Generate Piecewise ParametricPlot from Graphics

Question

Some Graphics in Mathematica could also be described by functions, in particular by ParametricPlots. For example, take this figure of a 'house' with a round 'roof'.

Graphics[{
    Line[{{-1,0},{-1,-1},{1,-1},{1,0}}],
    Circle[{0,0},1,{0,Pi}]
}]

Turning this manually into parametric functions gives:

Manipulate[
    ParametricPlot[
        Piecewise[{
            {{-1,-(tt-0)},0<=tt<1},
            {{-1+2(tt-1),-1},1<=tt<2},
            {{1,-1+(tt-2)},2<=tt<3},
            {{Cos[Pi(tt-3)],Sin[Pi(tt-3)]},3<=tt<4}
        }],
    {tt,0,t},Axes->False],
{{t,4},0,4}]

Is it possible to generate a (possibly Piecewise) parametric function which may be drawn using ParametricPlot automatically based on Graphics as input?

Background: imagine using Mathematica to export G-code for CNC laser engraving or cutting while giving you control over the order, direction and speed of each segment, and allowing you to verify the behaviour of your CNC before printing. Note that exporting an SVG will not give you control over direction nor speed.

Answer 1

ClearAll[toParametricLines]
toParametricLines[mcm_: .1][g_Graphics] := Module[{bsfs = 
    MeshPrimitives[DiscretizeGraphics[g, MaxCellMeasure -> {"Length" -> mcm}], 1] /. 
      Line -> (BSplineFunction[#, SplineDegree -> 1] &)}, 
  ParametricPlot[Through@bsfs@t, {t, 0, 1}]]

Examples:

g1 = Graphics[{Line[{{-1, 0}, {-1, -1}, {1, -1}, {1, 0}}], Circle[{0, 0}, 1, {0, Pi}]}];
toParametricLines[][g1]

SeedRandom[1]
g2 = Graphics[{RandomColor[], BezierCurve @ #} & /@ RandomReal[1, {10, 4, 2}]]

Show[toParametricLines[][g2], Axes -> False]

Update: For the case of a single closed curve, we can get a single BSplineFunction (as opposed to one function for each Line object returned by MeshPrimitives[...]) using

ClearAll[toParametricLines2]
toParametricLines2[mcm_: .1][g_Graphics] := Module[{bsf = 
     MeshPrimitives[DiscretizeGraphics[g, MaxCellMeasure -> {"Length" -> mcm}], 1] /. 

      l : {__Line} :> BSplineFunction[Join @@ l[[All, 1]], SplineDegree -> 1] }, 
    ParametricPlot[bsf@t, {t, 0, 1}]]

toParametricLines2[][g1]

same as the first picture above

Answer 2

I wrote a function which takes Graphics as input and outputs a Piecewise parametric function by replacing the Head. The list of implemented Graphics types can easily be extended.

graphicsToParametricFunctions::grphtp="Unknown graphics type: `1`";
graphicsToParametricFunctions=Function[{graphics},Module[{line=Function[{pointPairs,t},With[{len=Length[pointPairs]},Piecewise[MapIndexed[{{((#2[[1]]-len t))#1[[1,1]]+(1+len t-#2[[1]])#1[[2,1]],(#2[[1]]-len t)#1[[1,2]]+(1+len t-#2[[1]])#1[[2,2]]},(#2[[1]]-1)/len<t<#2[[1]]/len}&,pointPairs]]]]},Module[{rules={Line[args__/;Depth[args]==3]:>Function[{t},line[Subsequences[args,{2}],t]],Line[args__/;Depth[args]==4]:>Function[{t},line[Flatten[Map[Subsequences[#,{2}]&,args],1],t]],Circle[{centerx__,centery__},radius__,{arc1__,arc2__}]:>Function[{t},{centerx+radius Cos[(1-t) arc1+t arc2],centery+radius Sin[(1-t) arc1+t arc2]}],BezierCurve[args__]:>Function[{t},Map[BernsteinBasis[Length[args]-1,#,t]&,Range[0,Length[args]-1]].args]}},Function[{t},Evaluate[PiecewiseExpand[Piecewise[MapIndexed[If[AnyTrue[Keys[rules],Function[{rule},MatchQ[#1,rule]]],{Replace[#1,rules][t-#2[[1]]+1],#2[[1]]-1<t<#2[[1]]},Message[graphicsToParametricFunctions::grphtp,#1]]&,Flatten[Level[graphics,1]]]]]]]]]];

Case 1

Two different types of Graphics which are attached, shaped as a house.

houseGraphic=Graphics[{Line[{{-1,0},{-1,-1},{1,-1},{1,0}}],Circle[{0,0},1,{0,Pi}]}]

houseParametricFunction = graphicsToParametricFunctions[houseGraphic]

Manipulate[Show[ParametricPlot[houseParametricFunction[tt],{tt,0,t}],PlotRange->{{-1,1},{-1,1}},ImageSize->220,Axes->False],{t,0,2,0.1}]

Case 2

Two Lines which are separate.

twoLinesGraphic=Graphics[{Line[{{{-1,1},{-1,-1},{1,-1},{1,1},{-1,1}},{{-1/2,1/2},{-1/2,-1/2},{1/2,-1/2},{1/2,1/2},{-1/2,1/2}}}]}]

twoLinesParametricFunction=graphicsToParametricFunctions[twoLinesGraphic]

Manipulate[Show[ParametricPlot[twoLinesParametricFunction[tt],{tt,0,t}],PlotRange->{{-1,1},{-1,1}},ImageSize->220,Axes->False],{t,0,1,0.05}]

Case 3

Random BezierCurves.

SeedRandom[1]
bezierGraphic=Graphics[Map[BezierCurve[#]&,RandomReal[1,{10,4,2}]]];

bezierParametricFunction=graphicsToParametricFunctions[bezierGraphic]

Manipulate[Show[ParametricPlot[bezierParametricFunction[tt],{tt,0,t}],PlotRange->{{0,1},{0,1}},ImageSize->220,Axes->False],{t,0,10,0.5}]

Case 4

Not implemented Graphics type.

graphicsToParametricFunctions[Graphics[Disk[]]]