Here I join 3 figures with lines in a tricky way, where I plot vertical and horizontal lines separately and set them by Inset at appropriate positions in such a way that the lines vanish when they touch the end figures.
Figure~1 http://s14.postimage.org/or877n6e9/image.jpg
y[t_] := Sin[\[Pi]Sin[π t]/(\[Pi]π t);
p[t_] = t^2;
a = 0.7;
b = 5.2;
yRoots = t /. {ToRules@Reduce[{y[t] == 0, 0 <= t <= 5}, t]};
yDRoots = t /. {ToRules@N@Reduce[{y'[t] == 0, 0 <= t <= 5}, y]};
ranges = Append[Prepend[yDRoots, 0], 6];
\[Theta][t_]θ[t_] := Piecewise[Table[{ArcTan[y[t]/(p[t] y'[t])] + k Pi, ranges[[k]] < t <= ranges[[k + 1]]}, {k, Length@ranges - 1}]];
\[Rho][t_]ρ[t_] := Sqrt[(y[t])^2 + (p[t] y'[t])^2];
\[CurlyEpsilon]ε = 1/(10^7);
p1 = Plot[\[Theta][t]Plot[θ[t], {t, a, b}, Ticks -> {None, Join[Table[{k Pi, k \[Pi]π}, {k, 0, 5}], Table[{(2 k - 1) Pi/2, (2 k - 1) Pi/2}, {k, 1, 5}]]}, AxesLabel -> {"t", "\[Theta]""θ"}, AxesStyle -> Directive[14], AxesOrigin -> {0, 0}, PlotRange -> {{0, 5.3}, {0, 16}}, AspectRatio -> 1, ImagePadding -> 20,
Epilog -> {{Red, AbsolutePointSize@5, Point[{#, \[Theta][#]θ[#]}&/@yRoots]},
{Blue, AbsolutePointSize@5, Point[{#, \[Theta][#]θ[#]}&/@(yDRoots + \[CurlyEpsilon]ε)]},
{Black, AbsolutePointSize@5,Point[{{a, \[Theta][a]θ[a]}, {b, \[Theta][b]θ[b]}}]},
{Gray, Dashed, Line[{{0, \[Theta][#]θ[#]}, {#, \[Theta][#]θ[#]}, {#, -100}}&/@yRoots], Line[{{0, \[Theta][#]θ[#]}, {#, \[Theta][#]θ[#]}}&/@yRoots]},
{Gray, Dashed, Line[{{0, \[Theta][#]θ[#]}, {#, \[Theta][#]θ[#]}, {#, -100}}&/@(yDRoots+\[CurlyEpsilon]yDRoots+ε)]}}];
p2 = Plot[y[t], {t, a, b}, Ticks -> {Table[{k, ""}, {k, 1, 5}], {{1, ""}}}, AxesLabel -> {"t", "y"}, AxesStyle -> Directive[14], AxesOrigin -> {0, 0}, PlotRange -> {{0, 5.3}, {-0.3, 1}}, AspectRatio -> 1, ImagePadding -> 20,
Epilog -> {{Red, AbsolutePointSize@5, Point[{#, y[#]} & /@ yRoots]},
{Blue, AbsolutePointSize@5, Point[{#, y[#]} & /@ yDRoots]},
{Black, AbsolutePointSize@5, Point[{{a, y[a]}, {b, y[b]}}]},
{Gray, Dashed, Line[{{#, 0}, {#, 100}} & /@ yRoots]},
{Gray, Dashed, Line[{{100, y[#]}, {#, y[#]}, {#, 100}} & /@ yDRoots]}}];
p3 = ParametricPlot[{\[Rho][t]ρ[t] Cos[\[Theta][t]]Cos[θ[t]], \[Rho][t]ρ[t] Sin[\[Theta][t]]Sin[θ[t]]}, {t, a, b}, Ticks -> None, AxesLabel -> None, AxesStyle -> Directive[14], AxesOrigin -> {0, 0}, ImagePadding -> 20, PlotRange -> {{-6, 6}, {-0.3, 1}}, AspectRatio -> 1,
Epilog -> {{Red, AbsolutePointSize@5, Point[(\[Rho][#]*ρ[#]*{Cos[\[Theta][#]]Cos[θ[#]], Sin[\[Theta][#]]Sin[θ[#]]})&/@yRoots]},
{Blue, AbsolutePointSize@5,Point[(\[Rho][#]*ρ[#]*{Cos[\[Theta][#]]Cos[θ[#]], Sin[\[Theta][#]]Sin[θ[#]]}) & /@ (yDRoots + \[CurlyEpsilon]ε)]},
{Black, AbsolutePointSize@5, Point[{\[Rho][a]*ρ[a]*{Cos[\[Theta][a]]Cos[θ[a]], Sin[\[Theta][a]]Sin[θ[a]]}, \[Rho][b]*ρ[b]*{Cos[\[Theta][b]]Cos[θ[b]], Sin[\[Theta][b]]Sin[θ[b]]}}]},
{Gray, Dashed,Line[{({-100, \[Rho][#]*Sin[\[Theta][#]]ρ[#]*Sin[θ[#]]}), (\[Rho][#]*ρ[#]*{Cos[\[Theta][#]]Cos[θ[#]],Sin[\[Theta][#]]Sin[θ[#]]})}&/@(yDRoots + \[CurlyEpsilon]ε)]},
{Gray, Dotted, Line[{{0, 0}, (\[Rho][a]*ρ[a]*{Cos[\[Theta][a]]Cos[θ[a]],Sin[\[Theta][a]]Sin[θ[a]]})}]}}];
(* Vertical lines *)
pvl = Plot[2, {t, a, b}, Axes -> None, AxesOrigin -> {0, 0}, PlotRange -> {{0, 5.3}, {-1, 1}}, AspectRatio -> 1, ImagePadding -> 20,
Epilog -> {{Gray, Dashed, Line[{{#, 0.66}, {#, 1}} & /@ yRoots]},
{Gray, Dashed, Line[{{#, 0.66}, {#, 1}} & /@yDRoots]}}];
(* Horizontal lines *)
phl = Plot[2, {t, a, b}, Axes -> None, AxesOrigin -> {0, 0}, PlotRange -> {{0, 5.3}, {-0.3, 1}}, AspectRatio -> 1, ImagePadding -> 20,
Epilog -> {{Gray, Dashed, Line[{{0.03, y[#]}, {0.9, y[#]}} & /@yDRoots]}}];
(* Put the images together *)
Graphics[{Inset[p1, ImageScaled@{.05, 0.52}, {0, 0}, 1],
Inset[pvl, ImageScaled@{.05, .31}, {0, 0}, 1],
Inset[p2, ImageScaled@{.05, .12}, {0, 0}, 1],
Inset[phl, ImageScaled@{.48, .12}, {0, 0}, 1],
Inset[p3, ImageScaled@{.77, .12}, {0, 0}, 1]}, ImageSize -> 800, PlotRange -> All]
