Sphere with Great Circle by Oystein

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2013-01-30 20:22

Circle Geometric Construction

This example demonstrates the dynamic changes of a sphere and its great circle. By adjusting the sphere's position, radius, and points P and Q, one can observe the changes in the sphere and great circle. Key mathematical concepts include spheres, great circles, and geometric construction. The example uses commands such as Line, Circle, Intersect, Rotate, and Dilate in GeoGebra. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

Line(AA, BB) → a
OrthogonalLine(CC, a) → b
OrthogonalLine(CC, b) → c
Circle(CC, l_{11}) → d
Intersect(d, c) → D, E
OrthogonalLine(D, c) → g
OrthogonalLine(E, c) → i
Circle(CC, br) → e
Intersect(e, b) → F, G
OrthogonalLine(G, b) → f
Intersect(g, f) → I
Intersect(f, i) → H
OrthogonalLine(F, b) → h
Intersect(g, h) → J
Intersect(h, i) → K
Polygon(H, I, J, K) → mangekant1, h_1, i_1, j, k_{10}
OrthogonalLine(Rotate, h_1) → m
Intersect(h_1, m) → M
Distance(Rotate, M) → avstandLM
If((-π < ve) ∧ (ve < (-π) / 2) ∨ (0 < ve) ∧ (ve < π / 2), (-van * zp) / (size - zp) - 3, van * zp / (size - zp) + 3) → iz
Circle(O, size) → r_8
OrthogonalLine(O, a) → p
Intersect(r_8, p) → R, S_{17}
OrthogonalLine(S_{17}, p) → s
OrthogonalLine(R, p) → a_1
OrthogonalLine(O, p) → q
Intersect(r_8, q) → P_{13}, Q_{18}
OrthogonalLine(P_{13}, q) → t
Intersect(s, t) → U
Intersect(a_1, t) → V
OrthogonalLine(Rotate, k_{10}) → n
Intersect(k_{10}, n) → N
Distance(Rotate, N) → avstandLN
Rotate(U, ho, O) → U'
OrthogonalLine(U', q) → k_1
Intersect(q, k_1) → B_1
Rotate(V, ho, O) → V'
OrthogonalLine(V', q) → j_1
Intersect(q, j_1) → C_1
Dilate(U', l1, B_1) → U''
Dilate(V', l1, C_1) → V''
Midpoint(V'', U'') → E_1
Distance(O, E_1) → avstandOE1
If((0 < ho) ∧ (ho < π) ∧ ((-π) / 2 < ve) ∧ (ve < π / 2) ∨ (-π < ho) ∧ (ho < 0) ∧ ((-π < ve) ∧ (ve < (-π) / 2) ∨ (π / 2 < ve) ∧ (ve < π)), van * yp / (size - yp) + 3, (-van * yp) / (size - yp) - 3) → iy
OrthogonalLine(Q_{18}, q) → b_1
Intersect(a_1, b_1) → W
Rotate(W, ho, O) → W'
OrthogonalLine(W', q) → l_1
Intersect(l_1, q) → A_1
Dilate(W', l1, A_1) → W''
Midpoint(W'', V'') → D_1
Distance(O, D_1) → avstandOD1
If(((-π < ve) ∧ (ve < (-π) / 2) ∨ (π / 2 < ve) ∧ (ve < π)) ∧ ((-π) / 2 < ho) ∧ (ho < π / 2) ∨ ((-π) / 2 < ve) ∧ (ve < π / 2) ∧ ((-π < ho) ∧ (ho < (-π) / 2) ∨ (π / 2 < ho) ∧ (ho < π)), van * xp / (size - xp) + 3, (-van * xp) / (size - xp) - 3) → ix
Intersect(s, b_1) → T
Polygon(U, T, W, V) → mangekant2, u, t_1, w, v
Rotate(T, ho, O) → T'
Polygon(U', T', W', V') → mangekant2', u', t', w', v'
OrthogonalLine(T', q) → c_1
OrthogonalLine(W', q) → d_1
OrthogonalLine(V', q) → e_1
OrthogonalLine(U', q) → f_1
OrthogonalLine(T', q) → g_1
Intersect(g_1, q) → Z
Distance(Z, T') → avstandZT'
Distance(B_1, U') → avstandB1U'
Dilate(T', l1, Z) → T''
Polygon(T'', W'', V'', U'') → mangekant3, t'', w'', v'', u''
Dilate(D_1, ix, O) → F_1
Dilate(E_1, iy, O) → G_1
Circle(O, size) → p_1
Circle(O, zp) → q_1
Intersect(q_1, p) → J_1, K_1
Dilate(K_1, iz, O) → H_1
Dilate(D_1, xpos, O) → I_1
Dilate(D_1, xneg, O) → L_1
Dilate(E_1, ypos, O) → M_1
Dilate(E_1, yneg, O) → N_1
Line(M_1, F_1) → m_1
Line(N_1, F_1) → n_1
Line(L_1, G_1) → r_1
Line(I_1, G_1) → s_1
Intersect(n_1, r_1) → O_1
Intersect(m_1, r_1) → P_1
Intersect(s_1, m_1) → Q_1
