Curve synthesis - Adding two curves to get another one

Question

Case 1: Let's say I have a few coordinates (A), (B), (C), (W), (X), (Y), and (Z) (possibily obtained through intersection, rotation, and whatnots).

I draw two piecewise linear curves through (A), (B), (C) on one hand and (W), (X), (Y), and (Z) on the other hand.

Case 2: I have two curves defined by two functions, say f(x) = x + 3 and g(x) = - x + 2.

Case 3: One of the curves is defined by coordinates and the other by a function

In all these cases, is there a way to "add" both curves to obtain a third one (say h(x) = f(x) + g(x))?

For sake of simplicity, all curves I'm interested in at the moment have the nice property of being piecewise linear but feel free to relax this if there is an all encompassing solution.

Example for case 1:

\documentclass{minimal}
\usepackage{tikz}
\usetikzlibrary{calc,positioning}

\begin{document}

\begin{tikzpicture}
\newcounter{i}

% Curve 'one' .. is there a way to give it a name? 
% Coordinates saved for further labeling, and so on.
% How to do it smoothly?
\begin{scope}[red] 
  \setcounter{i}{0}%
  \foreach \point in {(0,4),(4,1),(10,6)} {%
    \node[coordinate] (one-\arabic{i}) at \point { };%
    \fill (one-\arabic{i}) circle (0.1);%
    \stepcounter{i}
  }
  \draw (one-0) -- (one-1) -- (one-2);
\end{scope}

% Curve 'two' .. 
% If I don't need to save the coordinates for later use, 
% there must be a better way to draw?
\begin{scope}[green] 
  \setcounter{i}{0}%
  \foreach \point in {(0,1),(3,5),(6,5),(10,2)} {%
    \node[coordinate] (two-\arabic{i}) at \point { };%
    \fill (two-\arabic{i}) circle (0.1);%
    \stepcounter{i}
  }
  \draw (two-0) -- (two-1) -- (two-2) -- (two-3);
\end{scope}

%%% What I want is the "sum" of curves one and two ...
\end{tikzpicture}

\end{document}

Answer 1

Here is a solution using intersections and calc libraries.

Method

1. Draw and name (via name path) a first continuous path (x ranges 0 to 10).

2. Draw and name (via name path) a second continuous path (x ranges 0 to 10).

3. Foreach x in 0,...,100 :

   1. Draw and name a vertical line at x/10 (y ranges y-min to y-max of your paths).
   2. Search intersections (via name intersections) between this line and the first path and between this line and the second path (you should find a single intersection for each path).
   3. Calculate the sum of y-coordinates (via let operation) of these two intersections and create a new coordinate at (x/10, sum) named sum-x.
4. Draw a line through points (sum-0) to (sum-100).

Result

Code

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{calc,intersections}

\begin{document}
\begin{tikzpicture}[line width=1pt]
  % a grid
  \draw[help lines] (0,-.5) grid (10,10);
  % x axis
  \draw[-latex,thick] (0,0) -- (10,0) node[right]{$x$};

  % red line
  \def\lineone{(0,4),(4,1),(8,6),(10,6)}
  \foreach \point[count=\c] in \lineone {%
    \coordinate[at=\point] (one-\c);%
    %\fill[red] (one-\c) circle (0.1);%
  }
  \draw[red,name path=one] (one-1) \foreach \i in {2,...,\c}{-- (one-\i)} node[right]{one};

  % blue line
  \def\linetwo{(0,1),(3,5),(4,2),(6,5),(10,2)}
  \foreach \point[count=\c] in \linetwo {%
    \coordinate[at=\point] (two-\c);%
    %\fill[blue] (two-\c) circle (0.1);%
  }
  \draw[blue,name path=two] (two-1) \foreach \i in {2,...,\c}{-- (two-\i)} node[right]{two};

  % one + two
  \foreach \c in {0,...,100} {
    \pgfmathsetmacro{\x}{\c/10}
    \path[name path=line] (\x,0) -- (\x,6);
    \path[name intersections={of=one and line,name=newone}];
    \path[name intersections={of=two and line,name=newtwo}];
    \path let \p1=(newone-1), \p2=(newtwo-1) in 
    (\x1,\y1+\y2) coordinate (sum-\c);
  }
  \draw[red!50!blue]
  (sum-0) \foreach \x in {1,...,100}{-- (sum-\x)} node[right]{one + two};

  % a green function
  \draw[green!50!black,name path=three]
  plot[domain=0:10,samples=100,smooth] (\x,{sin(3*\x r)+2}) node[right]{three};

  % one + three
  \foreach \c in {0,...,100} {
    \pgfmathsetmacro{\x}{\c/10}
    \path[name path=line] (\x,0) -- (\x,6);
    \path[name intersections={of=one and line,name=newone}];
    \path[name intersections={of=three and line,name=newthree}];
    \path let \p1=(newone-1), \p2=(newthree-1) in 
    (\x1,\y1+\y2) coordinate (sum-\c);
  }
  \draw[red!50!green!50!black]
  (sum-0) \foreach \x in {1,...,100}{-- (sum-\x)} node[right]{one + three};

  % a orange function
  \draw[orange,name path=four]
  plot[domain=0:10,samples=100,smooth] (\x,{cos(1*\x r)+4}) node[right]{four};

  % four + three
  \foreach \c in {0,...,100} {
    \pgfmathsetmacro{\x}{\c/10}
    \path[name path=line] (\x,0) -- (\x,6);
    \path[name intersections={of=four and line,name=newfour}];
    \path[name intersections={of=three and line,name=newthree}];
    \path let \p1=(newfour-1), \p2=(newthree-1) in 
    (\x1,\y1+\y2) coordinate (sum-\c);
  }
  \draw[orange!50!green!50!black]
  (sum-0) \foreach \x in {1,...,100}{-- (sum-\x)} node[right]{three + four};

\end{tikzpicture}
\end{document}