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Cell["NORMAL  LINES  DRAWN  TO  ELLIPSES  AND  ELLIPTIC  INTEGRALS", "Title",
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Cell[TextData[StyleBox[
"Tilak de Alwis\n\nDepartment of Mathematics\nSoutheastern Louisiana \
University\nHammond, LA 70402, USA\nEmail:FMAT1117@SELU.EDU",
  FontSize->14,
  FontWeight->"Bold"]], "Text"],

Cell[TextData[{
  StyleBox["ABSTRACT",
    FontWeight->"Bold"],
  "\nConsider the ellipse  ",
  Cell[BoxData[
      \(TraditionalForm\`x\^2/a\^2 + y\^2/b\^2 = 1\)]],
  "  where  ",
  Cell[BoxData[
      \(TraditionalForm\`a > b > 0\)]],
  ", and an arbitrary point  ",
  Cell[BoxData[
      \(TraditionalForm\`P\_0\ \((x\_0, y\_0)\)\)]],
  " on the plane.  Finding the minimum distance from  ",
  Cell[BoxData[
      \(TraditionalForm\`\(P\_0\ \)\)]],
  " to the ellipse is a well-known problem.  One can easily show that the \
minimum distance path lies along a normal line to the ellipse, passing \
through  ",
  Cell[BoxData[
      \(TraditionalForm\`P\_0\)]],
  ".  This paper deals with a study of all the normal lines drawn from point  \
",
  Cell[BoxData[
      \(TraditionalForm\`\(P\_0\ \)\)]],
  " to the ellipse.  We have used the computer algebra system ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " to illustrate and discover several aspects of these normal lines.  ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " can be used in contemporary mathematical research and educaiton in more \
than one way: as a computational, visualization, experimentation and a \
conjecture forming tool (see [4]-[14]).  The paper well illustrates such \
usage via a study of the normal lines to the ellipse.  We used ",
  StyleBox["Matheamtica",
    FontSlant->"Italic"],
  " version 3.0 on a ",
  StyleBox["Windows",
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  " are [2], [16], [18] and [19].   "
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Cell["1.  Introduction", "Section",
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Cell[TextData[{
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  ":\n\n                                                                      \
  ",
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  "/",
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  "                                                                       \
(1.1)\n\nSuppose ",
  Cell[BoxData[
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from ",
  Cell[BoxData[
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  " to the ellipse.  Most of the material in the introduction can also be \
found in [14], but in order to make this paper self contained as much as \
possible, we will also include it here.\n\nSuppose ",
  Cell[BoxData[
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the point ",
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  "                                                              ",
  StyleBox["Fig 1.1.",
    FontWeight->"Bold"],
  "  Normal Line to an Ellipse"
}], "Text",
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Cell[TextData[{
  "One can find the slope of the line ",
  Cell[BoxData[
      \(TraditionalForm\`\(P\_0\) P\)]],
  " in two different ways: \n\n                                               \
          slope of  ",
  Cell[BoxData[
      \(TraditionalForm\`\(P\_0\) P\)]],
  " = ",
  Cell[BoxData[
      \(TraditionalForm
      \`\((y\_0 - b\ Sin\ \[Theta])\)/\((x\_0 - a\ Cos\ \[Theta])\)\)]],
  "                                  \t       (1.2)\n                         \
                   \nOne can implicitly differentiate the equation (1.1) with \
respect to ",
  Cell[BoxData[
      \(TraditionalForm\`x\)]],
  " , to obtain the slope of the tangent line at ",
  Cell[BoxData[
      \(TraditionalForm\`\((x, \ y)\)\)]],
  ":\n\n                                                             ",
  Cell[BoxData[
