Tikz : Draw smooth convex

Question

I am looking for a command in tikz which takes points as arguments and, if these points are counter-clockwise the vertices of a convex polygon, draws a convex whose boundary is smooth and whose boundary contains the points I gave as arguments. I expect this to be rather simple (using Bézier curves for example) but I am a beginner and have never written such command on Latex before.

I have already tried commands like plot smooth or Hobby, and it often does great job, but sometimes in bad cases it does not give me a convex and I don't want to think too much each time I give my vertices.

Here is a bad example.

\documentclass[10pt,a4paper]{article}
\usepackage{tikz}
\usetikzlibrary{hobby}

\begin{document}
\begin{tikzpicture}[scale=5]
\coordinate (a) at (0,0);
\coordinate (b) at (3,0);
\coordinate (c) at (3,1);
\coordinate (d) at (1.5,1);
\coordinate (e) at (0,1);

\path[draw,use Hobby shortcut,closed=true]
(a)..(b)..(c)..(d)..(e);

\draw (a) node{$\bullet$};
\draw (b) node{$\bullet$};
\draw (c) node{$\bullet$};
\draw (d) node{$\bullet$};
\draw (e) node{$\bullet$};

\draw (a)--(b)--(c)--(d)--(e)--(a);
\end{tikzpicture}
\end{document}

I added the example because I was asked to, and it was a good remark : I realized that the algorithm I was thinking about would not work because it does not work with the extreme example I put above. So it might be more difficult than I thought, it might even be not worth look at, and better to look for a way not to ask Hobby to display bad settings of points.

A more complex command I am also looking for is a command which does almost the same, except that I can specify at each vertex :

1. If I want the curve between this vertex and the following to be a straight line (it can forces the convex not to be smooth at some vertices, for example if I ask this condition to be fulfilled for each vertex I get the polygon which does just link one by one the vertices counter-clockwise).

2. If I want the convex to a bit sharp at the vertex (for example giving a percentage of sharpness), the sharpest convex being of course the polygon which does just link the vertices one by one counter-clockwise.

In fact there cannot be a general answer to my question because one can consider the following very bad situation. The points are counter-clockwise

(0,0) (0.5,0) (1,0) (1,0.5) (1,1) (0.5,1) (0,1) (0,0.5)

Then there is only convex whose boundary contains the points : the square itself, which is not smooth (by the way we see here that convexity can be very rigid, far more than just smoothness since a finite number of points determine entirely the shape of my convex). Sorry for asking sort of non rigorous question. Still there must be a way to have a function which displays smooth convex when it is possible and else adds some singularities (the fewest possible).

Is there a good reference to learn quickly how to build commands like the one I need ?

Answer 1

UPDATE: 1st attempt to answer the updated question.

\documentclass[tikz,border=3.14mm]{standalone}

\begin{document}
\begin{tikzpicture}[scale=5]
\coordinate (a) at (0,0);
\coordinate (b) at (3,0);
\coordinate (c) at (3,1);
\coordinate (d) at (1.5,1);
\coordinate (e) at (0,1);

\foreach \X in {a,...,e}
{\fill (\X) circle (0.6pt);}

\draw (a) to[out=-90,in=-90] (b)--(c)--(d)--(e)-- cycle;
\draw[blue] (a) to[out=-60,in=-120] (b);
\draw[red] (a) to[out=-30,in=-150] (b);
\end{tikzpicture}
\end{document}

Just for fun, no competitor of the answers to this question. You can use any smooth plot through the coordinates and draw a contour around them. If the contour is very sharp, you may need to decrease contour step.

\documentclass[tikz,border=7pt]{standalone}
\usetikzlibrary{decorations,decorations.markings} 
\pgfkeys{/tikz/.cd,
    contour distance/.store in=\ContourDistance,
    contour distance=-10pt, % for the other orientation use a +
    contour step/.store in=\ContourStep,
    contour step=1pt,
}

