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Net of Cone: Dynamic Construction and Unfolding

Net of Cone: Dynamic Construction and Unfolding
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2019-12-03 02:42
AngleSolid GeometryTransformationParametric EquationGeometric Construction
① This example demonstrates how to dynamically construct the net of a cone in GeoGebra, showing the lateral surface unfolding into a sector and the base into a circle. ② Core mathematical concepts include solid geometry, surface development of a cone, relationships between sectors and circles, rotation and angle measurement, as well as the use of parametric equations and piecewise functions for surface rendering. ③ Construction highlights: from the summary, commands such as OrthogonalLine, Circle, Point, Intersect, Cone, Segment, Distance, Angle, Rotate, CircumcircleArc, Surface, and CurveCartesian are used, involving objects like t, a, c, A, B, b, d, e, C, l, r, α, A', f, B', β, C', C'_1, g, h, r', k, ϕ, j, m, D, E, F, n, p, with rotation and parameterization driving the unfolding animation. ④ Suitable for upper secondary or early university students learning solid geometry, spatial visualization, and dynamic geometry software, as well as for teachers creating interactive lessons on cone surface area derivation.

GeoGebra Commands & Parameters

  1. OrthogonalLine((1, 0, 0), xOyPlane) → a
  2. Circle(a, (0, 0, 0)) → c
  3. Point(a) → A
  4. Intersect(xAxis, a) → B
  5. Cone(B, A, 1) → b, d, e
  6. Intersect(yAxis, xAxis) → C
  7. Segment(C, A) → l
  8. Distance(B, C) → r
  9. Angle(A, C, B) → α
  10. Rotate(A, (t * α), yAxis) → A'
  11. OrthogonalLine(A', xOyPlane) → f
  12. Intersect(f, xOyPlane) → B'
  13. Rotate(C, β / 2, f) → C'
  14. Rotate(C, ((-β)) / 2, f) → C'_1
  15. CircumcircleArc(C', C, C'_1) → g
  16. Distance(A', B') → h
  17. Distance(B', C') → r'
  18. Surface(((k - ((v * r') * cos((u * β / 2)))) * cos(ϕ)) - (((1 - v) * h) * sin(ϕ)), ((v * r') * sin((u * β / 2))), (((1 - v) * h) * cos(ϕ)) + ((k - ((v * r') * cos((u * β / 2)))) * sin(ϕ)), u, -1, 1, v, 0, 1) → j
  19. CurveCartesian(((k - (r' * cos((u * β / 2)))) * cos(ϕ)), (r' * sin((u * β / 2))), ((k - (r' * cos((u * β / 2)))) * sin(ϕ)), u, -1, 1) → m
  20. Segment(E, F) → n
  21. Segment(D, F) → p

AI Drawing Prompt Reference

Draw a dynamic unfolding of a cone surface. Start by drawing a circle from the origin, then construct a cone. Next, unfold the cone's lateral surface into a sector while keeping the base as a circle. Use rotation and parameterization to show the relationship between the sector and the circle. Finally, present a graphic with rotation, angles, and parameterization to help understand the cone's surface unfolding process.
The prompt above was compiled from the original drawing commands and is for reference only.