# How to define where a line intersects its reflection

It is relatively easy to be able to find where two parametric functions intersect each other, but how about when one is a function and the second is its reflection? In the example below is it possible to define the value of intersection on the y axis?

H4a={{3.99932*10^-8, 2730.83720332692}, {4.04932*10^-8, 1630.73548351008},{4.09932*10^-8, 1277.35595196883}, {4.14932*10^-8, 1072.70360763529}, {4.19932*10^-8, 933.77632531279}, {4.24932*10^-8, 831.27749281985}, {4.29932*10^-8, 751.59481243293}, {4.34932*10^-8, 687.36617073509}, {4.39932*10^-8, 634.19609669597}, {4.44932*10^-8, 589.26940583597}, {4.49932*10^-8, 550.68537232488}, {4.54932*10^-8, 517.10668401615}, {4.59932*10^-8, 487.56071045906}, {4.64932*10^-8, 461.32048125234}, {4.69932*10^-8, 437.83006374543}, {4.74932*10^-8, 416.65596392318}, {4.79932*10^-8, 397.45444170215}, {4.84932*10^-8, 379.94891550006}, {4.89932*10^-8, 363.91396417635}, {4.94932*10^-8, 349.16376064501}, {4.99932*10^-8, 335.54355361763}, {5.04932*10^-8, 322.92328998620}, {5.09932*10^-8, 311.19276921945}, {5.14932*10^-8, 300.25791227827}, {5.19932*10^-8, 290.03785401278}, {5.24932*10^-8, 280.46265226146}, {5.29932*10^-8, 271.47146465654}, {5.34932*10^-8, 263.01108422334}, {5.39932*10^-8, 255.03475313929}, {5.44932*10^-8, 247.50119424772}, {5.49932*10^-8, 240.37381457968}, {5.54932*10^-8, 233.62004588761}, {5.59932*10^-8, 227.21079516706}, {5.64932*10^-8, 221.11998411525}, {5.69932*10^-8, 215.32416099476}, {5.74932*10^-8, 209.80217181913}, {5.79932*10^-8, 204.53488043315}, {5.84932*10^-8, 199.50492912035}, {5.89932*10^-8, 194.69653298106}, {5.94932*10^-8, 190.09530259197}, {5.99932*10^-8, 185.68809046291}};

Sd1s = Interpolation[H4a];
Xd1s =
Plot[{Sd1s[r3]},{r3, 40*10^-9, 60*10^-9}, PlotRange -> All, AxesOrigin -> {0,0}];
Show[
Xd1s,
Xd1s /.
L_Line :>
{Red,
GeometricTransformation[L, ReflectionTransform[{1, 0}, {45*10^-9, 0}]]},
PlotRange -> All, ImageSize -> 400]


• Interpolation[ReflectionTransform[{1, 0}, {45*10^-9, 0}] /@ H4a][0] – Dr. belisarius Jan 25 '16 at 22:47
• Hmmm the above is the y-value for x==0 – Dr. belisarius Jan 25 '16 at 22:52
• Don't a function and its reflection in a vertical axis meet each other at the reflection axis? In this case that would be Sd1s[45*10^(-9)] ==> 550.197 – Sjoerd C. de Vries Jan 25 '16 at 22:53
• @SjoerdC.deVries Iff the function is defined at that point :) – Dr. belisarius Jan 25 '16 at 22:57
• Thanks for your replies guys:) Is it possible to find a more general rule? They could intersect in more than 1 point. – George Jan 25 '16 at 23:05

Sd1s = Interpolation[H4a];
Sd1sR = Interpolation[{-#1 + 2*45*10^-9, #2} & @@@ H4a];
pt = {x /. #, Sd1s[x] /. #} &@ FindRoot[Sd1s[x] == Sd1sR[x], {x, 45*10^-9}]
(* {4.5*10^-8, 550.197} *)


and

Show[
Plot[{Sd1s[r3]}, Evaluate@{r3, Sequence @@ First@Sd1s["Domain"]}, PlotRange -> All, AxesOrigin -> {0, 0}]
, Plot[{Sd1sR[r3]}, Evaluate@{r3, Sequence @@ First@Sd1sR["Domain"]}, PlotRange -> All, AxesOrigin -> {0, 0}, PlotStyle -> Red]
, Epilog -> {Blue, PointSize[0.2], Point[pt]}
]