Intersect(s_1, n_1) → R_1
Polygon(P_1, O_1, R_1, Q_1) → mangekant4, p_2, o_1, r_2, q_2
Dilate(K_1, zpos, O) → S_1
Dilate(K_1, zneg, O) → T_1
Segment(I_1, L_1) → a_2
Segment(N_1, M_1) → b_2
Line(F_1, S_1) → c_2
Line(T_1, F_1) → d_2
Line(S_1, G_1) → e_2
Line(T_1, G_1) → f_2
Line(H_1, N_1) → g_2
Line(H_1, I_1) → h_2
Line(H_1, L_1) → i_2
Line(H_1, M_1) → j_2
Intersect(g_2, e_2) → V_1
Intersect(g_2, f_2) → W_1
Intersect(e_2, j_2) → Z_1
Intersect(f_2, j_2) → A_2
Line(V_1, F_1) → k_2
Line(Z_1, F_1) → l_2
Line(W_1, F_1) → m_2
Line(A_2, F_1) → n_2
Intersect(i_2, c_2) → B_2
Intersect(i_2, d_2) → C_2
Intersect(c_2, h_2) → D_2
Intersect(h_2, d_2) → E_2
Line(B_2, G_1) → s_2
Line(D_2, G_1) → t_2
Line(E_2, G_1) → a_3
Line(C_2, G_1) → b_3
Intersect(k_2, s_2) → F_2
Intersect(s_2, l_2) → G_2
Intersect(t_2, l_2) → H_2
Intersect(t_2, k_2) → I_2
Intersect(m_2, b_3) → J_2
Intersect(m_2, a_3) → K_2
Intersect(a_3, n_2) → L_2
Intersect(n_2, b_3) → M_2
Segment(F_2, G_2) → c_3
Segment(G_2, H_2) → d_3
Segment(H_2, I_2) → e_3
Segment(I_2, F_2) → f_3
Segment(F_2, J_2) → g_3
Segment(I_2, K_2) → h_3
Segment(L_2, H_2) → i_3
Segment(G_2, M_2) → j_3
Segment(M_2, L_2) → k_3
Segment(L_2, K_2) → l_3
Segment(K_2, J_2) → m_3
Segment(J_2, M_2) → n_3
Segment(T_1, S_1) → p_3
Dilate(I_1, enx, O) → N_2
Dilate(I_1, tox, O) → O_2
Dilate(I_1, trex, O) → P_2
Dilate(I_1, fix, O) → Q_2
Dilate(I_1, fex, O) → R_2
Dilate(I_1, enxm, O) → S_2
Dilate(I_1, six, O) → T_2
Dilate(I_1, sevx, O) → U_2
Dilate(I_1, eix, O) → V_2
Dilate(I_1, nix, O) → W_2
Dilate(I_1, tex, O) → Z_2
Dilate(I_1, toxm, O) → A_3
Dilate(I_1, trexm, O) → B_3
Dilate(I_1, fixm, O) → C_3
Dilate(I_1, fexm, O) → D_3
Dilate(I_1, sixm, O) → E_3
Dilate(I_1, sevxm, O) → F_3
Dilate(I_1, eixm, O) → G_3
Dilate(I_1, nixm, O) → H_3
Dilate(I_1, texm, O) → I_3
Line(Z_2, G_1) → q_3
Line(W_2, G_1) → r_3
Line(V_2, G_1) → s_3
Line(U_2, G_1) → t_3
Line(T_2, G_1) → a_4
Line(R_2, G_1) → b_4
Line(Q_2, G_1) → c_4
Line(P_2, G_1) → d_4
Line(O_2, G_1) → e_4
Line(N_2, G_1) → f_4
Line(S_2, G_1) → g_4
Line(A_3, G_1) → h_4
Line(B_3, G_1) → i_4
Line(C_3, G_1) → j_4
Line(D_3, G_1) → k_4
Line(E_3, G_1) → l_4
Line(F_3, G_1) → m_4
Line(G_3, G_1) → n_4
Line(H_3, G_1) → p_4
Line(I_3, G_1) → q_4
Intersect(q_2, q_4) → J_3
Intersect(q_2, p_4) → K_3
Intersect(q_2, n_4) → L_3
Intersect(q_2, m_4) → M_3
Intersect(q_2, l_4) → N_3
Intersect(q_2, k_4) → O_3
Intersect(q_2, j_4) → P_3
Intersect(q_2, i_4) → Q_3
Intersect(q_2, h_4) → R_3
Intersect(q_2, g_4) → S_3
Intersect(q_2, f_4) → T_3
Intersect(q_2, e_4) → U_3
Intersect(q_2, d_4) → V_3
Intersect(q_2, c_4) → W_3
Intersect(q_2, b_4) → Z_3
Intersect(q_2, a_4) → A_4
Intersect(q_2, t_3) → B_4
Intersect(q_2, s_3) → C_4
Intersect(q_2, r_3) → D_4
Intersect(q_2, q_3) → E_4
Intersect(o_1, q_3) → F_4
Intersect(o_1, r_3) → G_4
Intersect(o_1, s_3) → H_4
Intersect(o_1, t_3) → I_4
Intersect(o_1, a_4) → J_4
Intersect(o_1, b_4) → K_4
Intersect(o_1, c_4) → L_4
Intersect(o_1, d_4) → M_4
Intersect(o_1, e_4) → N_4
Intersect(o_1, f_4) → O_4
Intersect(o_1, g_4) → P_4
Intersect(o_1, h_4) → Q_4
Intersect(o_1, i_4) → R_4
Intersect(o_1, j_4) → S_4
Intersect(o_1, k_4) → T_4
Intersect(o_1, m_4) → W_4
Intersect(o_1, n_4) → Z_4
Intersect(o_1, p_4) → A_5
Intersect(o_1, q_4) → B_5
Segment(B_5, J_3) → r_4
Segment(A_5, K_3) → s_4
Segment(Z_4, L_3) → t_4
Segment(W_4, M_3) → a_5
