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Rolling Regular Polygon: Visualizing Rolling Motion and Geometric Transformation

Rolling Regular Polygon: Visualizing Rolling Motion and Geometric Transformation
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2012-10-10 15:05
PolygonSequenceTransformationSlider Animation
This example visualizes the rolling motion of a regular polygon, displaying the trajectories of its vertices and the contact points of its sides with the ground. It integrates key mathematical concepts such as polygon geometry, rotation and translation transformations, parametric equations, locus curves, and derivative analysis of rolling speed. The construction uses GeoGebra commands like Rotate, Polygon, LineBisector, Line, Intersect, and Sequence, defining objects n, length, roll, R, count, turns, B, flip, A, A', poly1, etc., and employs sliders to animate the rolling process. Designed for high‑school and college students, it serves as an interactive tool for exploring geometric motion, analytic geometry, and calculus applications.

GeoGebra Commands & Parameters

  1. Rotate(A, ((360 * ° / n) * (1 - flip)), B) → A'
  2. Polygon(B, A', n) → poly1, b, j, k, l, m, p, q, r, s, t, f_1, c, g_1, h_1, i_1, j_1, k_1, m_1, n_1, p_1, q_1, r_1, a, s_1, t_1, a_1, b_1, c_1, d_1, e_1, f_2, g_2, h_2, d, i_2, j_2, k_2, m_2, n_2, p_2, q_2, r_2, s_2, t_2, e, a_2, C, D, E, F, G, H, I, J, K, f, L, M, N, Q, S, T, U, V, W, Z, g, A_1, B_1, C_1, D_1, E_1, F_1, G_1, H_1, I_1, J_1, h, K_1, L_1, M_1, N_1, O_1, P_1, Q_1, R_1, S_1, T_1, i, U_1, V_1, W_1, Z_1, A_2, B_2, C_2, D_2, E_2
  3. Rotate(A', (((n - 2) * 90) * ° / n), B) → A''
  4. LineBisector(b) → l_1
  5. Line(B, A'') → l_2
  6. Intersect(l_1, l_2) → O
  7. Rotate(B, ((-2 * π / n) * (count - 1)), O) → P
  8. Sequence(((i * length), 0), i, 0, (turns * n)) → list1

AI Drawing Prompt Reference

Draw a dynamic illustration of a regular polygon rolling along a horizontal line, with sliders to control the number of sides, side length, roll angle, and number of turns; show the locus curves traced by each vertex and the contact points of each side with the ground; animate the motion using rotation and translation transformations, and display the derivative of the vertex speed as the polygon rolls.
The prompt above was compiled from the original drawing commands and is for reference only.