\pgfdeclaredecoration{closed contour}{initial}
{% 
\state{initial}[width=\ContourStep,next state=cont] {
    \pgfmoveto{\pgfpoint{\ContourStep}{\ContourDistance}}
    \pgfcoordinate{first}{\pgfpoint{\ContourStep}{\ContourDistance}}
    \pgfpathlineto{\pgfpoint{0.3\pgflinewidth}{\ContourDistance}}
    \pgfcoordinate{lastup}{\pgfpoint{1pt}{\ContourDistance}}
    \xdef\marmotarrowstart{0}
  }
  \state{cont}[width=\ContourStep]{
     \pgfmoveto{\pgfpointanchor{lastup}{center}}
     \pgfpathlineto{\pgfpoint{\ContourStep}{\ContourDistance}}
     \pgfcoordinate{lastup}{\pgfpoint{\ContourStep}{\ContourDistance}}
  }
  \state{final}[width=\ContourStep]
  { % perhaps unnecessary but doesn't hurt either
    \pgfmoveto{\pgfpointanchor{lastup}{center}}
    \pgfpathlineto{\pgfpointanchor{first}{center}}
  }
}

\begin{document}
  \begin{tikzpicture}
    \draw[decoration={closed contour},decorate] plot[smooth cycle] coordinates {(0,0) (2,0) (3,1) (0,2)};
    \draw plot[smooth cycle,mark=*] coordinates {(0,0) (2,0) (3,1) (0,2)};
  \end{tikzpicture}

  \begin{tikzpicture}
    \draw[decoration={closed contour},decorate] plot[smooth cycle,tension=1.5] coordinates {(0,0) (2,0) (3,1) (0,2)};
    \draw plot[smooth cycle,mark=*,tension=1.5] coordinates {(0,0) (2,0) (3,1) (0,2)};
  \end{tikzpicture}

    \begin{tikzpicture}
    \draw[decoration={closed contour},decorate] plot[smooth cycle,tension=0.5] coordinates {(0,0) (2,0) (3,1) (0,2)};
    \draw plot[smooth cycle,mark=*,tension=0.5] coordinates {(0,0) (2,0) (3,1) (0,2)};
  \end{tikzpicture}
\end{document}

A SECOND METHOD: Based on this answer. I don't know what the real life applications are like, but if this turns out to useful, I'll be happy to make it more user friendly.

\documentclass[margin=3.14mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{calc,decorations.pathreplacing,decorations.pathmorphing}

\makeatletter
% to produce automaticaly homothetic paths from https://tex.stackexchange.com/a/72753/121799
\newcounter{homothetypoints} % number of vertices of path
\tikzset{
  % homothety is a family...
  homothety/.style={homothety/.cd,#1},
  % ...with some keys
  homothety={
    % parameters
    scale/.store in=\homothety@scale,% scale of current homothetic transformation
    center/.store in=\homothety@center,% center of current homothetic transformation
    name/.store in=\homothety@name,% prefix for named vertices
    % default values
    scale=1,
    center={0,0},
    name=homothety,
    % initialization
    init memoize homothetic path/.code={
      \xdef#1{}
      \setcounter{homothetypoints}{0}
    },
    % incrementation
    ++/.code={\addtocounter{homothetypoints}{1}},
    % a style to store an homothetic transformation of current path into #1 macro
    store in/.style={
      init memoize homothetic path=#1,
      /tikz/postaction={
        decorate,
        decoration={
          show path construction,
          moveto code={
            % apply homothetic transformation to this segment and add result to #1
            \xdef#1{#1 ($(\homothety@center)!\homothety@scale!(\tikzinputsegmentfirst)$)}
            % name this vertex
            \coordinate[homothety/++](\homothety@name-\arabic{homothetypoints})
            at ($(\homothety@center)!\homothety@scale!(\tikzinputsegmentfirst)$);
          },
          lineto code={
            % apply homothetic transformation to this segment and add result to #1
            \xdef#1{#1 -- ($(\homothety@center)!\homothety@scale!(\tikzinputsegmentlast)$)}
            % name this vertex
            \coordinate[homothety/++] (\homothety@name-\arabic{homothetypoints})
            at ($(\homothety@center)!\homothety@scale!(\tikzinputsegmentlast)$);
          },
          curveto code={
            % apply homothetic transformation to this segment and add result to #1
            \xdef#1{#1
              .. controls ($(\homothety@center)!\homothety@scale!(\tikzinputsegmentsupporta)$)
              and ($(\homothety@center)!\homothety@scale!(\tikzinputsegmentsupportb)$)
              .. ($(\homothety@center)!\homothety@scale!(\tikzinputsegmentlast)$)}
            % name this vertex
            \coordinate[homothety/++] (\homothety@name-\arabic{homothetypoints})
            at ($(\homothety@center)!\homothety@scale!(\tikzinputsegmentlast)$);
          },
          closepath code={
            % apply homothetic transformation to this segment and add result to #1
            \xdef#1{#1 -- cycle ($(\homothety@center)!\homothety@scale!(\tikzinputsegmentlast)$)}
          },
        },
      },
     },
    store coordinates in/.style={
      init memoize homothetic path=#1,
      /tikz/postaction={
        decorate,
        decoration={
          show path construction,
          moveto code={
            % apply homothetic transformation to this segment and add result to #1
            \xdef#1{#1 ($(\homothety@center)!\homothety@scale!(\tikzinputsegmentfirst)$)}
            % name this vertex
            \coordinate[homothety/++](\homothety@name-\arabic{homothetypoints})
            at ($(\homothety@center)!\homothety@scale!(\tikzinputsegmentfirst)$);
          },
          lineto code={
            % apply homothetic transformation to this segment and add result to #1
            \xdef#1{#1 ($(\homothety@center)!\homothety@scale!(\tikzinputsegmentlast)$)}
            % name this vertex
            \coordinate[homothety/++] (\homothety@name-\arabic{homothetypoints})
            at ($(\homothety@center)!\homothety@scale!(\tikzinputsegmentlast)$);
          },
          curveto code={
            % apply homothetic transformation to this segment and add result to #1
            \xdef#1{#1 ($(\homothety@center)!\homothety@scale!(\tikzinputsegmentlast)$)}
            % name this vertex
            \coordinate[homothety/++] (\homothety@name-\arabic{homothetypoints})
            at ($(\homothety@center)!\homothety@scale!(\tikzinputsegmentlast)$);
          },
          closepath code={
            % apply homothetic transformation to this segment and add result to #1
            \xdef#1{#1 }
          },
        },
      },
    },
  },
}
\makeatother

