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Apple (Manzana) – Geometric Art with Rotation and Ellipses

Apple (Manzana) – Geometric Art with Rotation and Ellipses
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2023-03-15 20:44
SequenceTransformationParametric EquationGeometric ConstructionGraphing
In collections geometriadinamica.es
① This GeoGebra example demonstrates a geometric figure named 'Apple' (Manzana), composed of rotated ellipses, segments, circles, and intersection points, forming an apple-like silhouette. ② Core mathematical concepts include rotation, ellipse equations, standard deviation as a geometric measure, multiplication and arithmetic operations, as well as parametric curves (CurveCartesian) and sequences (Sequence). ③ Construction highlights: commands such as Segment, Point, OrthogonalLine, Circle, Intersect, Ellipse, Angle, CurveCartesian, and Sequence are used, with numerous auxiliary objects (u10, p, v10, Q4, Q3, p2, P3, etc.) to generate the apple shape through rotation and ellipse parameterization. ④ Suitable for high school to university students and teachers studying analytic geometry, parametric equations, rotation transformations, and advanced GeoGebra construction techniques; also valuable for mathematical art creation or demonstrating the beauty of combining geometry and algebra.

GeoGebra Commands & Parameters

  1. Segment(Q3, Q4) → p2
  2. Point(p2) → P3
  3. Segment(P3, Q4) → p4
  4. Point(p4) → P4
  5. Segment(Q1, Q2) → p1
  6. Point(p1) → P1
  7. Segment(P1, Q2) → p3
  8. Point(p3) → P2
  9. OrthogonalLine(C, xAxis) → zz2
  10. Circle(C, 0.62) → zz0
  11. Intersect(zz0, zz2, 1) → ZZ4
  12. Intersect(zz0, zz2, 2) → ZZ3
  13. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  14. Point(zElipse2) → Z
  15. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  16. Angle(ZZ8, C, Z) → zAngInc
  17. OrthogonalLine(C, yAxis) → zz1
  18. Intersect(zz0, zz1, 2) → ZZ2
  19. Intersect(zz0, zz1, 1) → ZZ1
  20. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  21. Point(zElipse1) → XY
  22. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  23. Angle(ZZ6, C, XY) → zAngRot
  24. Segment(Q5, Q6) → p5
  25. Point(p5) → P5
  26. Segment(ZZ5, ZZ6) → zz4
  27. Segment(ZZ7, ZZ8) → zz3
  28. Intersect(zElipse2, zElipse1, 1) → zPunto1
  29. Intersect(zElipse2, zElipse1, 4) → zPunto2
  30. Segment(zPunto1, zPunto2) → zSeg2
  31. CurveCartesian((r / 6 * (((cos(α) * (4 + (3.8 * cos(v)))) * sin(zAngRot)) + ((sin(α) * (4 + (3.8 * cos(v)))) * cos(zAngRot)))), (r / 6 * (((((-cos(α)) * (4 + (3.8 * cos(v)))) * cos(zAngRot)) * sin(zAngInc)) + (((sin(α) * (4 + (3.8 * cos(v)))) * sin(zAngRot)) * sin(zAngInc)) + (((((cos(v) + sin(v) - 1) * (1 + sin(v))) * ln(1 - (3.14159 * v / 10))) + (7.5 * sin(v))) * cos(zAngInc)))), v, v1, v2) → curva2
  32. CurveCartesian((r / 6 * (((cos(u) * (4 + 3.8)) * sin(zAngRot)) + ((sin(u) * (4 + 3.8)) * cos(zAngRot)))), (r / 6 * (((((-cos(u)) * (4 + 3.8)) * cos(zAngRot)) * sin(zAngInc)) + (((sin(u) * (4 + 3.8)) * sin(zAngRot)) * sin(zAngInc)))), u, u1, u2) → curva1
  33. CurveCartesian((r / 6 * (((cos(u) * (4 + (3.8 * cos(β - 3.14159)))) * sin(zAngRot)) + ((sin(u) * (4 + (3.8 * cos(β - 3.14159)))) * cos(zAngRot)))), (r / 6 * (((((-cos(u)) * (4 + (3.8 * cos(β - 3.14159)))) * cos(zAngRot)) * sin(zAngInc)) + (((sin(u) * (4 + (3.8 * cos(β - 3.14159)))) * sin(zAngRot)) * sin(zAngInc)) + (((((cos(β - 3.14159) + sin(β - 3.14159) - 1) * (1 + sin(β - 3.14159))) * ln(1 - (3.14159 * (β - 3.14159) / 10))) + (7.5 * sin(β - 3.14159))) * cos(zAngInc)))), u, u1, u2) → a1
  34. Sequence(CurveCartesian[(r / 6 * (((cos(u) * (4 + (3.8 * cos(v)))) * sin(zAngRot)) + ((sin(u) * (4 + (3.8 * cos(v)))) * cos(zAngRot)))), (r / 6 * (((((-cos(u)) * (4 + (3.8 * cos(v)))) * cos(zAngRot)) * sin(zAngInc)) + (((sin(u) * (4 + (3.8 * cos(v)))) * sin(zAngRot)) * sin(zAngInc)) + (((((cos(v) + sin(v) - 1) * (1 + sin(v))) * ln(1 - (3.14159 * v / 10))) + (7.5 * sin(v))) * cos(zAngInc)))), v, v1, v2], u, u1, u2, paso) → list2
  35. Sequence(CurveCartesian[(r / 6 * (((cos(u) * (4 + (3.8 * cos(v)))) * sin(zAngRot)) + ((sin(u) * (4 + (3.8 * cos(v)))) * cos(zAngRot)))), (r / 6 * (((((-cos(u)) * (4 + (3.8 * cos(v)))) * cos(zAngRot)) * sin(zAngInc)) + (((sin(u) * (4 + (3.8 * cos(v)))) * sin(zAngRot)) * sin(zAngInc)) + (((((cos(v) + sin(v) - 1) * (1 + sin(v))) * ln(1 - (3.14159 * v / 10))) + (7.5 * sin(v))) * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list1

AI Drawing Prompt Reference

Create a geometric “apple” silhouette in the plane using ellipses and rotated parametric curves. Start with a coordinate system, an origin-centered circle, and construct points from intersections between the circle and perpendicular directions given by the axes. Use those intersection points as parameters/foci to draw two intersecting ellipse curves, and mark their key intersection points. From the circle–ellipse configuration derive a rotation angle and a secondary inclination angle. Then define parametric Cartesian curves whose x–y coordinates depend on these two angles and on trigonometric expressions involving two parameters; sample the curves over specified parameter ranges to generate a sequence of points. Connect selected pairs of points with short segments and combine the resulting curve pieces from both ellipse-related constructions so the overall figure resembles an apple outline with many intersections and segment accents.
The prompt above was compiled from the original drawing commands and is for reference only.