (3.8 * cos(v)))), (0 * (p_1 + (3.8 * cos(v)))), (((cos(v) + sin(v) - 1) * (1 + sin(v))) * log(1 - (π * v / 10))) + (p_2 * sin(v)), v, b_1, b_2) → c_1
  14. CurveCartesian(p_1 + (3.8 * cos(v)), (0 * (p_1 + (3.8 * cos(v)))), (((cos(v) + sin(v) - 1) * (1 + sin(v))) * log(1 - (π * v / 10))) + (p_2 * sin(v)), v, b_1, b_2) → c_2
  15. CurveCartesian(p_1 + (3.8 * cos(v)), (((cos(v) + sin(v) - 1) * (1 + sin(v))) * log(1 - (π * v / 10))) + (p_2 * sin(v)), v, b_1, b_2) → c_4