Shades and Shadows of Saddle Surface 2

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2020-04-22 04:00

This example demonstrates the shades and shadows of a saddle surface (hyperbolic paraboloid), involving core mathematical concepts like surfaces, indefinite integrals, hyperbolas, integers, triangles, and step functions. It utilizes commands such as Segment, Line, Intersect, OrthogonalLine, Point, Vector, Translate, Locus, and Polygon. Suitable for high school and above, it helps learners understand spatial geometry and calculus applications in 3D surfaces.

GeoGebra Commands & Parameters

Segment(D, A') → f
Segment(A', B) → g
Line(B, f) → h
Line(D, g) → i
Intersect(h, i) → C'
Segment(D, C') → j
Segment(C', B) → k
OrthogonalLine(A', g) → l
OrthogonalLine(C', j) → m
Point(l) → A
Point(m) → C
Segment(A, A') → n
Segment(C, C') → p
Segment(D, A) → q
Segment(A, B) → r
Segment(B, C) → s
Segment(C, D) → t
Line(D, C) → s_2
Intersect(s_2, r) → E
Line(E, n) → b
Intersect(g, b) → G
Line(B, j) → e
Line(G, k) → i_1
Intersect(j, i_1) → I
Segment(G, I) → j_1
Line(I, n) → l_1
Intersect(t, l_1) → K
Segment(K, I) → q_1
Line(C, B) → c
Intersect(c, q) → F
OrthogonalLine(F, g) → g_1
Intersect(g_1, f) → H
Vector(A', H) → u
Translate(B, u) → B'
OrthogonalLine(B', g) → k_1
Intersect(s, k_1) → J
Segment(J, B') → m_1
Segment(I, B') → n_k
Point(n_k) → L
Line(L, j_1) → p_1
Intersect(p_1, j) → M
Intersect(g, p_1) → N
Intersect(p, p_1) → O
Intersect(p, n) → P
Line(M, n) → t_1
Line(N, t_1) → a_1
Intersect(t, t_1) → Q
Intersect(a_1, r) → R
Segment(Q, R) → c_1
Line(L, t_1) → d_1
Intersect(c_1, d_1) → S
Locus(S, L) → ponthalmaz1
Segment(D, C) → r_1
Segment(C, B) → s_1
Segment(A, D) → b_1
Intersect(r_1, b_1) → T
Intersect(r_1, s_1) → U
Point(r_1) → V
Segment(A, B) → e_1
Polygon(A, D, U, B) → q1, a_2, d_2, u_1, b_2
Point(ponthalmaz1) → W
Point(ponthalmaz1) → Z
Point(ponthalmaz1) → A_1
Point(ponthalmaz1) → B_1
Point(ponthalmaz1) → C_1
Point(ponthalmaz1) → D_1
Point(ponthalmaz1) → E_1
Point(ponthalmaz1) → F_1
Point(ponthalmaz1) → G_1
Point(ponthalmaz1) → H_1
Point(ponthalmaz1) → I_1
Point(ponthalmaz1) → J_1
Polygon(U, K, G_1, H_1, I_1, J_1, J, W, Z, A_1, B_1, C_1, D_1, E_1, F_1) → poligon1, u_2, l_2, m_2, n_2, p_2, q_2, j_3, k_2, w, z_1, f_2, g_2, h_2, i_2, j_2
Segment(A', A_s) → a
Intersect(b_1, e_1) → L_1
Segment(L_1, A_s) → d
Line(C', a) → f_1
Line(U, d) → h_1
Intersect(f_1, h_1) → C_s
Segment(C', C_s) → n_1
Segment(U, C_s) → r_2
Segment(D, A_s) → t_2
Segment(C_s, B) → e_2
Intersect(r_1, b_1) → N_1
Segment(C_s, N_1) → c_2
Segment(B, A_s) → f_3
Intersect(s_1, e_1) → R_1
Line(R_1, A_s) → k_7
Line(A_s, N_1) → l_7
Intersect(c_2, k_7) → S_5
Intersect(e_2, l_7) → T_1
Segment(S_5, A_s) → m_7
Segment(A_s, T_1) → n_7
Line(S_5, a) → p_7
Line(T_1, a) → q_7
Intersect(q_7, k) → U_1
Intersect(p_7, j) → G_2
Segment(G_2, S_5) → r_7
Segment(U_1, T_1) → s_7
Line(U_1, j) → g_3
Line(G_2, k) → h_3
Intersect(g_3, f) → S_1'
Intersect(g, h_3) → S_2'
Segment(U_1, S_1') → i_3
Segment(S_2', G_2) → k_3
Segment(S_1', S_2') → n_s
Line(S_1', n) → l_3
Line(S_2', n) → m_3
