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Cell[CellGroupData[{
Cell["Connecting Bezier with continuity of the curvature", "Title"],

Cell["\<\
Luc Barthelet
6/4/97\
\>", "Section"],

Cell[CellGroupData[{

Cell["\<\
The bezier curve is defined by 4 points p0,p1,p2,p3. In  the first example we \
put the four points on the corners of the unit square.\
\>", "Section"],

Cell[BoxData[{
    \(\n\(p[u_]\  := \ 
        p0\ \((u^3)\)\  + \ 3\ p1\ \((u^2)\) \((1 - u)\)\  + \ 
          3\ p2\ u\ \((\((1 - u)\)^2)\)\  + \ 
          p3\ \((\((1 - u)\)^3)\);\)\), "\n", 
    \(\(p0 = {0, 0};\)\), "\n", 
    \(\(p1 = {0, 1};\)\), "\n", 
    \(\(p2 = {1, 1};\)\), "\n", 
    \(\(p3 = {1, 0};\)\)}], "Input"],

Cell[CellGroupData[{

Cell["This calculate the curvature.", "Subsection"],

Cell[BoxData[
    \(\(c[u_] := Det[{\(p'\)[u], \(\(p'\)'\)[u]}]/\((norm[\(p'\)[u]]^3)\); 
    \)\)], "Input"],

Cell[BoxData[
    \(\(norm[u_] := Sqrt[u . u]; \)\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Let's see what it looks like", "Subsection"],

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Cell["\<\
We now build a second bezier segment. They share a point, and there is \
continuity of the tangent at the point of connection (p5 is aligned with p2 \
and (p3==p4)).\
\>", "Section"],

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Cell["\<\
In Cyan we show the curbature. In this case it is discontinue at the junction\
\
\>", "Subsection"],

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Cell[CellGroupData[{

Cell["\<\
The following few lines allowed me to find that the value of t that provide \
continuity (c[1]==cc[0]) is of the form coeffa + coeffb t == c[1].\
\>", "Section"],

Cell[BoxData[
    \(p0 := {p0x, p0y}; \np1 := {p1x, p1y}; \np2 := {p2x, p2y}; \n
    p3 := {p3x, p3y}; \np4 := p3; t =. ; \np5 := t \((p3 - p2)\) + p4; \n
    p6 := {p6x, p6y}; \np7 := {p7x, p7y}; \n{p0x, p0y} =. ; \n{p1x, p1y} =. ; 
    \n{p2x, p2y} =. ; \n{p3x, p3y}\  = \  . ; \n{p6x, p6y}\  = \  . ; 
    \n{p7x, p7y}\  =. ; \nt =. ; \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(cc[0]\)], "Input"],

Cell[BoxData[
    \(\((18\ p3y\ p6x - 18\ p3x\ p6y - 18\ p3y\ p7x + 18\ p6y\ p7x + 
          18\ p3x\ p7y - 18\ p6x\ p7y - 18\ p2y\ p6x\ t + 18\ p3y\ p6x\ t + 
          18\ p2x\ p6y\ t - 18\ p3x\ p6y\ t + 18\ p2y\ p7x\ t - 
          18\ p3y\ p7x\ t - 18\ p2x\ p7y\ t + 18\ p3x\ p7y\ t)\)/
      \((\((3\ p6x - 3\ p7x)\)\^2 + \((3\ p6y - 3\ p7y)\)\^2)\)\^\(3/2\)\)], 
  "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Extract the coefficients of the polynomial in t", "Subsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(firstCoeff\  = \ Simplify[\ Coefficient[\ cc[0], t]]\)], "Input"],

Cell[BoxData[
    \(\(-\(\(2\ 
            \((p2y\ \((p6x - p7x)\) + p3y\ \((\(-p6x\) + p7x)\) - 
                \((p2x - p3x)\)\ \((p6y - p7y)\))\)\)\/\(3\ 
            \((p6x\^2 + p6y\^2 - 2\ p6x\ p7x + p7x\^2 - 2\ p6y\ p7y + p7y\^2)
                \)\^\(3/2\)\)\)\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(zeroCoeff\  = \ Simplify[\ Coefficient[\ cc[0], t, 0]]\)], "Input"],

