mjhjmtu

This code does not work with named coordinates (such as the following code). How can I reflect the blue line over the red line by using coordinate names. And how do I reflect just a named coordinate?

documentclass[tikz]standalone

begindocument
begintikzpicture[scale=0.55]
coordinate (A) at (0,0);
coordinate (B) at (1,1);
coordinate (C) at (1,2);
coordinate (D) at (2,0);
coordinate (E) at (2,3);

draw[blue] (B)--(A)--(C);
draw[red] (D)--(E);
endtikzpicture
enddocument

This is a list of proposals. None of them is perfect. However, the aim is not to transform the points one by one, but the full line. (Transforming the points one by one is possible e.g. with the tkz-euclide or just with calc.) The ordering indicates a ranking of these options.

First option: (ab)use show path construction. (Problems: one has to cheat with the colors and also this is not one path but two of them.)

documentclass[tikz,border=3.14mm]standalone
usetikzlibrarydecorations.pathreplacing,calc
makeatletter
tikzsetreflect at/.style args=#1--#2decorate,decoration=
show path construction,
lineto code=draw[tikz@textcolor]
($2*($(#1)!(tikzinputsegmentfirst)!(#2)$)-(tikzinputsegmentfirst)$)
-- ($2*($(#1)!(tikzinputsegmentlast)!(#2)$)-(tikzinputsegmentlast)$);
makeatother
begindocument
begintikzpicture[scale=0.55]
coordinate (A) at (0,0);
coordinate (B) at (1,1);
coordinate (C) at (1,2);
coordinate (D) at (2,0);
coordinate (E) at (2,3);
draw[blue] (B)--(A)--(C);
draw[red] (D)--(E);
draw[blue,reflect at=D--E] (B)--(A)--(C);
endtikzpicture
enddocument

A slight modification thereof does point reflections.

documentclass[tikz,border=3.14mm]standalone
usetikzlibrarydecorations.pathreplacing,calc
makeatletter
tikzsetpoint reflect at/.style args=#1decorate,decoration=
show path construction,
lineto code=draw[tikz@textcolor]
($(tikzinputsegmentfirst)+2*($(#1)-(tikzinputsegmentfirst)$)$)
-- ($(tikzinputsegmentlast)+2*($(#1)-(tikzinputsegmentlast)$)$);
makeatother
begindocument
begintikzpicture[scale=0.55]
path (0,0) coordinate (A) (1,1) coordinate (B) (1,2) coordinate (C) 
(2,3) coordinate (D);
draw[blue] (B)--(A)--(C);
fill[red] (D) circle(1pt);
draw[blue,point reflect at=D] (B)--(A)--(C);
endtikzpicture
enddocument

Second option: Change the to path. (Problems: not one continuous path but separate ones and you need to draw segment by segment.)

documentclass[tikz]standalone
usetikzlibrarycalc
tikzsetreflect at/.style args=#1--#2to path=%
($2*($(#1)!(tikztostart)!(#2)$)-(tikztostart)$)
-- ($2*($(#1)!(tikztotarget)!(#2)$)-(tikztotarget)$)

begindocument
begintikzpicture[scale=0.55]
coordinate (A) at (0,0);
coordinate (B) at (1,1);
coordinate (C) at (1,2);
coordinate (D) at (2,0);
coordinate (E) at (2,3);

draw[blue] (B)--(A)--(C);
draw[red] (D)--(E);
draw[blue,reflect at=D--E] (B) to (A) (A) to (C);
endtikzpicture
enddocument

Third option: More core-level. (Problem: doesn't work with rescaling things.)

documentclass[tikz]standalone
makeatletter
tikzsetget mirror data/.code args=#1--#2%pgftransformreset 
pgfutil@tempdima=pgf@x
pgfutil@tempdimb=pgf@y
pgfpointanchor#1center
pgf@xa=pgf@x
pgf@ya=pgf@y
pgfpointanchor#2center
pgf@xb=pgf@x
pgf@yb=pgf@y
pgfmathsetmacrotmpt2*(-(pgf@ya*(pgf@xb-pgf@xa)) + pgfutil@tempdimb*(pgf@xb-pgf@xa) + (pgf@xa - pgfutil@tempdima)*(pgf@yb-pgf@ya))/((pgf@xb-pgf@xa)^2 + (pgf@yb-pgf@ya)^2)
advancepgf@xb by-pgf@xa
advancepgf@yb by-pgf@ya
pgfutil@tempdima=tmptpgf@yb
pgfutil@tempdimb=-tmptpgf@xb
,
mirror at/.style args=#1--#2get mirror data=#1--#2,xshift=pgfutil@tempdima,
yshift=pgfutil@tempdimb
makeatother
begindocument
begintikzpicture[scale=1]
coordinate (A) at (0,0);
coordinate (B) at (1,1);
coordinate (C) at (1,2);
path (2,0) coordinate (D) ++ (rnd*120:2) coordinate (E);
draw[blue] (B)--(A)--(C);
draw[blue] ([mirror at=D--E]B)--([mirror at=D--E]A)--([mirror at=D--E]C);
draw[red] (D)--(E);
endtikzpicture
enddocument

