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Parapluie Curve Length Problem

Parapluie Curve Length Problem
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2015-06-15 01:53
FunctionCoordinate GeometryParametric Equation
In collections My 3D Collection
This example demonstrates how to calculate the length of a curve through geometric construction. Key mathematical concepts include coordinate geometry and parametric equations. Using GeoGebra commands such as Surface, Segment, and CurveCartesian, the umbrella-shaped curve is constructed and its length is calculated. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Surface(((((cos((π / 4 * (1 + v))) * cos(((p - 1) * π / 2))) - (sin(((p - 1) * π / 2)) * (sin((π / 4 * (1 + v))) - 1))) * cos(u)) * (1 + ((((1 - p) * (1 - v)) * a) * cos(((n * j) * u))))), ((((cos((π / 4 * (1 + v))) * cos(((p - 1) * π / 2))) - (sin(((p - 1) * π / 2)) * (sin((π / 4 * (1 + v))) - 1))) * sin(u)) * (1 + ((((1 - p) * (1 - v)) * a) * cos(((n * j) * u))))), (cos((π / 4 * (1 + v))) * sin(((p - 1) * π / 2))) + (cos(((p - 1) * π / 2)) * (sin((π / 4 * (1 + v))) - 1)) + 1, u, 0, 6.28319, v, 0, 1) → toile
  2. Segment((0, 0, 1.05), (0, 0, 0.1)) → c
  3. Sequence(CurveCartesian[((((cos((π / 4 * (1 + v))) * cos(((p - 1) * π / 2))) - (sin(((p - 1) * π / 2)) * (sin((π / 4 * (1 + v))) - 1))) * cos(u)) * (1 + ((((1 - p) * (1 - v)) * a) * cos(((n * j) * u))))), ((((cos((π / 4 * (1 + v))) * cos(((p - 1) * π / 2))) - (sin(((p - 1) * π / 2)) * (sin((π / 4 * (1 + v))) - 1))) * sin(u)) * (1 + ((((1 - p) * (1 - v)) * a) * cos(((n * j) * u))))), (cos((π / 4 * (1 + v))) * sin(((p - 1) * π / 2))) + (cos(((p - 1) * π / 2)) * (sin((π / 4 * (1 + v))) - 1)) + 1, v, 0, 1], u, 0, 6.28319, (2 * π / n)) → baleines
  4. CurveCartesian(0.1 + (0.1 * cos(t)), 0, 0.1 + (0.1 * sin(t)), t, 3.141592653589793, 6.28319) → e
  5. Element(baleines, 1) → b
  6. Point(b, 0.5) → B
  7. Intersect(c, Sphere[B, d]) → A, C
  8. Point(Segment[(0, 0, 0.15), (0, 0, 0.6)]) → D
  9. Sequence(Segment[Point[Element[baleines, i], 0.5], C], i, 1, n) → segments
  10. CurveCartesian(((((cos(π / 4) * cos(((p - 1) * π / 2))) - (sin(((p - 1) * π / 2)) * (sin(π / 4) - 1))) * cos(u)) * (1 + (((1 - p) * a) * cos(((n * j) * u))))), ((((cos(π / 4) * cos(((p - 1) * π / 2))) - (sin(((p - 1) * π / 2)) * (sin(π / 4) - 1))) * sin(u)) * (1 + (((1 - p) * a) * cos(((n * j) * u))))), (cos(π / 4) * sin(((p - 1) * π / 2))) + (cos(((p - 1) * π / 2)) * (sin(π / 4) - 1)) + 1, u, 0, 6.28319) → f_1