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Pointed Hats

Pointed Hats
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2020-03-15 16:13
CircleCoordinate GeometryGeometric Construction
This example demonstrates the geometric shape of pointed hats. It involves concepts of cones and circles. Using GeoGebra commands such as Sequence, CurveCartesian, and Join, the pointed hat can be constructed. Suitable for high school geometry and calculus courses.

GeoGebra Commands & Parameters

  1. Sequence((x(O), y(O)) - (R * (((sin(0°) * sin(i°)) * w) + ((cos(0°) * sin(i°)) * u) - (cos(i°) * v))), i, 0, k, n) → liste1
  2. CurveCartesian(x(O - (R * (((0 * sin(i°)) * w) + (sin(i°) * u) - (cos(i°) * v)))), y(O - (R * (((0 * sin(i°)) * w) + (sin(i°) * u) - (cos(i°) * v)))), i, 0, k) → d
  3. Join(liste1) → liste2
  4. Sequence(Polygon[O', Element[liste1, i], Element[liste1, i + 1]], i, 0, Length[liste1] + 2, 1) → liste3
  5. Sequence(Polygon[O', Element[liste1, i], Element[liste1, i + 1]], i, 0, Length[liste1] + 2, 3) → liste4
  6. Sequence((x(O), y(O)) - ((R + h / 2.5) * (((sin(0°) * sin(i°)) * w) + ((cos(0°) * sin(i°)) * u) - (cos(i°) * v))), i, 0, k, n) → liste5
  7. Sequence(Polygon[Element[liste1, i], Element[liste1, i + 1], Element[liste5, i + 1], Element[liste5, i]], i, 0, Length[liste1] + 2, 1) → liste6
  8. Sequence(Polygon[Element[liste1, i], Element[liste1, i + 1], Element[liste5, i + 1], Element[liste5, i]], i, 0, Length[liste1] + 2, 3) → liste7
  9. CurveCartesian(x(O - ((R + h / 2.5) * (((0 * sin(i°)) * w) + (sin(i°) * u) - (cos(i°) * v)))), y(O - ((R + h / 2.5) * (((0 * sin(i°)) * w) + (sin(i°) * u) - (cos(i°) * v)))), i, 0, k) → e
  10. Sequence((x(A), y(A)) - (R * (((sin(0°) * sin(i°)) * w) + ((cos(0°) * sin(i°)) * u) - (cos(i°) * v))), i, 0, k, n) → liste8
  11. Sequence((x(A'), y(A')) - (R * (((sin(0°) * sin(i°)) * w) + ((cos(0°) * sin(i°)) * u) - (cos(i°) * v))), i, 0, k, n) → liste9
  12. Sequence((x(A), y(A)) - ((R + h / 2.5) * (((sin(0°) * sin(i°)) * w) + ((cos(0°) * sin(i°)) * u) - (cos(i°) * v))), i, 0, k, n) → liste10
  13. Sequence(Polygon[Element[liste8, i], Element[liste8, i + 1], Element[liste9, i + 1], Element[liste9, i]], i, 0, Length[liste9] + 2, 1) → liste11
  14. Sequence(Polygon[Element[liste8, i], Element[liste8, i + 1], Element[liste9, i + 1], Element[liste9, i]], i, 0, Length[liste9] + 2, 3) → liste12
  15. Sequence(Polygon[Element[liste8, i], Element[liste8, i + 1], Element[liste10, i + 1], Element[liste10, i]], i, 0, Length[liste8] + 2, 1) → liste13
  16. Sequence(Polygon[Element[liste8, i], Element[liste8, i + 1], Element[liste10, i + 1], Element[liste10, i]], i, 0, Length[liste8] + 2, 3) → liste14
  17. CurveCartesian(x(A - (R * (((0 * sin(i°)) * w) + (sin(i°) * u) - (cos(i°) * v)))), y(A - (R * (((0 * sin(i°)) * w) + (sin(i°) * u) - (cos(i°) * v)))), i, 0, k) → f
  18. CurveCartesian(x(A' - (R * (((0 * sin(i°)) * w) + (sin(i°) * u) - (cos(i°) * v)))), y(A' - (R * (((0 * sin(i°)) * w) + (sin(i°) * u) - (cos(i°) * v)))), i, 0, k) → i
  19. CurveCartesian(x(A - ((R + h / 2.5) * (((0 * sin(i°)) * w) + (sin(i°) * u) - (cos(i°) * v)))), y(A - ((R + h / 2.5) * (((0 * sin(i°)) * w) + (sin(i°) * u) - (cos(i°) * v)))), i, 0, k) → j
  20. Sequence(A' + ((R * cos(i° + 90°)), (R * sin(i° + 90°)), 0), i, 0, k, n) → liste17
  21. Sequence(Polygon[Element[liste17, i], Element[liste17, i + 1], Element[liste9, i + 1], Element[liste9, i]], i, 0, Length[liste9] + 2, 1) → l2
  22. Sequence(Polygon[Element[liste17, i], Element[liste17, i + 1], Element[liste9, i + 1], Element[liste9, i]], i, 0, Length[liste9] + 2, 3) → l1
  23. Sequence(Polygon[Element[liste17, i], Element[liste17, i + 1], A'], i, 0, Length[liste8] + 2, 1) → l3
  24. Sequence(Polygon[Element[liste17, i], Element[liste17, i + 1], A'], i, 0, Length[liste8] + 2, 3) → l4
  25. Point(j) → B
  26. Mirror(B, A) → B'
  27. Segment(B, B') → l
  28. Segment(A, A') → m
  29. Point(e) → C
  30. Mirror(C, O) → C'
  31. Segment(C, C') → p

AI Drawing Prompt Reference

Draw a geometric pointed hat shape involving cones and circles. First, create the curve for the hat's top, then construct the curve for the brim, fill the top and brim areas with polygons, and finally connect the top and brim to form the complete pointed hat.
The prompt above was compiled from the original drawing commands and is for reference only.