I have this ParametricPlot3D
:
generated by this code:
x[α_, β_, γ_, t_] :=
Sin[α] Cos[β] Cos[γ] Cos[t] + Sin[α] Sin[γ] Sin[t] - Cos[α] Sin[β] Cos[γ];
y[α_, β_, γ_, t_] :=
Sin[α] Cos[β] Sin[γ] Cos[t] + Sin[α] Cos[γ] Sin[t] + Cos[α] Sin[β] Sin[γ];
z[α_, β_, t_] :=
Sin[α] Sin[β] Cos[t] + Cos[α] Cos[β]
α = π/3;
β = +π/3;
γ = 0;
Show[
ParametricPlot3D[{Cos[u] Sin[v], Cos[u] Cos[v], Sin[u]}, {u, 0, π}, {v, -π/2, π/2},
Mesh -> None,
PlotStyle -> Opacity[.25, Blue], PlotPoints -> 80,
MaxRecursion -> 4,
ExclusionsStyle -> ({Directive[Opacity[1], Thick, Red]}),
Boxed -> False, Axes -> False],
ParametricPlot3D[Normalize[{x[α, β, γ, t], y[α, β, γ, t], z[α, β, t]}], {t, 0, π}],
ParametricPlot3D[Normalize[{-x[α, β, γ, -t], -y[α, β, γ, -t], z[α, β, -t]}], {t, 2 π, π}],
Graphics3D[{PointSize[0.025], Point[{0, 0, 1}]}],
ViewPoint -> Front]
--
I want to project it down to a 2D plane so that I see something like this:
Any ideas?
EDIT:
The result of
Plot[Normalize[{x[α, β, γ, t], y[α, β, γ, t], z[α, β, t]}][[ ;; 2]], {t, 0, 2 π}]
is
EDIT 2
Trying to set $z=0$ in the parameterisation:
Show[
ParametricPlot3D[{Cos[u] Sin[v], Cos[u] Cos[v], Sin[u]}, {u, 0, π}, {v, -π/2, π/2},
Mesh -> None,
PlotStyle -> Opacity[.25, Blue], PlotPoints -> 80,
MaxRecursion -> 4,
ExclusionsStyle -> ({Directive[Opacity[1], Thick, Red]}),
Boxed -> False, Axes -> False],
ParametricPlot3D[Normalize[{x[α, β, γ, t], y[α, β, γ, t], 0}], {t, 0, π}],
ParametricPlot3D[Normalize[{-x[α, β, γ, -t], -y[α, β, γ, -t], 0}], {t, 2 π, π}],
Graphics3D[{PointSize[0.025], Point[{0, 0, 1}]}],
ViewPoint -> Front]
x
,y
, andz
, so we cannot reproduce. In principle, though, couldn't you just plotNormalize[{x[α, β, γ, t], y[α, β, γ, t], z[α, β, t]}][[;;2]]
? $\endgroup$ParametricPlot
withNormalize[{x[α, β, γ, t], y[α, β, γ, t], z[α, β, t]}][[;;2]]
$\endgroup$