How to recover coordinates from a Point

I'm doing some different engineering and geometric manipulation in MMA and am trying to figure out a way to recover the coordinates from the points I translated to the blue line to do further manipulations on them (like drawing interconnecting lines between the two horizontal lines). I've done a pretty good job searching the documentation for such a function, and also approached the problem from a different angle of perhaps using a geometric transform on the original list of pairs but it always seems to come down to the need to convert the list to Point that becomes a problem to continuing on. Any help would be appreciated. Thanks.

bottom = Line[{{x1,y1},{x2,y2}}];
\[Phi] = ArcTan[(y2-y1)/(x2-x1)]// N; (*angle off horizontal*)
\[Theta] = Pi-(\[Phi]+Pi/2); (*offset angle*)
vec = {Cos[\[Theta] ]*depth,Sin[\[Theta] ]*depth} // N (*offset vector*)
list= Prepend[
{(Cos[\[Phi]]*#)+x1,(Sin[\[Phi]]*#)+y1} &/@ Table[unit * x, {x,n}],
{x1,y1}]      ;        (*points*)
pts = Point /@ list;(*points*)
pts2 = Translate[#,vec] &/@ pts;(*points*)
Graphics[{{Red,bottom},{Blue,Translate[bottom,vec]},
{Red, pts},{Blue,pts2}}]


• is this what you need: Cases[Graphics[{{Red, bottom}, {Blue, Translate[bottom, vec]}, {Red, pts}, {Blue, pts2}}], Translate[Point[x_], t_] :> x + t, Infinity]?
– kglr
May 8, 2018 at 5:28
• .. or Cases[Graphics[{{Red, bottom}, {Blue, Translate[bottom, vec]}, {Red, pts}, {Blue, pts2}}], _Point | Translate[Point[_], _], Infinity] /. {Translate[Point[x_], t_] :> x + t, Point[y_] :> y}?
– kglr
May 8, 2018 at 5:33

Using unit = 1; depth = 1; n = 10; {{x1, y1}, {x2, y2}} = {{0, 0}, {10, 5}}; before the posted code in OP and

g = Graphics[{Red, bottom, Blue, Translate[bottom, vec], Red, pts, Blue, pts2}];


we can recover the coordinates of all Points using

{bluepoints, redpoints} =  GatherBy[SortBy[Cases[g, _Point | Translate[Point[_], _], ∞] /.
{Translate[Point[x_], t_]:>{1, x + t},  Point[y_]:>{2, y}}, First], First][[;;, ;;, 2]];


Showing the recovered points together with the original g:

Show[g, Prolog -> {Opacity[.5, Green], AbsolutePointSize[10],
Point @ bluepoints, Opacity[.5, Yellow], Point @ redpoints}]


• I'm having a bit of trouble finding the time to run this thru. I would have never thought of using Cases in this way but it's something I want to try. May 9, 2018 at 3:03
• Spent awhile reverse engineering your answer (especially the 2nd part) and it indeed does the job. I'm curious to know if you think this is 1) an inconvenient characteristic of the MMA language (having no function to directly recover the points as is found in other coding languages), b) a trivial workaround, or c) if I've found some edge case that's clumsy to workaround? And why? May 12, 2018 at 3:49