Fourth option: A style that computes the reflected coordinates. (Problems: Unfortunately, the syntax in this version requires to specify the coordinate twice, e.g. there are two Bs in ([reflect=B at D--E]B), and it does not work well with global transformations like scale=0.55. Other than that it uses this answer which shows how to compute the orthogonal projection of a point on a line.)

documentclass[tikz]standalone
usetikzlibrarycalc
tikzsetreflect/.style args=#1 at #2--#3shift=%
($2*($(#2)!(#1)!(#3)$)-2*(#1)$)

begindocument
begintikzpicture
coordinate (A) at (0,0);
coordinate (B) at (1,1);
coordinate (C) at (1,2);
coordinate (D) at (2,0);
coordinate (E) at (2,3);

draw[blue] (B)--(A)--(C);
draw[red] (D)--(E);

draw[orange] ([reflect=B at D--E]B) -- ([reflect=A at D--E]A)
-- ([reflect=C at D--E]C);
endtikzpicture
enddocument

Side-remark: Paul Gaborit's solution seems to work.

documentclass[tikz]standalone
usetikzlibraryspy,decorations.fractals
tikzset
 mirror scope/.is family,
 mirror scope/angle/.store in=mirrorangle,
 mirror scope/center/.store in=mirrorcenter,
 mirror setup/.code=tikzsetmirror scope/.cd,#1,
 mirror scope/.style=mirror setup=#1,spy scope=
 rectangle,lens=rotate=mirrorangle,yscale=-1,rotate=-1*mirrorangle,size=80cm,

newcommandmirror[1]spy[overlay,#1] on (mirrorcenter) in node at (mirrorcenter)

begindocument
begintikzpicture
coordinate (A) at (0,0);
coordinate (B) at (1,1);
coordinate (C) at (1,2);
coordinate (D) at (2,0);
coordinate (E) at (2,3);
 draw [help lines] (0,0) grid (4,3);
 beginscope[mirror scope=center=2,0,angle=90]
 draw[blue] (B) -- (A) -- (C);
 draw[red] (D) -- (E);
 mirror;
 endscope
endtikzpicture
enddocument

One possibility is using the tkz-euclide package.

To define A1 the mirror image of the point A with respect to the line DE use: tkzDefPointBy[reflection=over D--E](A) tkzGetPointA1

documentclass[border=1cm,tikz]standalone
usepackagetkz-euclide
begindocument
begintikzpicture
draw[help lines,dashed](0,0)grid(4,4);
coordinate (A) at (0,0);
coordinate (B) at (1,1);
coordinate (C) at (1,2);
coordinate (D) at (2,0);
coordinate[label=E] (E) at (2,3);

tkzDefPointBy[reflection=over D--E](A) tkzGetPointA1
tkzDefPointBy[reflection=over D--E](B) tkzGetPointB1
tkzDefPointBy[reflection=over D--E](C) tkzGetPointC1

draw[blue] (B)--(A)--(C);
draw[red] (D)--(E);

draw [green] (B1)--(A1)--(C1);
endtikzpicture
enddocument

A PSTricks solution only for comparison purposes.

documentclass[pstricks,border=12pt]standalone
usepackagepst-eucl
begindocument
pspicture[PointName=none,PointSymbol=none](8,3)
pstGeonode(1,3)A(0,0)B(2,2)C(4,3)X(4,0)Y
pstOrtSymXYA,B,C[A',B',C']
psline[linecolor=blue](X)(Y)
psline[linecolor=red](A)(B)(C)
psline[linecolor=red](A')(B')(C')
endpspicture
enddocument