π

Hyperboloid

Polosférický model

Poincarého kruhový model

Beltrami-Kleinův model

Poincarého polorovinný model

Absolutní kružnice

Přechod Hyp - PS

Přechod PS - BK

Přechod PS - PR

Přímky v hyperboloidickém modelu

μ: x = -2
a_1: x^(2) + y^(2) - z^(2) + 1 = 0
O = Intersect(yAxis, zAxis)
b_1: x^(2) + y^(2) + z^(2) = 1
A_{hyp} = PointIn(a_1)
B_{hyp} = PointIn(a_1)
ρ = Plane(A_{hyp}, B_{hyp}, O)
l_{hyp} = IntersectPath(ρ, a_1)
S = Point(zAxis)
a_{ps} = Line(S, A_{hyp})
b_{ps} = Line(S, B_{hyp})
A_{pk} = Intersect(a_{ps}, xOyPlane)
B_{pk} = Intersect(b_{ps}, xOyPlane)
d: x^(2) + y^(2) - z^(2) = 0
f = Plane(S, ρ)
U_∞, V_∞ = Intersect(f, k_a)
l_{pk} = CircumcircleArc(U_∞, B_{pk}, V_∞)
I, A_{ps} = Intersect(a_{ps}, b_1)
ρ_{bk} = Plane(U_∞, V_∞, A_{ps})
j = OrthogonalLine(A_{ps}, xOyPlane)
A_{bk} = Intersect(j, xOyPlane)
l_{bk} = Segment(V_∞, U_∞)
p: z = 0
G, B_{ps} = Intersect(b_{ps}, b_1)
l_{ps} = CircumcircleArc(U_∞, B_{ps}, V_∞)
g = OrthogonalLine(B_{ps}, p)
B_{bk} = Intersect(g, p)
E: (1, 0, 0)
a_{pr} = Line(E, A_{ps})
u_{pr} = Line(E, U_∞)
v_{pr} = Line(E, V_∞)
A_{pr} = Intersect(a_{pr}, μ)
V'_{∞} = Intersect(v_{pr}, μ)
U'_{∞} = Intersect(u_{pr}, μ)
b_{pr} = Line(E, B_{ps})
B_{pr} = Intersect(b_{pr}, μ)
l_{pr} = CircumcircleArc(U'_{∞}, B_{pr}, V'_{∞})
hyp = true
ps = false
pk = false
bk = false
pr = false
ak = false
hypps = false
psbk = false
pspr = false
P_{hyp} = PointIn(a_1)
D_{hyp} = PointIn(a_1)
a = Plane(O, P_{hyp}, D_{hyp})
r_{hyp} = IntersectPath(a, a_1)
C_1, D_1 = Intersect(l_{hyp}, r_{hyp})
as = IntersectPath(ρ, d)
A = Point(as)
B = Point(as)
asy1 = Line(O, A)
asy2 = Line(O, B)
b = Plane(P_{hyp}, asy1)
u_{hyp} = IntersectPath(b, a_1)
l = Plane(P_{hyp}, asy2)
v_{hyp} = IntersectPath(l, a_1)
k = false
h = Line(S, P_{hyp})
i = Line(S, D_{hyp})
C, D_{ps} = Intersect(i, b_1)
H, P_{ps} = Intersect(h, b_1)
m = OrthogonalLine(P_{ps}, p)
n = OrthogonalLine(D_{ps}, p)
D_{bk} = Intersect(n, p)
P_{bk} = Intersect(m, p)
q = Line(P_{bk}, D_{bk})
W_∞, Q_∞ = Intersect(q, b_1)
J = Midpoint(W_∞, Q_∞)
r_{ps} = CircumcircleArc(W_∞, P_{ps}, Q_∞)
r_{bk} = Segment(W_∞, Q_∞)
t = Line(V_∞, P_{bk})
D, F = Intersect(t, k_a)
c_1 = CircumcircleArc(V_∞, P_{ps}, D)
f_1 = Segment(V_∞, D)
g_1 = Line(U_∞, P_{bk})
K, L = Intersect(g_1, b_1)
d_1 = CircumcircleArc(U_∞, P_{bk}, L)
e_1 = CircumcircleArc(U_∞, P_{ps}, L)
D_{pk} = Intersect(i, p)
P_{pk} = Intersect(h, p)
r_{pk} = CircumcircleArc(W_∞, P_{pk}, Q_∞)
r = Line(E, W_∞)
s = Line(E, Q_∞)
W'_∞ = Intersect(r, μ)
Q'_∞ = Intersect(s, μ)
c = Line(E, P_{ps})
P_{pr} = Intersect(c, μ)
e = CircumcircleArc(Q'_∞, P_{pr}, W'_∞)
h_1 = Line(E, D_{ps})
D_{pr} = Intersect(h_1, μ)
i_1 = Tangent(V_∞, k_a)
j_1 = LineBisector(P_{pk}, V_∞, xOyPlane)
M = Intersect(i_1, j_1)
k_1 = Circle(M, P_{pk}, xOyPlane)
N = Intersect(k_a, k_1, 1)
l_1 = Tangent(U_∞, k_a)
m_1 = LineBisector(P_{pk}, U_∞, xOyPlane)
P = Intersect(m_1, l_1)
q_1 = Circle(P, P_{pk}, xOyPlane)
Q = Intersect(k_a, q_1, 1)
R = Intersect(k_a, q_1, 2)
T = Intersect(k_a, k_1, 2)