Cell[BoxData[
    \(\(-\(\(2\ 
            \((\(-p6y\)\ p7x + p3y\ \((\(-p6x\) + p7x)\) + 
                p3x\ \((p6y - p7y)\) + p6x\ p7y)\)\)\/\(3\ 
            \((p6x\^2 + p6y\^2 - 2\ p6x\ p7x + p7x\^2 - 2\ p6y\ p7y + p7y\^2)
                \)\^\(3/2\)\)\)\)\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Verify that we have well factorized cc[0] as a function of t.\
\>", "Subsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Simplify[cc[0]\  == \ zeroCoeff\  + \ t\ firstCoeff]\)], "Input"],

Cell[BoxData[
    \(True\)], "Output"]
}, Open  ]]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
The code that calculates t for any given value of the points defining the \
bezier curve\
\>", "Section"],

Cell[CellGroupData[{

Cell[BoxData[
    \(p0 := {p0x, p0y}; \np1 := {p1x, p1y}; \np2 := {p2x, p2y}; \n
    p3 := {p3x, p3y}; \np4 := p3; t =. ; \np5 := t \((p3 - p2)\) + p4; \n
    p6 := {p6x, p6y}; \np7 := {p7x, p7y}; \n
    Coeffa := \ 
      \(-\(\(2\ \((
                \(-p6y\)\ p7x + p3y\ \((\(-p6x\) + p7x)\) + 
                  p3x\ \((p6y - p7y)\) + p6x\ p7y)\)\)\/\(3\ 
              \((p6x\^2 + p6y\^2 - 2\ p6x\ p7x + p7x\^2 - 2\ p6y\ p7y + 
                    p7y\^2)\)\^\(3/2\)\)\)\); \n
    Coeffb\  := 
      \(-\(\(2\ \((
                p2y\ \((p6x - p7x)\) + p3y\ \((\(-p6x\) + p7x)\) - 
                  \((p2x - p3x)\)\ \((p6y - p7y)\))\)\)\/\(3\ 
              \((p6x\^2 + p6y\^2 - 2\ p6x\ p7x + p7x\^2 - 2\ p6y\ p7y + 
                    p7y\^2)\)\^\(3/2\)\)\)\); \n
    t := \ \((c[1] - Coeffa)\)/Coeffb; \)], "Input"],

Cell[BoxData[
    \(General::"spell1" \( : \ \) 
      "Possible spelling error: new symbol name \"\!\(Coeffb\)\" is similar \
to existing symbol \"\!\(Coeffa\)\"."\)], "Message"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["An application", "Section"],

Cell[BoxData[
    \({p0x, p0y} = {0, 0}; \n{p1x, p1y} = {0, 1}; 
    \n{p2x, p2y} = { .75,  .75}; \n{p3x, p3y}\  = \ {1, 0}; 
    \n{p6x, p6y}\  = \ {2, \(-1\)}; \n{p7x, p7y}\  = \ {3, 0}; \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(t\)], "Input"],

Cell[BoxData[
    \(4.12132034355964282`\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["The graphic showing the result.", "Section"],

Cell[CellGroupData[{

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      Graphics[{PointSize[0.02], \n\t\tRGBColor[0, 1, 0], Point[p0], 
          Point[p3], Point[p7], \n\t\t\tRGBColor[0, 0, 1], Point[p1], 
          Point[p2], Point[p5], Point[p6], \n\t\t\tRGBColor[1, 0, 0], \n\t\t\t
          Line[{p0, p1}], Line[{p3, p2}], Line[{p4, p5}], Line[{p6, p7}], \n
          \t\t\tRGBColor[0, 0, 0], \n\t\t\t
          Line[Table[p[u], {u, 0, 1, 1/100}]], \n\t\t\t\t
          Line[Table[pp[u], {u, 0, 1, 1/100}]], \n\t\t\tRGBColor[0, 1, 1], \n
          \t\t\tLine[Table[{u, c[u]}, {u, 0, 1, 1/100}]], \n\t\t\t
          Line[Table[{1 + \((p7 - p4)\)[\([1]\)] u, cc[u]}, {u, 0, 1, 1/100}]]
            \n\t\t}], PlotRange -> All]; \)\)], "Input"],

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