Intersect(b_1, l_3) → S_1
Intersect(e_1, m_3) → S_2
Segment(S_1, S_1') → n_3
Segment(S_2', S_2) → p_3
Point(n_s) → Q_1
Line(Q_1, f) → q_3
Intersect(g, q_3) → V_1
Intersect(j, q_3) → W_1
Line(V_1, n) → r_3
Line(W_1, n) → s_3
Intersect(r_1, s_3) → Z_1
Intersect(r_3, e_1) → A_2
Segment(A_2, Z_1) → t_3
Line(Q_1, n) → a_3
Intersect(t_3, a_3) → B_2
Locus(B_2, Q_1) → ponthalmaz2
Line(S_2', a) → b_3
Line(S_1', a) → c_3
Intersect(t_2, c_3) → C_2
Intersect(b_3, f_3) → D_2
Segment(S_1', C_2) → d_3
Segment(S_2', D_2) → e_3
Line(Q_1, a) → f_4
Line(B_2, d) → g_4
Intersect(g_4, f_4) → E_2
Locus(E_2, Q_1) → ponthalmaz3
Intersect(i_3, k_3) → A_p'
Line(A_p', n) → h_4
Intersect(d, h_4) → A_p
Segment(A_p, A_p') → i_4
Segment(S_1, A_p) → j_4
Segment(A_p, S_2) → k_4
Line(G_2, i_4) → l_4
Line(U_1, i_4) → m_4
Intersect(l_4, r_1) → I_2
Intersect(m_4, s_1) → J_2
Segment(I_2, G_2) → n_4
Segment(U_1, J_2) → p_4
Segment(A_p, I_2) → q_4
Segment(A_p, J_2) → r_4
Point(ponthalmaz3) → K_1
Point(ponthalmaz3) → M_1
Point(ponthalmaz3) → O_1
Point(ponthalmaz3) → P_1
Point(ponthalmaz3) → F_2
Point(ponthalmaz3) → H_2
Point(ponthalmaz3) → C_4
Point(ponthalmaz3) → D_4
Point(ponthalmaz3) → E_4
Point(ponthalmaz3) → F_4
Point(ponthalmaz3) → G_4
Point(ponthalmaz3) → H_4
Point(ponthalmaz3) → I_4
Point(ponthalmaz3) → J_4
Polygon(R_1, C_s, C_4, D_4, E_4, F_4, G_4, H_4, I_4, J_4, D_2, N_1, C_2, K_1, M_1, O_1, P_1, F_2, H_2) → poligon5, j_7, g_8, h_8, i_8, j_8, k_8, l_8, m_8, n_8, p_8, q_8, c_s, t_7, a_7, b_7, c_7, d_7, e_7, f_8
Point(ponthalmaz2) → K_2
Point(ponthalmaz2) → L_2
Point(ponthalmaz2) → M_2
Point(ponthalmaz2) → N_2
Point(ponthalmaz2) → O_2
Point(ponthalmaz2) → P_2
Point(ponthalmaz2) → Q_2
Point(ponthalmaz2) → R_2
Point(ponthalmaz2) → S_4
Point(ponthalmaz2) → T_2
Point(ponthalmaz2) → U_2
Point(ponthalmaz2) → V_2
Point(ponthalmaz2) → W_2
Point(ponthalmaz2) → Z_2
Point(ponthalmaz2) → A_3
Point(ponthalmaz2) → B_3
Point(ponthalmaz2) → C_3
Point(ponthalmaz2) → D_3
Point(ponthalmaz2) → E_3
Point(ponthalmaz2) → F_3
Polygon(L_1, S_2, S_4, T_2, U_2, V_2, W_2, Z_2, A_3, B_3, C_3, D_3, K_2, E_3, F_3, S_1, L_2, M_2, N_2, O_2, P_2, Q_2, R_2) → poligon2, s_4, g_5, h_5, i_5, j_5, v_2, w_2, z_2, k_5, l_5, m_5, t_4, n_5, p_5, q_5, o_1, a_4, b_4, c_4, d_4, o_2, e_4, f_5
Polygon(A_p, S_2, S_4, T_2, U_2, V_2, W_2, Z_2, A_3, B_3, C_3, D_3, K_2, E_3, F_3, S_1, L_2, M_2, N_2, O_2, P_2, Q_2, R_2) → poligon3, r_5, g_6, h_6, i_6, j_6, k_6, l_6, m_6, n_6, p_6, q_6, s_5, r_6, s_6, t_6, a_6, t_5, a_5, b_5, c_5, d_5, e_5, f_6
Point(ponthalmaz1) → G_3
Point(ponthalmaz1) → H_3
Point(ponthalmaz1) → I_3
Point(ponthalmaz1) → J_3
Point(ponthalmaz1) → K_3
Point(r_4) → L_3
Point(ponthalmaz1) → M_3
Point(r_1) → N_3
Polygon(U, G_3, H_3, I_3, J_3, K_3, L_3, M_3, N_3) → poligon4, u_3, b_6, c_6, d_6, e_6, f_7, g_7, h_7, i_7
Point(ponthalmaz3) → O_3
Point(ponthalmaz3) → P_3
Point(ponthalmaz3) → Q_3
Point(ponthalmaz3) → R_3
Point(ponthalmaz3) → S_3
Point(ponthalmaz3) → T_3
Point(ponthalmaz3) → U_3
Point(ponthalmaz3) → V_3
Point(ponthalmaz3) → W_3
Point(ponthalmaz3) → Z_3
Point(ponthalmaz3) → A_4
Point(ponthalmaz3) → B_4