fgridc[p_]=Surround[Graphics[{{{GrayLevel[.5],Thickness[.02],Line[{{-10-10p,-10},{0,0},{10-10p,-10}}]}},Table[Point[{x+tp,t}],{x,-10,10},{t,-10,10}]},AspectRatioAutomatic,PlotRange{{-10,10},{-10,10}}]];
fgrid[p_]=Surround[Graphics[{{GrayLevel[.5],Thickness[.02],Line[{{10,-10},{0,0},{-10,-10}}]},Table[Point[{xCosh[p]+tSinh[p],tCosh[p]+xSinh[p]}],{x,-10,10},{t,-10,10}]},AspectRatioAutomatic,PlotRange{{-10,10},{-10,10}}]];
Show[GraphicsColumn[{GraphicsRow[(LabelWrapper[
fgridc[#1], StringForm["v = `` c", InputForm[#1]]] &
) /@ Range[0, 1/2, 1/4]],
GraphicsRow[(LabelWrapper[fgrid[#1],
StringForm["v = `` c", InputForm[#]]] & ) /@
Range[0, 1/2, 1/4]]}, 0.03]];
Show[fgrid[1/3]];
GalileanGraphic[p_,list_]:=Surround[Graphics[{{{GrayLevel[.5],Thickness[.02],Line[{{-10-10p,-10},{0,0},{10-10p,-10}}]}},Module[{t,x},{t,x}=#;Point[{x+tp,t}]]&/@list},AspectRatioAutomatic,PlotRange{{-10,10},{-10,10}}]]
LorentzGraphic[p_,list_]:=Surround[Graphics[{{GrayLevel[.5],Thickness[.02],Line[{{10,-10},{0,0},{-10,-10}}]},Module[{t,x},{t,x}=#;Point[{xCosh[p]+tSinh[p],tCosh[p]+xSinh[p]}]]&/@list},AspectRatioAutomatic,PlotRange{{-10,10},{-10,10}}]]
LorentzGraphicLC[p_,list_,lcsize_:.5]:=Surround[Graphics[{Module[{t,x,v},{t,x}=#;v={xCosh[p]+tSinh[p],tCosh[p]+xSinh[p]};{{GrayLevel[.5],Polygon[{v-{1,1}lcsize,v,v+{1,-1}lcsize}]},Point[v]}]&/@list},AspectRatioAutomatic,PlotRange{{-10,10},{-10,10}}]]
LorentzGraphicLC[p_,list_,lcsize_:.5]:=Surround[Graphics[{Module[{t,x,v},{t,x}=#;v={xCosh[p]+tSinh[p],tCosh[p]+xSinh[p]};{{GrayLevel[.5],Polygon[{v-{1,1}lcsize,v,v+{1,-1}lcsize}]},Point[v]}]&/@list},AspectRatioAutomatic,PlotRange{{-10,10},{-10,10}}]]
TransformGraphicLC[f_,list_,lcsize_:.5]:=Surround[Graphics[{{GrayLevel[.5],Polygon[f/@{#-{1,1}lcsize,#,#+{1,-1}lcsize}]},Point[f[#]]}&/@list,AspectRatioAutomatic,PlotRange{{-10,10},{-10,10}}]]
Lorentz[{x_,t_},p_]:={xCosh[p]+tSinh[p],tCosh[p]+xSinh[p]}
Galileo[{x_,t_},p_]:={x+tp,t}
randpts=Table[Random[Real,{-10,10}],{1000},{2}];
rpts=Table[Random[Real,{-10,10}],{100},{2}];
Show[TransformGraphicLC[Lorentz[#,.6]&,randpts,.7]];
Show[GraphicsRow[Function[p,TransformGraphicLC[Lorentz[#,p]&,randpts,.5]]/@{0,.25,.5,.75}]];
Show[GraphicsRow[Function[p,TransformGraphicLC[Galileo[#,p]&,randpts,.5]]/@{0,.25,.5,.75}]];
Show[TransformGraphicLC[Galileo[#,.6]&,randpts,.7]];
Show[TransformGraphicLC[Lorentz[#,0]&,rpts,1]];
Show[LorentzGraphic[.1,randpts]];