Tikz: Shifting a curve to get a tangent condition

Question

I want to achieve a very similar thing like in this post Line tangent to a a curve (using plot or controls), starting at a point. But I need a more simple solution. I can create a first tangent curve by connecting three points. That's enough for my purpose (visual judgement).

Is there a way to 'shift' the gray curve and make it hit the red line as a tangent? Maybe with some intersection package? I don't want to calculate further points of intersection by hand.

\begin{center}
\begin{tikzpicture}[thick]
    % Axis and coordinates
    \coordinate (y) at (0,6);
    \coordinate (o) at (0,0);
    \coordinate (x) at (6,0);
    \draw[<->,line width=1.5pt] (y) node[above]{$C_{t+1}$} -- (o) -- (x) node[below,yshift=-2mm]{$C_t$};

        \draw[thick, blue] (0,5) -- (5,0);
        \draw[dash pattern=on 6pt off 3pt,thick, red] (0,2.5) -- (5,0);
        \draw[dash pattern=on 6pt off 3pt,thick, green] (0,3.5) -- (3.5,0);

        \filldraw [black] (3,2) circle (2pt) node[right]{A};
        \coordinate (A) at (3,2);

        \filldraw [black] (1.5,4) circle (2pt);
        \filldraw [black] (5,0.5) circle (2pt);
        \coordinate (ul) at (1.5,4);
        \coordinate (ur) at (5,0.5);

        \draw [ultra thick,gray] (ul) to[out=-60,in=133] (A) to[out=90-133,in=152] (ur);
        \end{tikzpicture}
\end{center}

PS: I know the system is atm relative small to illustrate the tangent point anyway. I may adjust overall size to 8 units later. Thank you very much, Mac.

Update: There is a small but crucial mistake in my request ^.° of course I need more than a y-shift but a parallel curve, with identical perpendicular distance.

Update 2: To make matters worse, both suggested solutions give a good solution to my conrecte example diagram. But it's hard (for me) to modify them in order to show different aspects. I need another version of a tangent condition (of the same parallel curves) in which there a curve which both tangents the red and the green line.

Maybe it would be a better idea to start with the lower curve, constructing it from two given point on the red/green line?

Answer 1

The following may need some fine tuning, and also relies on you not needing exactly those coordinates for ul and ur.

So what I do is, instead of drawing the grey lines with a to[in=...,out=...], I draw a circular arc. Well, actually two. To avoid having to calculate the start of the arc, I draw two arcs starting in A, using (A) arc[start angle=225,delta angle=20,radius=9cm]; The start angle is due to the fact that the blue line has an angle of 45 degrees with the horizontal. The delta angle decides how far to draw the line, and I draw one with a positive delta angle, one with a negative. The radius is just trial and error.

The second arc is drawn mostly the same way, but with a slightly different delta angle on one side, and a larger radius, so the arcs are from concentric circles. The starting point is shifted by a certain distance (0.41cm - trial and error) away from A in a direction of 225 degrees, and this distance is the same as the difference in radius.

\documentclass[border=5mm]{standalone}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}[thick]
% Axis and coordinates
\coordinate (y) at (0,6);
\coordinate (o) at (0,0);
\coordinate (x) at (6,0);
\draw[<->,line width=1.5pt] (y) node[above]{$C_{t+1}$} -- (o) -- (x) node[below,yshift=-2mm]{$C_t$};
%
\draw[thick, blue] (0,5) -- (5,0);
\draw[dash pattern=on 6pt off 3pt,thick, red] (0,2.5) -- (5,0);
\draw[dash pattern=on 6pt off 3pt,thick, green] (0,3.5) -- (3.5,0);
%
\filldraw [black] (3,2)   coordinate[label=right:A] (A) circle[radius=2pt];

\pgfmathsetlengthmacro{\radA}{9cm}
\pgfmathsetlengthmacro{\RadOffset}{0.41cm}
\pgfmathsetlengthmacro{\radB}{\radA + \RadOffset}
\draw [ultra thick,gray] (A)
  arc[start angle=225,delta angle=-15,radius=\radA] coordinate (ul)
  (A)
  arc[start angle=225,delta angle=15,radius=\radA] coordinate (ur);