      \(TraditionalForm\`d\ y/d\ x = \(-b\^2\) x/\((\(a\^2\) y)\)\)]],
  "                                                   \t\t       (1.3)\n  \n\
One can evaluate equation (1.3) at ",
  Cell[BoxData[
      \(TraditionalForm\`P(a\ Cos\ \[Theta], \ b\ Sin\ \[Theta])\)]],
  " to find the slope of the tangent line to the ellipse at  ",
  Cell[BoxData[
      \(TraditionalForm\`P\)]],
  ".  The slope of the normal line ",
  Cell[BoxData[
      \(TraditionalForm\`\(P\_0\) P\)]],
  " at  ",
  Cell[BoxData[
      \(TraditionalForm\`P\)]],
  "  can be obtained by taking the negative reciprocal of this: \n\n          \
                         slope of  ",
  Cell[BoxData[
      \(TraditionalForm\`\(P\_0\) P\)]],
  "  =  ",
  Cell[BoxData[
      \(TraditionalForm
      \`\(a\^2\) 
        y/\((\(b\^2\) x)\) |\_\((a\ Cos\ \[Theta], \ b\ Sin\ \[Theta])\)\)]],
  "  =  ",
  Cell[BoxData[
      \(TraditionalForm\`a\ Sin\ \[Theta]\ /\ \((b\ Cos\ \[Theta])\)\)]],
  "                          \t       (1.4)\n\nOne can set the right-hand \
sides of equations (1.2) and (1.4) equal to each other and simplify to obtain \
the following equation:\n\n                                                  \
",
  Cell[BoxData[
      \(TraditionalForm
      \`\(x\_0\)(a\ Sin\ \[Theta]) - \(y\_0\)(b\ Cos\ \[Theta]) = 
        \((a\^2 - b\^2)\)\ Sin\ \[Theta]\ Cos\ \[Theta]\)]],
  "                                  \t       (1.5)\n\nIt is important to \
realize that for given values ",
  Cell[BoxData[
      \(TraditionalForm\`a, \ b, \ x\_0\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`y\_0\)]],
  ", the number of distinct solutions for \[Theta] of the above equation \
(1.5) where 0 \[LessEqual] \[Theta] < 2\[Pi] , correspond to the number of \
distinct normal lines that can be drawn from ",
  Cell[BoxData[
      \(TraditionalForm\`\(P\_0\)(x\_0, y\_0)\)]],
  " to the ellipse.  In order to solve the equation (1.5), we will consider \
two cases:\n\n",
  StyleBox["Case 1: ",
    FontWeight->"Bold"],
  Cell[BoxData[
      \(TraditionalForm\`y\_0 = 0\)]],
  "   \n    In this case, the equation (1.5) will imply that ",
  Cell[BoxData[
      \(TraditionalForm\`\(\ Sin\ \[Theta] = 0\)\)]],
  "  or  ",
  Cell[BoxData[
      \(TraditionalForm
      \`Cos\ \[Theta] = a\ x\_0/\((a\^2 - b\^2)\)\_\(\  \)\)]],
  ". The equation   Sin \[Theta] = 0  implies that ",
  Cell[BoxData[
      \(TraditionalForm\`P = \((a, 0)\)\)]],
  " or ",
  Cell[BoxData[
      \(TraditionalForm\`\((\(-a\), 0)\)\)]],
  ".   Therefore, these two values for  ",
  Cell[BoxData[
      \(TraditionalForm\`P\)]],
  " will correspond to two distinct normal lines (along the ",
  Cell[BoxData[
      \(TraditionalForm\`X\)]],
  "-axis).  However, the equation ",
  Cell[BoxData[
      \(TraditionalForm\`Cos\ \[Theta] = a\ x\_0/\((a\^2 - b\^2)\)\)]],
  " will produce real values for \[Theta] if and only if ",
  Cell[BoxData[
      \(TraditionalForm
      \`\(| x\)\_0 | \( \[LessEqual] \((a\^2 - b\^2)\)/a\) = a\ e\^2\)]],
  " where ",
  Cell[BoxData[
      \(TraditionalForm\`e = \@\(a\^2 - b\^2\)/a\)]],
  " is the eccentricity of the ellipse.  If ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(| x\)\_0 | \( < \((a\^2 - b\^2)\)/a\), \)\)]],
  " then we obtain two distinct values of  \[Theta]  such that ",
  Cell[BoxData[
      \(TraditionalForm\`Cos\ \[Theta] = a\ x\_0/\((a\^2 - b\^2)\)\)]],
  " with 0 \[LessEqual] \[Theta] < 2\[Pi]  and ",
  Cell[BoxData[
      \(TraditionalForm\`\[Theta] \[NotEqual] \[Pi]\)]],
  ", corresponding to two more distinct normal lines.  On the other hand, if \
",
  Cell[BoxData[
      \(TraditionalForm\`\(\(| x\)\_0 | \) = \((a\^2 - b\^2)\)/a\)]],
  ", then we obtain the previous normal lines ",
  Cell[BoxData[
      \(TraditionalForm\`\(P\_0\) P\)]],
  " along the ",
  Cell[BoxData[
      \(TraditionalForm\`X\)]],
  "-axis with ",
  Cell[BoxData[
      \(TraditionalForm\`P = \((a, 0)\)\)]],
  " or ",
  Cell[BoxData[
      \(TraditionalForm\`\((\(-a\), 0)\)\)]],
