Composite Construction of Tangent Function and Cylinder Geometry (Part 3)

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2024-01-23 20:46

In collections Exploration: One Special Constant

① This example demonstrates a dynamic geometric system in GeoGebra that combines the tangent function graph with rectangles, line segments, and cylinders through transformations such as rotation, reflection, and translation. ② Core mathematical concepts include the properties of the tangent function, geometric mean, cylindrical solid geometry, and intersections of lines and circles. ③ Construction highlights: Based on the construction summary, commands used include Rotate, CircleArc, Mirror, Line, Intersect, Angle, Segment, Circle, OrthogonalLine, Point, Vector, Dilate, Translate, Ray, and If, involving objects like M, a, r, N, N', c, c', C, C', C''₁, O, t, g₁, H, h, i, D, β, D', D'', s₁, t₁, R, H', R', H'', C'', n₂, p₂, q₂, U, eq1, f, B, B'₂, b, f₁, d₁, I, J, forming a complex network of geometric relations. ④ Suitable for high school or college students studying trigonometry and geometric transformations, useful for exploring interactions between tangent functions and solid geometry, and as a teaching demonstration of dynamic mathematical models.

GeoGebra Commands & Parameters

Rotate(N, If[1 < a ≤ 2, (-(a - 1)), -1], M) → N'
CircleArc(M, N', N) → c
Mirror(c, yAxis) → c'
Rotate(C, If[1 < a ≤ 2, (-(a - 1)), -1], N') → C'
Rotate(C', -2, M) → C''_{1}
Rotate(C''_{1}, -2, M) → O
Line(O, C''_{1}) → t
Line(M, t) → g_1
Intersect(c', g_1) → H,
Line(C', C''_{1}) → h
Line(M, h) → i
Intersect(c, i, 1) → D
Angle(D, M, H) → β
Rotate(D, If[5 < a ≤ 6, ((a - 5) * β), β], M) → D'
Mirror(D', M) → D''
Segment(D', D'') → s_1
Circle(H, s_1 / 7) → t_1
Intersect(t_1, s_1, 1) → R
Mirror(H, R) → H'
Mirror(R, H') → R'
Mirror(H', R') → H''
Mirror(C', yAxis) → C''
Line(M, C'') → n_2
OrthogonalLine(C'', n_2) → p_2
Line(M, p_2) → q_2
Point(q_2) → U
Segment(((-r), 0), (r, 0)) → f
Rotate(B, If[7.5 < a ≤ 8.5, ((-(a - 7.5)) * 2), -2], (0, 0)) → B'_{2}
Line((0, 0), B'_{2}) → b
OrthogonalLine(B'_{2}, b) → f_1
Circle((0, 0), r) → d_1
Vector(I, J) → v
Rotate((r, 0), 1, (0, 0)) → K
Rotate((r, 0), 3, (0, 0)) → L
Mirror(K, y = 0) → K'
Dilate(M, If[0 < a ≤ 1, 1 - a, 0], N) → M'
Vector(M, M') → w
Translate(f, w) → f'_{2}
Mirror(N', yAxis) → N''
Mirror(A, yAxis) → A'
Circle(N', r - c) → d
Segment(N', C') → g
Mirror(N', yAxis) → N''_{1}
Segment(N''_{1}, C'') → g'
Rotate(C', -1, M) → C''_{2}
Angle(B, M, D) → α
Rotate(B, If[2 < a ≤ 3, ((a - 2) * α), α], M) → B'
Mirror(B', M) → B''
Segment(B'', B') → j
Dilate(M, If[3 < a ≤ 4, 4 - a, 0], C''_{2}) → M'_{1}
Vector(M, M'_{1}) → u
Translate(B'', u) → B'''
Translate(B', u) → B''_{1}
Segment(B''', B''_{1}) → j'
Rotate(C''_{2}, If[4 < a ≤ 5, a - 4, 1], M) → E
Rotate(C''_{2}, If[4 < a ≤ 5, (-(a - 4)), -1], M) → F
CircleArc(M, C''_{2}, E) → e
CircleArc(M, F, C''_{2}) → k
Circle(E, j / 2 - e) → p
Line(M, E) → l
OrthogonalLine(E, l) → m