Segment(T_4, O_3) → c_5
Segment(S_4, P_3) → d_5
Segment(R_4, Q_3) → e_5
Segment(Q_4, R_3) → f_5
Segment(P_4, S_3) → g_5
Segment(O_4, T_3) → h_5
Segment(N_4, U_3) → i_5
Segment(M_4, V_3) → j_5
Segment(L_4, W_3) → k_5
Segment(K_4, Z_3) → l_5
Segment(J_4, A_4) → m_5
Segment(I_4, B_4) → n_5
Segment(H_4, C_4) → p_5
Segment(G_4, D_4) → q_5
Segment(F_4, E_4) → r_5
Dilate(M_1, eny, O) → C_5
Dilate(M_1, toy, O) → D_5
Dilate(M_1, trey, O) → E_5
Dilate(M_1, firy, O) → F_5
Dilate(M_1, femy, O) → G_5
Dilate(M_1, sixy, O) → H_5
Dilate(M_1, sevy, O) → I_5
Dilate(M_1, eiy, O) → J_5
Dilate(M_1, niy, O) → K_5
Dilate(M_1, tey, O) → L_5
Dilate(M_1, toym, O) → M_5
Dilate(M_1, enym, O) → N_5
Dilate(M_1, treym, O) → O_5
Dilate(M_1, firym, O) → P_5
Dilate(M_1, sixym, O) → Q_5
Dilate(M_1, femym, O) → R_5
Dilate(M_1, sevym, O) → S_5
Dilate(M_1, eiym, O) → T_5
Dilate(M_1, niym, O) → U_5
Dilate(M_1, teym, O) → V_5
Line(V_5, F_1) → s_5
Line(U_5, F_1) → t_5
Line(T_5, F_1) → a_6
Line(S_5, F_1) → b_6
Line(Q_5, F_1) → c_6
Line(R_5, F_1) → d_6
Line(P_5, F_1) → e_6
Line(O_5, F_1) → f_6
Line(M_5, F_1) → g_6
Line(N_5, F_1) → h_6
Line(C_5, F_1) → i_6
Line(D_5, F_1) → j_6
Line(E_5, F_1) → k_6
Line(F_5, F_1) → l_6
Line(G_5, F_1) → m_6
Line(H_5, F_1) → n_6
Line(I_5, F_1) → p_6
Line(J_5, F_1) → q_6
Line(K_5, F_1) → r_6
Line(L_5, F_1) → s_6
Intersect(p_2, s_5) → W_5
Intersect(p_2, t_5) → Z_5
Intersect(p_2, a_6) → A_6
Intersect(p_2, b_6) → B_6
Intersect(p_2, c_6) → C_6
Intersect(p_2, d_6) → D_6
Intersect(p_2, e_6) → E_6
Intersect(p_2, f_6) → F_6
Intersect(p_2, g_6) → G_6
Intersect(p_2, h_6) → H_6
Intersect(p_2, i_6) → I_6
Intersect(p_2, j_6) → J_6
Intersect(p_2, k_6) → K_6
Intersect(p_2, l_6) → L_6
Intersect(p_2, m_6) → M_6
Intersect(p_2, n_6) → N_6
Intersect(p_2, p_6) → O_6
Intersect(p_2, q_6) → P_6
Intersect(p_2, r_6) → Q_6
Intersect(p_2, s_6) → R_6
Intersect(r_2, s_6) → S_6
Intersect(r_2, r_6) → T_6
Intersect(r_2, q_6) → U_6
Intersect(r_2, p_6) → V_6
Intersect(r_2, n_6) → W_6
Intersect(r_2, m_6) → Z_6
Intersect(r_2, l_6) → A_7
Intersect(r_2, k_6) → B_7
Intersect(r_2, j_6) → C_7
Intersect(r_2, i_6) → D_7
Intersect(r_2, h_6) → E_7
Intersect(r_2, g_6) → F_7
Intersect(r_2, f_6) → G_7
Intersect(r_2, e_6) → H_7
Intersect(r_2, d_6) → I_7
Intersect(r_2, c_6) → J_7
Intersect(r_2, b_6) → K_7
Intersect(r_2, a_6) → L_7
Intersect(r_2, t_5) → M_7
Intersect(r_2, s_5) → N_7
Segment(N_7, W_5) → t_6
Segment(M_7, Z_5) → a_7
Segment(L_7, A_6) → b_7
Segment(K_7, B_6) → c_7
Segment(J_7, C_6) → d_7
Segment(I_7, D_6) → e_7
Segment(H_7, E_6) → f_7
Segment(G_7, F_6) → g_7
Segment(F_7, G_6) → h_7
Segment(E_7, H_6) → i_7
Segment(D_7, I_6) → j_7
Segment(C_7, J_6) → k_7
Segment(B_7, K_6) → l_7
Segment(A_7, L_6) → m_7
Segment(Z_6, M_6) → n_7
Segment(W_6, N_6) → p_7
Segment(V_6, O_6) → q_7
Segment(U_6, P_6) → r_7
Segment(T_6, Q_6) → s_7
Segment(S_6, R_6) → t_7
Intersect(o_1, l_4) → U_4
Intersect(q_2, l_4) → V_4
Segment(U_4, V_4) → b_5
Dilate(S_1, enz, O) → O_7
Vector(O, N_2) → z
Vector(O, C_5) → u_1
Vector(O, O_7) → v_1
Dilate(I_1, px, O) → A
Dilate(M_1, py, O) → B
Dilate(S_1, pz, O) → C
Line(H_1, A) → a_8
Line(H_1, B) → b_8
Line(A, G_1) → c_8
Line(C, G_1) → d_8
Line(B, F_1) → e_8
Line(C, F_1) → f_8
Intersect(a_8, f_8) → P_7
Intersect(b_8, d_8) → Q_7
Line(Q_7, F_1) → g_8
Line(P_7, G_1) → h_8
Intersect(g_8, h_8) → R_7
Locus(R_7, α) → loc1