\begin{document}
\begin{tikzpicture}[font=\bfseries\sffamily]

  % some styles

  % draw a path (and memomize its definition into \mypath with points named A-1, A-2,...)
  \draw[homothety={store in=\mypath,name=A}]
  plot[mark=*] coordinates {(0,0) (2,0) (3,1) (0,2)} -- cycle;
  % compute the barycentric coordinate (can be automatized)
  \coordinate (A-center) at (barycentric cs:A-1=0.25,A-2=0.25,A-3=0.25,A-4=0.25);
  % compute the homothetic hull
  \path[homothety={store coordinates in=\secondpath,scale=1.2,center=A-center}] \mypath;
  % draw a smooth version of the hull
  \draw[blue] plot [smooth cycle] coordinates {\secondpath};
\end{tikzpicture}
\end{document}

Answer 2

Ok after some work I managed to write some code (probably one of the most badly written code in history, I already begin to apologize) which does what I want (I hope). I am going to write it here so that I can say that my question is answered. However I am aware it is not a satisfactory and not easy to understand ; I hope it is better than nothing.

\documentclass[10pt,a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}

\usepackage{tikz}
\usetikzlibrary{calc}
\usepackage{xstring}
\usepackage{intcalc}
\usepackage{comment}


\newcounter{mycount}

%a command to compute the length of a list
\newcommand{\length}[1]{
    \setcounter{mycount}{0}
    \foreach \x in {#1}{\stepcounter{mycount}}}


%a command to compute the ith element of a list
\makeatletter
\newcommand{\listnb}[3]{
    \foreach \temp@a [count=\temp@i] in {#1} {
        \IfEq{\temp@i}{#2}{\global\let#3\temp@a\breakforeach}{}}
    \par}
\makeatother


%a command to compute the (i modulo the length of the list)-th element of a list
\newcommand{\listnbmod}[2]{
    \listnb{#1}{\intcalcInc{\intcalcMod{\intcalcDec{#2}}{\themycount}}}}


%a command to get angles
\newcommand{\getanglepoints}[3]{
    \pgfmathanglebetweenpoints{\pgfpointanchor{#2}{center}}
                              {\pgfpointanchor{#3}{center}}
    \global\let#1\pgfmathresult}


