X(u,v)=

Y(u,v)=

Z(u,v)=

2. Given P in surface: P=X(t,s)

3.  Given v_0∈T_PS :

1. Surface S:

change curve Yes/No

Run ODE Yes/No

5. 解ODE求Geodesic:

4. 給解ODE的t的範圍: t_0=0 到

input curve

Z_1(u, v) = sin(v)
cur: true
Z_2 = If(cur, Z_1)
Y_1(u, v) = (sin(u) * (2 + cos(v)))
Y_2 = If(cur, Y_1)
X_1(u, v) = (cos(u) * (2 + cos(v)))
X_2 = If(cur, X_1)
inputBox1 = Textfield(X_1)
inputBox2 = Textfield(Y_1)
inputBox3 = Textfield(Z_1)
Y_v = Derivative(Y_1, v)
u_0 = 3.630284844148205 rad (slider from 0 to 360°, step 0.5°)
v_0 = 1.7802358370342162 rad (slider from 0 to 6.283185307179586, step 0.017453292519943295)
Y_u = Derivative(Y_2, u)
Z_u = Derivative(Z_2, u)
X_u = Derivative(X_2, u)
β = 1.9931660057775245 rad (slider from 0 to 6.283185307179586, step 0.2°)
X_v = Derivative(X_2, v)
Z_v = Derivative(Z_2, v)
E(u, v) = (X_u(u, v) * X_u(u, v)) + (Y_u(u, v) * Y_u(u, v)) + (Z_u(u, v) * Z_u(u, v))
F(u, v) = (X_u(u, v) * X_v(u, v)) + (Y_u(u, v) * Y_v(u, v)) + (Z_u(u, v) * Z_v(u, v))
G(u, v) = (X_v(u, v) * X_v(u, v)) + (Y_v(u, v) * Y_v(u, v)) + (Z_v(u, v) * Z_v(u, v))
E_u = Derivative(E, u)
E_v = Derivative(E, v)
F_v = Derivative(F, v)
F_u = Derivative(F, u)
G_u = Derivative(G, u)
G_v = Derivative(G, v)
D(u, v) = (E(u, v) * G(u, v)) - F(u, v)^(2)
N_x(u, v) = (Y_u(u, v) * Z_v(u, v)) - (Z_u(u, v) * Y_v(u, v))
N_y(u, v) = (Z_u(u, v) * X_v(u, v)) - (X_u(u, v) * Z_v(u, v))
N_z(u, v) = (X_u(u, v) * Y_v(u, v)) - (Y_u(u, v) * X_v(u, v))
geo: true
A_{111}(u, v) = ((G(u, v) * E_u(u, v)) - ((2 * F(u, v)) * F_u(u, v)) + (F(u, v) * E_v(u, v))) / ((2 * D(u, v)))
A_{11}1 = Simplify(A_{111})
A_{121}(u, v) = ((G(u, v) * E_v(u, v)) - (F(u, v) * G_u(u, v))) / ((2 * D(u, v)))
A_{12}1 = Simplify(A_{121})
A_{221}(u, v) = (((2 * G(u, v)) * F_v(u, v)) - (G(u, v) * G_u(u, v)) - (F(u, v) * G_v(u, v))) / ((2 * D(u, v)))
A_{22}1 = Simplify(A_{221})
A_{112}(u, v) = (((2 * E(u, v)) * F_u(u, v)) - (E(u, v) * E_v(u, v)) - (F(u, v) * E_u(u, v))) / ((2 * D(u, v)))
A_{11}2 = Simplify(A_{112})
A_{122}(u, v) = ((E(u, v) * G_u(u, v)) - (F(u, v) * E_v(u, v))) / ((2 * D(u, v)))
A_{222}(u, v) = ((E(u, v) * G_v(u, v)) - ((2 * F(u, v)) * F_v(u, v)) + (F(u, v) * G_u(u, v))) / ((2 * D(u, v)))
A_{12}2 = Simplify(A_{122})
A_{22}2 = Simplify(A_{222})
P: (X_2(u_0, v_0), Y_2(u_0, v_0), Z_2(u_0, v_0))
nn: (N_x(u_0, v_0), N_y(u_0, v_0), N_z(u_0, v_0))
n = UnitVector(nn)
u'(t, u, v, u1, v1) = u1
v'(t, u, v, u1, v1) = v1
u1'(t, u, v, u1, v1) = (-((A_{11}1(u, v) * u1^(2)) + (((2 * A_{12}1(u, v)) * u1) * v1) + (A_{22}1(u, v) * v1^(2))))
v1'(t, u, v, u1, v1) = (-((A_{11}2(u, v) * u1^(2)) + (((2 * A_{12}2(u, v)) * u1) * v1) + (A_{22}2(u, v) * v1^(2))))
V_{00}: (cos(β), sin(β))
t_1: 15
t_0 = If(geo, 0)
t1 = 17.7 (slider from 5 to 35, step 0.1)
numericalIntegral1, numericalIntegral2, numericalIntegral3, numericalIntegral4 = NSolveODE({u', v', u1', v1'}, t_0, {u_0, v_0, x(V_{00}), y(V_{00})}, t1)
len = Length(numericalIntegral1)
L = 0.51 (slider from 0.01 to 0.99, step 0.01)
vv: ((X_u(u_0, v_0), Y_u(u_0, v_0), Z_u(u_0, v_0)) * cos(β)) + ((X_v(u_0, v_0), Y_v(u_0, v_0), Z_v(u_0, v_0)) * sin(β))
v_1 = UnitVector(vv)
v = Vector(P, P + v_1)
c = Circle(P, 1, n)
A: (X_2(50°, 100°), Y_2(50°, 100°), Z_2(50°, 100°))
a = Surface(X_2(u, v), Y_2(u, v), Z_2(u, v), u, 0, (2 * π), v, 0, (2 * π))
list1 = Sequence(Segment[(X_2(y(Point[numericalIntegral1, s1]), y(Point[numericalIntegral2, s1])), Y_2(y(Point[numericalIntegral1, s1]), y(Point[numericalIntegral2, s1])), Z_2(y(Point[numericalIntegral1, s1]), y(Point[numericalIntegral2, s1]))), (X_2(y(Point[numericalIntegral1, s1 + 1 / len]), y(Point[numericalIntegral2, s1 + 1 / len])), Y_2(y(Point[numericalIntegral1, s1 + 1 / len]), y(Point[numericalIntegral2, s1 + 1 / len])), Z_2(y(Point[numericalIntegral1, s1 + 1 / len]), y(Point[numericalIntegral2, s1 + 1 / len])))], s1, 0, L - 1 / len, 1 / len)
list2 = Sequence(Segment[(X_2(y(Point[numericalIntegral1, s1]), y(Point[numericalIntegral2, s1])), Y_2(y(Point[numericalIntegral1, s1]), y(Point[numericalIntegral2, s1])), Z_2(y(Point[numericalIntegral1, s1]), y(Point[numericalIntegral2, s1]))), (X_2(y(Point[numericalIntegral1, s1 + 1 / len]), y(Point[numericalIntegral2, s1 + 1 / len])), Y_2(y(Point[numericalIntegral1, s1 + 1 / len]), y(Point[numericalIntegral2, s1 + 1 / len])), Z_2(y(Point[numericalIntegral1, s1 + 1 / len]), y(Point[numericalIntegral2, s1 + 1 / len])))], s1, 0, L - 1 / len, 1 / len)