Locus(R_7, β) → loc2
Dilate(I_1, px1, O) → S_7
Dilate(M_1, py1, O) → T_7
Line(H_1, T_7) → i_8
Line(H_1, S_7) → j_8
Line(T_7, F_1) → k_8
Line(S_7, G_1) → l_8
Intersect(i_8, d_8) → U_7
Intersect(j_8, f_8) → V_7
Line(V_7, G_1) → m_8
Line(U_7, F_1) → n_8
Intersect(n_8, m_8) → W_7
Locus(W_7, α) → loc3
Dilate(I_1, px0, O) → Z_7
Dilate(M_1, py0, O) → A_8
Line(H_1, Z_7) → p_8
Line(H_1, A_8) → q_8
Line(Z_7, G_1) → s_8
Line(A_8, F_1) → t_8
Intersect(q_8, d_8) → B_8
Intersect(p_8, f_8) → C_8
Line(B_8, F_1) → a_9
Line(C_8, G_1) → b_9
Intersect(b_9, a_9) → D_8
Locus(D_8, α) → loc4
Dilate(I_1, px2, O) → E_8
Dilate(M_1, py2, O) → F_8
Line(H_1, F_8) → c_9
Line(H_1, E_8) → d_9
Intersect(c_9, d_8) → G_8
Intersect(f_8, d_9) → H_8
Line(G_8, F_1) → e_9
Line(H_8, G_1) → f_9
Intersect(f_9, e_9) → I_8
Locus(I_8, α) → loc5
Dilate(I_1, px3, O) → J_8
Dilate(M_1, py3, O) → K_8
Line(H_1, K_8) → g_9
Line(H_1, J_8) → h_9
Intersect(h_9, f_8) → L_8
Intersect(g_9, d_8) → M_8
Line(L_8, G_1) → i_9
Line(M_8, F_1) → j_9
Intersect(i_9, j_9) → N_8
Locus(N_8, α) → loc6
Dilate(I_1, px4, O) → O_8
Dilate(M_1, py4, O) → P_8
Line(H_1, P_8) → k_9
Line(H_1, O_8) → l_9
Intersect(k_9, d_8) → Q_8
Intersect(l_9, f_8) → R_8
Line(Q_8, F_1) → m_9
Line(R_8, G_1) → n_9
Intersect(n_9, m_9) → S_8
Locus(S_8, α) → loc7
Dilate(I_1, px5, O) → T_8
Dilate(M_1, py5, O) → U_8
Line(H_1, U_8) → p_9
Line(H_1, T_8) → q_9
Intersect(p_9, d_8) → V_8
Intersect(q_9, f_8) → W_8
Line(V_8, F_1) → r_9
Line(W_8, G_1) → s_9
Intersect(r_9, s_9) → Z_8
Locus(Z_8, α) → loc8
Dilate(I_1, px6, O) → A_9
Dilate(M_1, py6, O) → B_9
Line(H_1, B_9) → t_9
Line(H_1, A_9) → a_{10}
Intersect(t_9, d_8) → C_9
Intersect(a_{10}, f_8) → D_9
Line(C_9, F_1) → b_{10}
Line(D_9, G_1) → c_{10}
Intersect(b_{10}, c_{10}) → E_9
Locus(E_9, α) → loc9
Dilate(I_1, px7, O) → F_9
Dilate(M_1, py7, O) → G_9
Line(H_1, G_9) → d_{10}
Line(H_1, F_9) → e_{10}
Intersect(d_{10}, d_8) → H_9
Intersect(e_{10}, f_8) → I_9
Line(H_9, F_1) → f_{10}
Line(I_9, G_1) → g_{10}
Intersect(g_{10}, f_{10}) → J_9
Locus(J_9, α) → loc10
Dilate(I_1, px8, O) → K_9
Dilate(M_1, py8, O) → L_9
Dilate(I_1, px8, O) → M_9
Dilate(M_1, py8, O) → N_9
Line(H_1, N_9) → h_{10}
Line(H_1, M_9) → i_{10}
Intersect(h_{10}, d_8) → O_9
Intersect(i_{10}, f_8) → P_9
Line(O_9, F_1) → j_{10}
Line(P_9, G_1) → l_{10}
Intersect(j_{10}, l_{10}) → Q_9
Locus(Q_9, α) → loc11
Dilate(I_1, px9, O) → R_9
Dilate(M_1, py9, O) → S_9
Line(H_1, S_9) → m_{10}
Line(H_1, R_9) → n_{10}
Intersect(m_{10}, d_8) → T_9
Intersect(n_{10}, f_8) → U_9
Line(T_9, F_1) → p_{10}
Line(U_9, G_1) → q_{10}
Intersect(p_{10}, q_{10}) → V_9
Locus(V_9, α) → loc12
Dilate(I_1, qx1, O) → W_9
Dilate(M_1, qy1, O) → Z_9
Dilate(S_1, qz1, O) → A_{10}
Line(H_1, Z_9) → r_{10}
Line(H_1, W_9) → s_{10}
Line(W_9, G_1) → t_{10}
Line(A_{10}, G_1) → a_{11}
Line(Z_9, F_1) → b_{11}
Line(A_{10}, F_1) → c_{11}
Intersect(s_{10}, c_{11}) → B_{10}
Intersect(r_{10}, a_{11}) → C_{10}
Line(C_{10}, F_1) → d_{11}
Line(B_{10}, G_1) → e_{11}
Intersect(d_{11}, e_{11}) → D_{10}
Locus(D_{10}, β) → loc13
Dilate(I_1, qx2, O) → E_{10}
Dilate(M_1, qy2, O) → F_{10}
Dilate(S_1, qz2, O) → G_{10}
Line(H_1, F_{10}) → f_{11}
Line(H_1, E_{10}) → g_{11}
Line(G_{10}, G_1) → h_{11}
Line(G_{10}, F_1) → i_{11}
Intersect(g_{11}, i_{11}) → H_{10}
Intersect(f_{11}, h_{11}) → I_{10}
Line(I_{10}, F_1) → j_{11}