%another command to get angles
\newcommand{\getanglelines}[5]{
    \pgfmathanglebetweenlines{\pgfpointanchor{#2}{center}}
                             {\pgfpointanchor{#3}{center}}
                             {\pgfpointanchor{#4}{center}}
                             {\pgfpointanchor{#5}{center}}
    \global\let#1\pgfmathresult}


% a command to get distances
\makeatletter
\newcommand{\getdistance}[3]{
    \pgfpointdiff{\pgfpointanchor{#2}{center}} 
                 {\pgfpointanchor{#3}{center}} 
    \pgf@xa=\pgf@x
    \pgf@ya=\pgf@y
    \pgfmathparse{veclen(\pgf@xa,\pgf@ya)/28.45274/2} 
    \global\let#1\pgfmathresult}
\makeatother


%a command to choose good angles which control the Béziers curves I will glue together
\makeatletter
\newcommand{\chooseangle}[6]{
    \getanglelines{\temp@a}{#1}{#2}{#2}{#3}
    \getanglelines{\temp@b}{#2}{#3}{#3}{#4}
    \getanglelines{\temp@c}{#3}{#4}{#4}{#5}
    \getanglepoints{\temp@d}{#3}{#4}
    \pgfmathsetmacro\temp@e{ifthenelse(greater(\temp@a +\temp@c ,0.001),-\temp@c /(max(\temp@a +\temp@c ,0.001))*\temp@b+\temp@d,-\temp@b /2 +\temp@d)}
    \global\let#6=\temp@e}
\makeatother


%a command to choose good distances to control the Béziers curves
\makeatletter
\newcommand{\choosecontrolpoints}[6]{
    \getanglepoints{\temp@c}{#1}{#2}
    \getdistance{\temp@l}{#1}{#2}
    \pgfmathsetmacro\temp@a{Mod(\temp@c -#3 ,360)}
    \pgfmathsetmacro\temp@b{Mod(#4-\temp@c ,360)}
    \pgfmathsetmacro\temp@la{
        ifthenelse(less(\temp@a,90),
            ifthenelse(less(\temp@b,90),abs(sin(\temp@b))*\temp@l,\temp@l),
            ifthenelse(less(\temp@b,90),abs(sin(\temp@b))*\temp@l,\temp@l))}
    \pgfmathsetmacro\temp@lb{
        ifthenelse(less(\temp@b,90),
            ifthenelse(less(\temp@a,90),abs(sin(\temp@a))*\temp@l,\temp@l),
            ifthenelse(less(\temp@a,90),abs(sin(\temp@a))*\temp@l,\temp@l))}
    \global\let#5=\temp@la
    \global\let#6=\temp@lb}
\makeatother


%the final command
\makeatletter
\newcommand{\cvx}[1]{
    \foreach \i in {1,...,\themycount} {
        \listnbmod{#1}{\intcalcSub{\i}{2}}{\A}
        \listnbmod{#1}{\intcalcSub{\i}{1}}{\B}
        \listnbmod{#1}{\i}{\C}
        \listnbmod{#1}{\intcalcAdd{\i}{1}}{\D}
        \listnbmod{#1}{\intcalcAdd{\i}{2}}{\E}
        \listnbmod{#1}{\intcalcAdd{\i}{3}}{\F}
        \chooseangle{\A}{\B}{\C}{\D}{\E}{\a}
        \chooseangle{\B}{\C}{\D}{\E}{\F}{\b}
        \choosecontrolpoints{\C}{\D}{\a}{\b}{\la}{\lb} 
        \coordinate (G) at ($(\C)+(\a:\la)$);
        \coordinate (H) at ($(\D)+(\b+180:\lb)$);
        \draw (\C) .. controls (G) and (H) .. (\D);}}
\makeatother


%examples where it works
\begin{document}
\begin{tikzpicture}
\coordinate (A) at (0,0);
\coordinate (B) at (1,0);
\coordinate (C) at (0.55,0.55);
\coordinate (D) at (0,1);
\length{A,B,C,D}
\cvx{A,B,C,D}
\end{tikzpicture}

\begin{tikzpicture}
\coordinate (A) at (0,0);
\coordinate (B) at (1,0);
\coordinate (C) at (0.5,0.5);
\coordinate (D) at (0,1);
\coordinate (E) at (0,0.5);
\length{A,B,C,D,E}
\cvx{A,B,C,D,E}
\end{tikzpicture}
\end{document}

And I still don't know how to put images. I am going to look at that soon but not now.