\fill (ul) circle[radius=2pt] (ur) circle[radius=2pt];

\draw [ultra thick,gray] (A) ++(225:\RadOffset) arc[start angle=225,delta angle=-15,radius=\radB]  (A)++(225:\RadOffset) arc[start angle=225,delta angle=20,radius=\radB];
\end{tikzpicture}
\end{document}

Old answer

You can always add [yshift=-0.5cm] at the start of each of the coordinates, e.g. ([yshift=-0.5cm]ur). To shift both horizontally and vertically use e.g. ([shift={(x,y)}]ur)

\documentclass{article}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}[thick]
% Axis and coordinates
\coordinate (y) at (0,6);
\coordinate (o) at (0,0);
\coordinate (x) at (6,0);
\draw[<->,line width=1.5pt] (y) node[above]{$C_{t+1}$} -- (o) -- (x) node[below,yshift=-2mm]{$C_t$};
%
\draw[thick, blue] (0,5) -- (5,0);
\draw[dash pattern=on 6pt off 3pt,thick, red] (0,2.5) -- (5,0);
\draw[dash pattern=on 6pt off 3pt,thick, green] (0,3.5) -- (3.5,0);
%
\filldraw [black] (3,2)   coordinate[label=right:A] (A);
\filldraw [black] (1.5,4) coordinate (ul) circle[radius=2pt];
\filldraw [black] (5,0.5) coordinate (ur) circle[radius=2pt];

\draw [ultra thick,gray] (ul) to[out=-60,in=133] (A) to[out=90-133,in=152] (ur);

\newcommand{\curveshift}{(0cm,-0.5cm)}
\draw [ultra thick,gray] ([shift=\curveshift]ul) to[out=-60,in=133] ([shift=\curveshift]A) to[out=90-133,in=152] ([shift=\curveshift]ur);


\end{tikzpicture}
\end{document}

Answer 2

Here's an Asymptote version. The idea is to rotate everything virtually so that the red line is horizontal, and then compute the bounding box of the gray curve. Its minimum coordinate (i.e., "height") is the desired shifting distance.

Update: Abstracted shifting/tangent code into function. Added a curve tangent to the green dashed line as well.

\documentclass{standalone}
\usepackage{asymptote}
\begin{document}
\begin{asy}
unitsize(1cm);
pair y = (0,6);
pair o = (0,0);
pair x = (6,0);
draw(y -- o -- x, arrow=Arrows(TeXHead), p=linewidth(1.5pt));
label("$C_{t+1}$", position=y, align=N);
label("$C_t$", position=x, align=2S);

pen tkzthick = linewidth(0.6pt);
pen tkzultrathick = linewidth(1.6pt);
pen mydashpattern = linetype(new real[] {6pt, 3pt});

draw((0,5) -- (5,0), tkzthick + blue);
pair redstart = (0,2.5), redend = (5,0);
pair greenstart = (0,3.5), greenend = (3.5,0);
draw(redstart -- redend, mydashpattern + tkzthick + red);
draw(greenstart -- greenend, mydashpattern + tkzthick + green);

pair A = (3,2);
dot(A, L="$A$", align=E);

pair ul = (1.5,4);
pair ur = (5,0.5);
dot(ul);
dot(ur);

path curve = ul {dir(-60)} .. A {dir(90-133)} .. {-dir(152)} ur;
draw(curve, tkzultrathick + gray);

path touchline(path curve, pair a, pair b) {
  pair vector = b - a;
  // Compute the angle of the line segment.
  real angle = degrees(atan2(vector.y, vector.x));
  // Find the transformation to make the line coincide with the x-axis.
  transform T = rotate(-angle) * shift(-a);
  // What is the y-coordinate of the bottom of the curve after this transformation?
  real height = min(T * curve).y;
  // Apply the transformation, lower the curve to touch the line, and then put
  // it all back in place.
  return inverse(T) * shift((0,-height)) * T * curve;
}

draw(touchline(curve, redstart, redend), tkzultrathick + gray);
// Uncomment next line to add tangent to the green line:
// draw(touchline(curve, greenstart, greenend), tkzultrathick + gray);
\end{asy}
\end{document}