Line(H_{10}, G_1) → k_{11}
Intersect(k_{11}, j_{11}) → J_{10}
Locus(J_{10}, β) → loc14
Dilate(I_1, qx3, O) → K_{10}
Dilate(M_1, qy3, O) → L_{10}
Dilate(S_1, qz3, O) → M_{10}
Line(H_1, K_{10}) → m_{11}
Line(H_1, L_{10}) → n_{11}
Line(M_{10}, G_1) → p_{11}
Line(M_{10}, F_1) → q_{11}
Intersect(m_{11}, q_{11}) → N_{10}
Intersect(n_{11}, p_{11}) → O_{10}
Line(O_{10}, F_1) → r_{11}
Line(N_{10}, G_1) → s_{11}
Intersect(r_{11}, s_{11}) → P_{10}
Locus(P_{10}, β) → loc15
Dilate(I_1, qx4, O) → Q_{10}
Dilate(M_1, qy4, O) → R_{10}
Dilate(S_1, qz4, O) → S_{10}
Line(H_1, Q_{10}) → t_{11}
Line(H_1, R_{10}) → a_{12}
Line(S_{10}, G_1) → b_{12}
Line(S_{10}, F_1) → c_{12}
Intersect(t_{11}, c_{12}) → T_{10}
Intersect(a_{12}, b_{12}) → U_{10}
Line(U_{10}, F_1) → d_{12}
Line(T_{10}, G_1) → e_{12}
Intersect(e_{12}, d_{12}) → V_{10}
Locus(V_{10}, β) → loc16
Dilate(I_1, qx5, O) → W_{10}
Dilate(M_1, qy5, O) → Z_{10}
Dilate(S_1, qz5, O) → A_{11}
Line(H_1, W_{10}) → f_{12}
Line(H_1, Z_{10}) → g_{12}
Line(A_{11}, G_1) → h_{12}
Line(A_{11}, F_1) → i_{12}
Intersect(g_{12}, h_{12}) → B_{11}
Intersect(f_{12}, i_{12}) → C_{11}
Line(B_{11}, F_1) → j_{12}
Line(C_{11}, G_1) → k_{12}
Intersect(j_{12}, k_{12}) → D_{11}
Locus(D_{11}, β) → loc17
Dilate(I_1, qx6, O) → E_{11}
Dilate(M_1, qy6, O) → F_{11}
Dilate(S_1, qz6, O) → G_{11}
Line(H_1, F_{11}) → l_{12}
Line(H_1, E_{11}) → m_{12}
Line(G_{11}, G_1) → n_{12}
Line(G_{11}, F_1) → p_{12}
Intersect(l_{12}, n_{12}) → H_{11}
Intersect(m_{12}, p_{12}) → I_{11}
Line(H_{11}, F_1) → q_{12}
Line(I_{11}, G_1) → r_{12}
Intersect(q_{12}, r_{12}) → J_{11}
Locus(J_{11}, β) → loc18
Dilate(I_1, qx7, O) → K_{11}
Dilate(M_1, qy7, O) → L_{11}
Dilate(S_1, qz7, O) → M_{11}
Line(H_1, L_{11}) → s_{12}
Line(H_1, K_{11}) → t_{12}
Line(M_{11}, G_1) → a_{13}
Line(M_{11}, F_1) → b_{13}
Intersect(s_{12}, a_{13}) → N_{11}
Intersect(t_{12}, b_{13}) → O_{11}
Line(N_{11}, F_1) → c_{13}
Line(O_{11}, G_1) → d_{13}
Intersect(d_{13}, c_{13}) → P_{11}
Locus(P_{11}, β) → loc19
Dilate(I_1, qx8, O) → Q_{11}
Dilate(M_1, qy8, O) → R_{11}
Dilate(S_1, qz8, O) → S_{11}
Line(H_1, Q_{11}) → e_{13}
Line(H_1, R_{11}) → f_{13}
Line(S_{11}, G_1) → g_{13}
Line(S_{11}, F_1) → h_{13}
Intersect(f_{13}, g_{13}) → T_{11}
Intersect(e_{13}, h_{13}) → U_{11}
Line(T_{11}, F_1) → i_{13}
Line(U_{11}, G_1) → j_{13}
Intersect(i_{13}, j_{13}) → V_{11}
Locus(V_{11}, β) → loc20
Dilate(I_1, qx9, O) → W_{11}
Dilate(M_1, qy9, O) → Z_{11}
Dilate(S_1, qz9, O) → A_{12}
Line(H_1, W_{11}) → k_{13}
Line(H_1, Z_{11}) → l_{13}
Line(A_{12}, G_1) → m_{13}
Line(A_{12}, F_1) → n_{13}
Intersect(l_{13}, m_{13}) → B_{12}
Intersect(k_{13}, n_{13}) → C_{12}
Line(B_{12}, F_1) → p_{13}
Line(C_{12}, G_1) → q_{13}
Intersect(p_{13}, q_{13}) → D_{12}
Locus(D_{12}, β) → loc21
Dilate(I_1, xx1, O) → E_{12}
Dilate(M_1, yy1, O) → F_{12}
Dilate(S_1, zz1, O) → G_{12}
Line(H_1, F_{12}) → r_{13}
Line(H_1, E_{12}) → s_{13}
Line(G_{12}, G_1) → t_{13}
Line(G_{12}, F_1) → a_{14}
Intersect(r_{13}, t_{13}) → H_{12}
Intersect(s_{13}, a_{14}) → I_{12}
Line(H_{12}, F_1) → b_{14}
Line(I_{12}, G_1) → c_{14}
Intersect(c_{14}, b_{14}) → S
Line(E_{12}, G_1) → d_{14}
Line(F_{12}, F_1) → e_{14}
Intersect(d_{14}, e_{14}) → K_{12}
Segment(F_{12}, K_{12}) → f_{14}
Segment(K_{12}, E_{12}) → g_{14}
Segment(K_{12}, S) → h_{14}
Segment(S, G_{12}) → i_{14}
Dilate(I_1, sx1, O) → L
