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Polar Rose Curves: r = a·cos(bθ) and r = a·sin(bθ) with Dynamic Construction

Polar Rose Curves: r = a·cos(bθ) and r = a·sin(bθ) with Dynamic Construction
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2020-04-11 14:59
FunctionTrigonometryParametric EquationSlider AnimationGraphing
① This example demonstrates the classic polar rose curves defined by r = a·cos(bθ) or r = a·sin(bθ), showing how the number and shape of petals change as parameters a and b vary. ② Core mathematical concepts include polar coordinates, trigonometric periodicity, parametric representation, symmetry, and the relationship between the integer b and petal count (b petals if b is odd, 2b petals if b is even). ③ Construction details: The GeoGebra commands include CurveCartesian to plot the polar equation in Cartesian parametric form, along with sliders for a, b, and θ; points A and B, lines, and angle objects are used to visualize the radius r as θ changes, with text labels text1 and text2 displaying the equation and parameters. ④ This is suitable for high school or early college courses in analytic geometry, trigonometry, or pre-calculus, and serves as an interactive tool for exploring polar graphs and parametric curves.

GeoGebra Commands & Parameters

  1. CurveCartesian((r(i) * cos(i)), (r(i) * sin(i)), i, 0, θ) → b
  2. Point(b) → A
  3. CurveCartesian(cos(i), sin(i), i, 0, 6.283185307179586) → c
  4. Line((0, 0), (1, 1)) → e
  5. Line((0, 0), (1, 1.7320508075688772)) → f
  6. Line((0, 0), (1.7320508075688772, 1)) → g
  7. Line((0, 0), (-1.73205, 1)) → h
  8. Line((0, 0), (1, -1.73205)) → i
  9. Line((0, 0), (1, -1)) → j
  10. Line((0, -0.02), (0.9659258262890683, 0.23881904510252072)) → k
  11. Line((0, 0), (0.25881904510252074, 0.9659258262890683)) → l
  12. Line((0, -0.02), (-0.25881904510252085, 0.9659258262890683)) → m
  13. Line((0, -0.02), (-0.9659258262890682, 0.258819045102521)) → n
  14. CurveCartesian((2 * cos(i)), (2 * sin(i)), i, 0, 6.28319) → o
  15. CurveCartesian((3 * cos(i)), (3 * sin(i)), i, 0, 6.28319) → p
  16. CurveCartesian((4 * cos(i)), (4 * sin(i)), i, 0, 6.28319) → q
  17. CurveCartesian((5 * cos(i)), (5 * sin(i)), i, 0, 6.28319) → s
  18. CurveCartesian((6 * cos(i)), (6 * sin(i)), i, 0, 6.28319) → t
  19. CurveCartesian((7 * cos(i)), (7 * sin(i)), i, 0, 6.28319) → u
  20. CurveCartesian((v(i) * cos(i)), (v(i) * sin(i)), i, 0, θ) → w
  21. Line(A, B) → d
  22. Angle(xAxis, d) → α

AI Drawing Prompt Reference

Draw polar rose curves defined by r = a·cos(bθ) and r = a·sin(bθ), where a and b are adjustable parameters. Use sliders to change the values of a and b and observe how the number and shape of petals change. Plot sine and cosine curves from 0 to 2π to visualize how the radius r varies with angle θ.
The prompt above was compiled from the original drawing commands and is for reference only.