Answer 3

My try of using the tzplot package:

\documentclass[tikz]{standalone}
    
\usepackage{tzplot}

\begin{document}

\begin{tikzpicture}
\tzhelplines[step=.5](6,6)
\tzaxes*(6,6){$C_t$}{$C_{t+1}$}
% budget lines
\tzcoors(5,0)(A0)(0,5)(B0)(0,2.5)(C0);
\tzline[blue]"bgtA"(A0)(B0)
\tzline[red,dashed]"bgtB"(A0)(C0)
% upper curve
\tzvXpointat*{bgtA}{3}(A){$A$}[45] % point A
\tzcoors($(A)+(-1,2.5)$)(AL)($(A)+(2.5,-1)$)(AR);
\tzplotcurve[thick,gray](AL)(A)(AR);
% lower curve
\tzvXpointat*{bgtB}{3.5}(E){$E$}   % point E
\tzpointangle(A0)(C0){\angE}
\tzcoors($(E)+(-2.5,3.5)$)(EL)($(E)+(2,-.4)$)(ER);
\tztos[thick,gray](EL)[out=-80,in=\angE](E)
                      [out=\angE-180,in=180](ER);
% shifted budget line: with some trial and errors
\clip (0,0) rectangle (6,6);
\tzline[green,densely dashed]<0,-1.135>"bgtAA"(A0)(B0) % shift
\tzvXpointat*[fill=none]{bgtAA}{2.25}
\end{tikzpicture}

\end{document}

• Step 1: Draw blue and red budget lines.
• Step 2: Pin down the point A on the blue line and draw the upper curve.
• Step 3: Pin down the point E on the red line and draw the lower curve.
• Step 4: Shift the blue line until it touches the lower curve.
  • You may need some trial and errors to get the (final) green line.

Answer 4

If you don't have exact requirements, cheat.

Wwe can use the nfold library to draw a parallel curve (offset) and we can place a picture at any position along a curve where TikZ sets up the coordinate system so that it is parallel (sloped) to the curve.

We can also instruct both nodes and pics along a curve to be yshifted by the same amount.

Code

\documentclass[tikz]{standalone}
\usetikzlibrary{arrows.meta, nfold, quotes}
\makeatletter
\tikzset{offset/.code=\tikz@addoption
  {\pgfgetpath\tikz@temp\pgfsetpath\pgfutil@empty\pgfoffsetpath\tikz@temp{#1}}}
\makeatother
\begin{document}
\begin{tikzpicture}[
  >=Latex,
  dot/.style={edge node={
    node[shape=circle, sloped, allow upside down, fill,
      inner sep=+.1em, #1, anchor=center]{}}},
  pics/tangent line/.style={
    /tikz/sloped, /tikz/allow upside down, background code={
    \draw[thin, pic actions] (intersection of O--x and {0,0}--right:1)
                          -- (intersection of O--y and {0,0}--left:1);}}]
\coordinate["$C_t$"     below] (x) at (right:5)
 coordinate["$C_{t+1}$" above] (y) at (   up:5)
 coordinate (O) at (0, 0)
 coordinate (ul) at (1.5, 3) coordinate (ur) at (4, 1);
\path[draw=gray, thick, postaction={draw, offset=1cm}, bend left]
  (ur) to["$A$"', dot/.list={at start, midway, at end,
                             {near start, yshift=1cm}, {near end, yshift=1cm}}]
  pic [blue]                                  {tangent line}
  pic [green, dashed, yshift=1cm, near   end] {tangent line}
  pic [red,   dashed, yshift=1cm, near start] {tangent line} (ul);
\draw[thick, <->] (y)  |- (x);
\end{tikzpicture}
\end{document}

Output