Dilate(M_1, sy1, O) → L_{12}
Dilate(S_1, sz1, O) → M_{12}
Line(H_1, L_{12}) → j_{14}
Line(H_1, L) → k_{14}
Line(M_{12}, F_1) → l_{14}
Line(L_{12}, F_1) → m_{14}
Line(L, G_1) → n_{14}
Line(M_{12}, G_1) → p_{14}
Intersect(j_{14}, p_{14}) → N_{12}
Intersect(k_{14}, l_{14}) → O_{12}
Line(O_{12}, G_1) → q_{14}
Line(N_{12}, F_1) → r_{14}
Intersect(q_{14}, r_{14}) → P
Intersect(n_{14}, m_{14}) → Q_{12}
Segment(L_{12}, Q_{12}) → s_{14}
Segment(Q_{12}, L) → t_{14}
Segment(Q_{12}, P) → a_{15}
Segment(P, M_{12}) → b_{15}
Vector(O, S) → w_1
Vector(S, P) → z_1
Vector(O, P) → u_2
Textfield(α) → inputBox1
Textfield(β) → inputBox2
Textfield(r) → inputBox3
Textfield(x1) → inputBox4
Textfield(y1) → inputBox5
Textfield(z1) → inputBox6
Dilate(I_1, px10, O) → P_{12}
Dilate(M_1, py10, O) → R_{12}
Dilate(S_1, pz, O) → S_{12}
Line(H_1, R_{12}) → c_{15}
Line(H_1, P_{12}) → d_{15}
Line(M_{12}, F_1) → e_{15}
Line(M_{12}, G_1) → f_{15}
Intersect(c_{15}, f_{15}) → T_{12}
Intersect(d_{15}, e_{15}) → U_{12}
Line(U_{12}, G_1) → g_{15}
Line(T_{12}, F_1) → h_{15}
Intersect(g_{15}, h_{15}) → V_{12}
Locus(V_{12}, α) → loc22
Dilate(I_1, px11, O) → W_{12}
Dilate(M_1, py11, O) → Z_{12}
Line(H_1, Z_{12}) → i_{15}
Line(H_1, W_{12}) → j_{15}
Intersect(j_{15}, e_{15}) → A_{13}
Intersect(i_{15}, f_{15}) → B_{13}
Line(B_{13}, F_1) → k_{15}
Line(A_{13}, G_1) → l_{15}
Intersect(l_{15}, k_{15}) → C_{13}
Locus(C_{13}, α) → loc23
Dilate(I_1, px12, O) → D_{13}
Dilate(M_1, py12, O) → E_{13}
Line(H_1, E_{13}) → m_{15}
Line(H_1, D_{13}) → n_{15}
Intersect(n_{15}, e_{15}) → F_{13}
Intersect(m_{15}, f_{15}) → G_{13}
Line(G_{13}, F_1) → p_{15}
Line(F_{13}, G_1) → q_{15}
Intersect(q_{15}, p_{15}) → H_{13}
Locus(H_{13}, α) → loc24
Dilate(I_1, px13, O) → I_{13}
Dilate(M_1, py13, O) → J_{13}
Line(H_1, J_{13}) → r_{15}
Line(H_1, I_{13}) → s_{15}
Intersect(r_{15}, f_{15}) → K_{13}
Intersect(s_{15}, e_{15}) → L_{13}
Line(K_{13}, F_1) → t_{15}
Line(L_{13}, G_1) → a_{16}
Intersect(a_{16}, t_{15}) → M_{13}
Locus(M_{13}, α) → loc25
Dilate(I_1, px14, O) → N_{13}
Dilate(M_1, py14, O) → O_{13}
Line(H_1, O_{13}) → b_{16}
Line(H_1, N_{13}) → c_{16}
Intersect(b_{16}, f_{15}) → Q_{13}
Intersect(c_{16}, e_{15}) → R_{13}
Line(R_{13}, G_1) → d_{16}
Line(Q_{13}, F_1) → e_{16}
Intersect(d_{16}, e_{16}) → S_{13}
Locus(S_{13}, α) → loc26
Dilate(I_1, px15, O) → T_{13}
Dilate(M_1, py15, O) → U_{13}
Line(H_1, U_{13}) → f_{16}
Line(H_1, T_{13}) → g_{16}
Intersect(g_{16}, e_{15}) → V_{13}
Intersect(f_{16}, f_{15}) → W_{13}
Line(V_{13}, G_1) → h_{16}
Line(W_{13}, F_1) → i_{16}
Intersect(i_{16}, h_{16}) → Z_{13}
Locus(Z_{13}, α) → loc27
Dilate(I_1, px16, O) → A_{14}
Dilate(M_1, py16, O) → B_{14}
Line(H_1, B_{14}) → j_{16}
Line(H_1, A_{14}) → k_{16}
Intersect(k_{16}, e_{15}) → C_{14}
Intersect(j_{16}, f_{15}) → D_{14}
Line(D_{14}, F_1) → l_{16}
Line(C_{14}, G_1) → m_{16}
Intersect(m_{16}, l_{16}) → E_{14}
Locus(E_{14}, α) → loc28
Dilate(I_1, px17, O) → F_{14}
Dilate(M_1, py17, O) → G_{14}
Line(H_1, G_{14}) → n_{16}
Line(H_1, F_{14}) → p_{16}
Intersect(n_{16}, f_{15}) → H_{14}
Intersect(p_{16}, e_{15}) → I_{14}
Line(I_{14}, G_1) → q_{16}
Line(H_{14}, F_1) → r_{16}
Intersect(q_{16}, r_{16}) → J_{14}
Locus(J_{14}, α) → loc29
Dilate(I_1, qx10, O) → K_{14}
Dilate(M_1, qy10, O) → L_{14}
Dilate(S_1, qz10, O) → M_{14}
Line(H_1, L_{14}) → s_{16}
