[Tex/LaTex] Tikz: Shifting a curve to get a tangent condition

curve fittingintersectionstikz-pgf

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?

Best Answer

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}

Related Solutions

[Tex/LaTex] Tangent to circle given point on circle

The voodoo effect comes from the fact that when a coordinate is omitted origin is assumed and this happens to be the center of the circle. If you change the starting point the mystery goes away rather quickly.

I've placed more arrows to show the effect when the paths start from different coordinates.

\documentclass[tikz]{standalone}
\usetikzlibrary{calc}
\def\rad{2cm}

\begin{document}
\begin{tikzpicture}
 [point/.style = {draw, circle, fill=black, inner sep=0.5pt}]
\draw[style=help lines] (0,-1) grid[step=1cm] (5,4);

\node (C) at (2,1) [point,label=0:C]{};
\draw (C) circle (\rad);
\path (2,1) node[point,label={180:P}] (P)  at +(120:\rad){};

\foreach \x in {0,10,...,90}{
\draw[-latex,draw=blue,thick] (2,1) -- (P) -- ([turn]\x:2cm);
\draw[-latex,draw=red] (P) -- ([turn]\x:2cm);% You can add (0,0) -- as an initial point too
}
\end{tikzpicture}
\end{document}

enter image description here

As you can see, when initiated from a different point the red arrows loose the magic tangentiality but rather follow the incoming angle to that point (though blue arrows still preserve since the initial point is given). When the initial point is omitted it assumes that the path starts from (0,0) hence there is an inherent illusion of guessing the tangent.

[Tex/LaTex] TikZ: Drawing an arc from an intersection to an intersection

A solution which allows to draw intersection segments of any two intersections is available as tikz library fillbetween.

This library works as general purpose tikz library, but it is shipped with pgfplots and you need to load pgfplots in order to make it work:

\documentclass{standalone}

\usepackage{tikz}
\usepackage{pgfplots}
\usetikzlibrary{fillbetween}

\begin{document}

\begin{tikzpicture}

\draw [name path=red,red] (120:1.06) circle (1.9);
%\draw [name path=yellow,yellow] (0:1.06) circle (2.12);
\draw [name path=green,green!50!black] (0:0.77) circle (2.41);
\draw [name path=blue,blue] (0:0) circle (1.06);

% substitute this temp path by `\path` to make it invisible:
\draw[name path=temp1, intersection segments={of=red and blue,sequence=L1}];
\draw[red,-stealth,ultra thick, intersection segments={of=temp1 and green,sequence=L3}];

\end{tikzpicture}

\end{document}

enter image description here

The key intersection segments is described in all detail in the pgfplots reference manual section "5.6.6 Intersection Segment Recombination"; the key idea in this case is to

create a temporary path temp1 which is the first intersection segment of red and blue, more precisely, it is the first intersection segment in the Left argument in red and blue : red. This path is drawn as thin black path. Substitute its \draw statement by \path to make it invisible.
Compute the desired intersection segment by intersecting temp1 and green and use the correct intersection segment. By trial and error I figured that it is the third segment of path temp1 which is written as L3 (L = left argument in temp1 and green and 3 means third segment of that path).

The argument involves some trial and error because fillbetween is unaware of the fact that end and startpoint are connected -- and we as end users do not see start and end point.

Note that you can connect these path segments with other paths. If such an intersection segment should be the continuation of another path, use -- as before the first argument in sequence. This allows to fill paths segments:

\documentclass{standalone}

\usepackage{tikz}
\usepackage{pgfplots}
\usetikzlibrary{fillbetween}

\begin{document}

\begin{tikzpicture}

\draw [name path=red,red] (120:1.06) circle (1.9);
%\draw [name path=yellow,yellow] (0:1.06) circle (2.12);
\draw [name path=green,green!50!black] (0:0.77) circle (2.41);
\draw [name path=blue,blue] (0:0) circle (1.06);

% substitute this temp path by `\path` to make it invisible:
\draw[name path=temp1, intersection segments={of=red and blue,sequence=L1}];
\draw[red,fill=blue,-stealth,ultra thick, intersection segments={of=temp1 and green,sequence=L3}]  
    [intersection segments={of=temp1 and green, sequence={--R2}}]
;

\end{tikzpicture}

\end{document}

enter image description here

Best Answer

Old answer

Related Solutions

[Tex/LaTex] Tangent to circle given point on circle

[Tex/LaTex] TikZ: Drawing an arc from an intersection to an intersection

Related Question