Rotate(M, ((-pi)) / 2, E) → M'_{2}
Ray(E, M'_{2}) → n
Intersect(p, n, 1) → G
Rotate(M, pi / 2, E) → M'_{3}
Segment(E, G) → q
Line(M, C''_{2}) → s
Mirror(E, s) → E'
Mirror(G, s) → G'
Segment(E', G') → q'
Rotate(C''_{1}, -1, M) → P
Segment(D', D'') → h_1
Dilate(M, If[6 < a ≤ 7, 7 - a, 0], P) → M'_{4}
Vector(M, M'_{4}) → u_1
Translate(D', u_1) → D''_{1}
Translate(D'', u_1) → D'''
Segment(D''_{1}, D''') → i_1
Rotate(P, If[7 < a ≤ 8, a - 7, 1], M) → P'
Line(M, P) → j_1
Mirror(P', j_1) → P''
CircleArc(M, P, P') → c_1
CircleArc(M, P'', P) → e_1
Circle(P', h_1 / 2 - c_1) → k_1
Circle(P'', h_1 / 2 - c_1) → p_1
Line(M, P') → l_1
OrthogonalLine(P', l_1) → m_1
Rotate(M, ((-pi)) / 2, P') → M'_{5}
Ray(P', M'_{5}) → n_1
Intersect(k_1, n_1) → Q, W
Segment(P', Q) → q_1
Mirror(Q, j_1) → Q'
Segment(P'', Q') → r_1
Mirror(R', H'') → R''
Mirror(H'', R'') → H'''
Mirror(R'', H''') → R'''
Dilate(H, If[8 < a ≤ 9, 9 - a, 0], R) → H'_{1}
Segment(H, H'_{1}) → a_1
Rotate(H, If[9 < a ≤ 10, ((a - 9) * pi), pi], R) → H'_{2}
Segment(R, H'_{2}) → b_1
Rotate(R, If[10 < a ≤ 11, ((a - 10) * pi), pi], H') → R'_{1}
Segment(H', R'_{1}) → f_2
Rotate(H', If[11 < a ≤ 12, ((a - 11) * pi), pi], R') → H''_{1}
Segment(R', H''_{1}) → g_2
Rotate(R', If[12 < a ≤ 13, ((a - 12) * pi), pi], H'') → R''_{1}
Segment(H'', R''_{1}) → h_2
Rotate(H'', If[13 < a ≤ 14, ((a - 13) * pi), pi], R'') → H'''_{1}
Segment(R'', H'''_{1}) → i_2
Rotate(R'', If[14 < a ≤ 15, ((a - 14) * pi), pi], H''') → R'''_{1}
Segment(H''', R'''_{1}) → j_2
Line(M, C'') → k_2
Angle(U, M, H'') → γ
Dilate(H'', If[15.5 < a ≤ 16.5, 16.5 - a, 0], C'') → H'''_{2}
Vector(H'', H'''_{2}) → v_1
Dilate(R', If[15.5 < a ≤ 16.5, 16.5 - a, 0], C'') → R''_{2}
Vector(R', R''_{2}) → w_1
Translate(R', w_1) → R''_{3}
Translate(H''_{1}, w_1) → S
Segment(R''_{3}, S) → l_2
Rotate(R''_{3}, If[15.5 < a ≤ 16.5, ((-(a - 15.5)) * γ), (-γ)], R''_{2}) → T
Rotate(S, If[15.5 < a ≤ 16.5, ((-(a - 15.5)) * γ), (-γ)], R''_{2}) → S'
Segment(T, S') → m_2
Rotate(C'', If[16.5 < a ≤ 17.5, ((a - 16.5) * 0.28), 0.28], M) → C'''
CircleArc(M, C'', C''') → c_2
Circle(C''', (0.14 * s_1) - c_2) → d_2
Line(C''', M) → r_2
OrthogonalLine(C''', r_2) → s_2
Rotate(M, ((-pi)) / 2, C''') → M'_{6}
Ray(C''', M'_{6}) → t_2
Intersect(d_2, t_2, 1) → V
Segment(C''', V) → a_2
If(8 < a ≤ 15, 0.75, 0.25) → o_1

AI Drawing Prompt Reference

Create a dynamic geometric system combining a tangent function graph, rectangles, line segments, and cylinders through transformations like rotation, reflection, and translation. Highlight the properties of the tangent function, geometric mean, cylindrical solid geometry, and intersections of lines and circles. Start by drawing the tangent function graph, then create related rectangles and line segments, generate cylinders through rotations and reflections, and finally display the dynamic relationships between all geometric elements.

The prompt above was compiled from the original drawing commands and is for reference only.