Line(H_1, K_{14}) → t_{16}
Line(M_{14}, F_1) → a_{17}
Line(M_{14}, G_1) → b_{17}
Intersect(t_{16}, a_{17}) → N_{14}
Intersect(s_{16}, b_{17}) → O_{14}
Line(N_{14}, G_1) → c_{17}
Line(O_{14}, F_1) → d_{17}
Intersect(c_{17}, d_{17}) → P_{14}
Locus(P_{14}, β) → loc30
Dilate(I_1, qx11, O) → Q_{14}
Dilate(M_1, qy11, O) → R_{14}
Dilate(S_1, qz11, O) → S_{14}
Line(H_1, Q_{14}) → e_{17}
Line(H_1, R_{14}) → f_{17}
Line(S_{14}, F_1) → g_{17}
Line(S_{14}, G_1) → h_{17}
Intersect(e_{17}, g_{17}) → T_{14}
Intersect(f_{17}, h_{17}) → U_{14}
Line(T_{14}, G_1) → i_{17}
Line(U_{14}, F_1) → j_{17}
Intersect(i_{17}, j_{17}) → V_{14}
Locus(V_{14}, β) → loc31
Dilate(I_1, qx12, O) → W_{14}
Dilate(M_1, qy12, O) → Z_{14}
Dilate(S_1, qz12, O) → A_{15}
Line(H_1, W_{14}) → k_{17}
Line(H_1, Z_{14}) → l_{17}
Line(A_{15}, F_1) → m_{17}
Line(A_{15}, G_1) → n_{17}
Intersect(l_{17}, n_{17}) → B_{15}
Intersect(k_{17}, m_{17}) → C_{15}
Line(C_{15}, G_1) → p_{17}
Line(B_{15}, F_1) → q_{17}
Intersect(p_{17}, q_{17}) → D_{15}
Locus(D_{15}, β) → loc32
Dilate(I_1, qx13, O) → E_{15}
Dilate(M_1, qy13, O) → F_{15}
Dilate(S_1, qz13, O) → G_{15}
Line(H_1, F_{15}) → r_{17}
Line(H_1, E_{15}) → s_{17}
Line(G_{15}, F_1) → t_{17}
Line(G_{15}, G_1) → a_{18}
Intersect(s_{17}, t_{17}) → H_{15}
Intersect(r_{17}, a_{18}) → I_{15}
Line(H_{15}, G_1) → b_{18}
Line(I_{15}, F_1) → c_{18}
Intersect(b_{18}, c_{18}) → J_{15}
Locus(J_{15}, β) → loc33
Dilate(I_1, qx14, O) → K_{15}
Dilate(M_1, qy14, O) → L_{15}
Dilate(S_1, qz14, O) → M_{15}
Line(H_1, K_{15}) → d_{18}
Line(H_1, L_{15}) → e_{18}
Line(M_{15}, F_1) → f_{18}
Line(M_{15}, G_1) → g_{18}
Intersect(d_{18}, f_{18}) → N_{15}
Intersect(e_{18}, g_{18}) → O_{15}
Line(N_{15}, G_1) → h_{18}
Line(O_{15}, F_1) → i_{18}
Intersect(i_{18}, h_{18}) → P_{15}
Locus(P_{15}, β) → loc34
Dilate(I_1, qx15, O) → Q_{15}
Dilate(M_1, qy15, O) → R_{15}
Dilate(S_1, qz15, O) → S_{15}
Line(H_1, Q_{15}) → j_{18}
Line(H_1, R_{15}) → k_{18}
Line(S_{15}, F_1) → l_{18}
Line(S_{15}, G_1) → m_{18}
Intersect(j_{18}, l_{18}) → T_{15}
Intersect(k_{18}, m_{18}) → U_{15}
Line(T_{15}, G_1) → n_{18}
Line(U_{15}, F_1) → p_{18}
Intersect(p_{18}, n_{18}) → V_{15}
Locus(V_{15}, β) → loc35
Dilate(I_1, qx16, O) → W_{15}
Dilate(M_1, qy16, O) → Z_{15}
Dilate(S_1, qz16, O) → A_{16}
Line(H_1, W_{15}) → q_{18}
Line(H_1, Z_{15}) → r_{18}
Line(A_{16}, F_1) → s_{18}
Line(A_{16}, G_1) → t_{18}
Intersect(q_{18}, s_{18}) → B_{16}
Intersect(r_{18}, t_{18}) → C_{16}
Line(B_{16}, G_1) → a_{19}
Line(C_{16}, F_1) → b_{19}
Intersect(a_{19}, b_{19}) → D_{16}
Locus(D_{16}, β) → loc36
Dilate(I_1, qx17, O) → E_{16}
Dilate(M_1, qy17, O) → F_{16}
Dilate(S_1, qz17, O) → G_{16}
Line(H_1, E_{16}) → c_{19}
Line(H_1, F_{16}) → d_{19}
Line(G_{16}, F_1) → e_{19}
Line(G_{16}, G_1) → f_{19}
Intersect(c_{19}, e_{19}) → H_{16}
Intersect(d_{19}, f_{19}) → I_{16}
Line(H_{16}, G_1) → g_{19}
Line(I_{16}, F_1) → h_{19}
Intersect(h_{19}, g_{19}) → J_{16}
Locus(J_{16}, β) → loc37
Dilate(S_1, gz1, O) → K_{16}
Line(K_{16}, G_1) → i_{19}
Line(K_{16}, F_1) → j_{19}
Intersect(s_{13}, j_{19}) → L_{16}
Intersect(r_{13}, i_{19}) → M_{16}
Line(M_{16}, F_1) → k_{19}
Line(L_{16}, G_1) → l_{19}
Intersect(l_{19}, k_{19}) → N_{16}
Dilate(S_1, gz2, O) → O_{16}
Line(O_{16}, G_1) → m_{19}
Line(O_{16}, F_1) → n_{19}
Intersect(s_{13}, n_{19}) → P_{16}
Intersect(r_{13}, mangekant1) → Q_{16}, R_{16}
Intersect(m_{19}, r_{13}) → S_{16}
Line(S_{16}, F_1) → p_{19}
Line(P_{16}, G_1) → q_{19}
Intersect(q_{19}, p_{19}) → T_{16}
Line(G_{12}, G_1) → r_{19}
Line(G_{12}, F_1) → s_{19}
Intersect(r_{13}, r_{19}) → U_{16}
Intersect(s_{13}, s_{19}) → V_{16}
Line(U_{16}, F_1) → t_{19}
Line(V_{16}, G_1) → a_{20}
Dilate(I_1, hx1, O) → W_{16}
Dilate(I_1, hx2, O) → Z_{16}
Line(H_1, Z_{16}) → b_{20}
Line(H_1, W_{16}) → c_{20}
Intersect(b_{20}, s_{19}) → A_{17}
Intersect(c_{20}, s_{19}) → B_{17}
Line(B_{17}, G_1) → d_{20}
Line(A_{17}, G_1) → e_{20}
Intersect(t_{19}, e_{20}) → C_{17}
Intersect(t_{19}, d_{20}) → D_{17}
Dilate(M_1, iy1, O) → E_{17}
Dilate(M_1, iy2, O) → F_{17}
Line(H_1, E_{17}) → f_{20}
Line(H_1, F_{17}) → g_{20}
Intersect(f_{20}, r_{19}) → G_{17}
Intersect(g_{20}, r_{19}) → H_{17}
Line(G_{17}, F_1) → h_{20}
Line(H_{17}, F_1) → i_{20}
Intersect(a_{20}, i_{20}) → I_{17}
Intersect(h_{20}, a_{20}) → J_{17}
Segment(N_{16}, T_{16}) → j_{20}
Segment(C_{17}, D_{17}) → k_{20}
Segment(I_{17}, J_{17}) → l_{20}
Locus(Q_{12}, β) → loc38
Dilate(I_1, sx2, O) → K_{17}
Dilate(M_1, sy2, O) → L_{17}
Dilate(S_1, sz2, O) → M_{17}
Line(H_1, L_{17}) → m_{20}
Line(H_1, K_{17}) → n_{20}
Line(M_{17}, G_1) → p_{20}
Line(K_{17}, G_1) → q_{20}
Line(M_{17}, F_1) → r_{20}
Line(L_{17}, F_1) → s_{20}
Intersect(n_{20}, r_{20}) → N_{17}
Intersect(m_{20}, p_{20}) → O_{17}
Line(O_{17}, F_1) → t_{20}
Line(N_{17}, G_1) → a_{21}
Intersect(a_{21}, t_{20}) → Q
Intersect(q_{20}, s_{20}) → Q_{17}
Segment(L_{17}, Q_{17}) → b_{21}
Segment(Q_{17}, K_{17}) → c_{21}
Segment(Q_{17}, Q) → d_{21}
Segment(Q, M_{17}) → e_{21}
Vector(S, Q) → v_2
Locus(Q, γ) → loc39
Locus(Q, δ) → loc40
Dilate(I_1, sx3, O) → J_{12}
Dilate(M_1, sy3, O) → P_{17}
Dilate(S_1, sz3, O) → R_{17}
Line(H_1, P_{17}) → f_{21}
Line(H_1, J_{12}) → g_{21}
Line(R_{17}, G_1) → h_{21}
Line(R_{17}, F_1) → i_{21}
Intersect(g_{21}, i_{21}) → T_{17}
Intersect(f_{21}, h_{21}) → U_{17}
Line(U_{17}, F_1) → j_{21}
Line(T_{17}, G_1) → k_{21}
Intersect(k_{21}, j_{21}) → V_{17}
Vector(S, V_{17}) → w_2
Dilate(I_1, mx1, O) → W_{17}
Dilate(M_1, my1, O) → Z_{17}
Dilate(S_1, mz1, O) → A_{18}
Line(H_1, Z_{17}) → l_{21}
Line(H_1, W_{17}) → m_{21}
Line(A_{18}, G_1) → n_{21}
Line(A_{18}, F_1) → p_{21}
Intersect(m_{21}, p_{21}) → B_{18}
Intersect(l_{21}, n_{21}) → C_{18}
Line(C_{18}, F_1) → q_{21}
Line(B_{18}, G_1) → r_{21}
Intersect(r_{21}, q_{21}) → D_{18}
Locus(D_{18}, ε) → loc41
Textfield(γ) → inputBox7
Textfield(δ) → inputBox8
Locus(Q_{17}, δ) → loc42
Angle(mangekant1) → ζ, η, θ, ι

AI Drawing Prompt Reference

Draw a sphere and its great circle. Start by creating a sphere center point O and a plane a passing through O. Then, construct a line b and c perpendicular to a, and a circle d with O as its center. Next, use D and E, the intersection points of d and c, to construct lines g and i perpendicular to c, and circles e and f with D and E as their centers. Through the intersection points F and G of e and b, construct a line h perpendicular to b, and lines g and h to find points I and H. Continue by constructing a line j perpendicular to b, and lines g and j to find points K. Finally, use H, I, J, and K to construct a quadrilateral and rotate and scale it to match the sphere and great circle.

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