  "   Hence one can conclude that the case ",
  Cell[BoxData[
      \(TraditionalForm\`y\_0 = 0\)]],
  "  corresponds to two or four distinct normal lines according as ",
  Cell[BoxData[
      \(TraditionalForm\`\(| x\)\_0 | \( \[GreaterEqual] a\ e\^2\)\)]],
  "  or  ",
  Cell[BoxData[
      \(TraditionalForm\`\(| x\)\_0 | \( < \ a\ e\^2\)\)]],
  ".\n\n",
  StyleBox["Case 2:",
    FontWeight->"Bold"],
  "  ",
  Cell[BoxData[
      \(TraditionalForm\`y\_0 \[NotEqual] 0\)]],
  ".\n     This case implies, via equation (1.5) that \[Theta] \[NotEqual] \
\[Pi].  Therefore, ",
  Cell[BoxData[
      \(TraditionalForm\`Tan\ \((\[Theta]/2)\)\)]],
  " is well defined.  Therefore, let us use the trigonometric substitution ",
  Cell[BoxData[
      \(TraditionalForm\`t = Tan\ \((\[Theta]/2)\)\)]],
  "  to solve equation (1.5) (see [3]).  One can easily show that ",
  Cell[BoxData[
      \(TraditionalForm\`Sin\ \[Theta] = 2  t/\((1 + t\^2)\)\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`Cos\ \[Theta] = \((1 - t\^2)\)/\((1 + t\^2)\)\)]],
  ". Substitute these expressions back in equation (1.5) and simplify to \
obtain\n     \n                                        ",
  Cell[BoxData[
      \(TraditionalForm
      \`\(t\^4\)(b\ y\_0) + 
          2 \( t\^\(3\ \)\)[a\ x\_0 + \((a\^2 - b\^2)\)]\  + 
          2  t\ [a\ x\_0 - \((a\^2 - b\^2)\)] - b\ y\_0 = 0\_\(\  \)\)]],
  "                                 (1.6)\n     \nThe ",
  StyleBox["\"Solve\"",
    FontWeight->"Bold"],
  " command of ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " can certainly solve the above equation (1.6) for ",
  Cell[BoxData[
      \(TraditionalForm\`t\)]],
  " .  However, as the reader can verify, these solutions are almost useless, \
because of their complexity.  Rather than the actual solutions themselves, at \
this point we are more interested in the nature or the number of solutions.  \
Therefore, we will proceed as follows:  One can further simplify equation \
(1.6).  Since ",
  Cell[BoxData[
      \(TraditionalForm\`a\^2 - b\^2 = a\^2\ e\^2\)]],
  ",  the equation (1.6) will read\n\n                                        \
       ",
  Cell[BoxData[
      \(TraditionalForm
      \`\(t\^4\)(b\ y\_0) + 2\ a\ \(t\^\(3\ \)\)[\ x\_0 + a\ e\^2]\  + 
          2\ a\ t\ [\ x\_0 - a\ e\^2] - b\ y\_0 = 0\_\(\  \)\)]],
  "                                       (1.7)\n\nSince  ",
  Cell[BoxData[
      \(TraditionalForm\`y\_0 \[NotEqual] 0\)]],
  " , the above equation (1.7) represents a quartic equation with real \
coefficients.  Notice that we have transformed the trigonometric equation \
(1.5) to a polynomial equation (1.7).  The discriminant of the quartic \
equation (1.7) reveals the nature of its roots.\n\nRecall that the \
discriminant of the quartic ",
  Cell[BoxData[
      \(TraditionalForm
      \`\[Phi](x) = 
        \(a\_0\) x\^4 + 4 \( a\_1\) x\^3 + 6 \( a\_2\) x\^2 + 4 \( a\_3\) x + 
          a\_4\)]],
  " with roots  ",
  Cell[BoxData[
      \(TraditionalForm\`p, q, r\)]],
  "  and  ",
  Cell[BoxData[
      \(TraditionalForm\`s\)]],
  " are given by (see [1]) \n                                            ",
  Cell[BoxData[
      \(TraditionalForm
      \`d = a\_0\%6\ \(\((p - q)\)\^2\) \(\((p - r)\)\^2\) \(\((p - s)\)\^2\) 
          \(\((q - r)\)\^2\) \(\((q - s)\)\^2\) \((r - s)\)\^2\)]],
  "              \t\t       (1.8)\n                                   \nOne \
can also obtain the following version for the discriminant in terms of the \
coefficients of the quartic (see [1]):\n\n                              ",
  Cell[BoxData[
      \(TraditionalForm
      \`d = 256\ [
          \((\(a\_0\) a\_4 - 4 \( a\_1\) a\_3 + 3  a\_2\^2)\)\^3 - 
            27 \((\(a\_0\) \(a\_2\) a\_4 + 2 \( a\_1\) \(a\_2\) a\_3 - 
                    \(a\_0\) a\_3\^2 - \(a\_4\) a\_1\^2 - a\_2\^3)\)\^2]\)]],
  "         (1.9)\n\nUsing the Intermediate Value Theorem in calculus, one \
can easily see that our equation (1.7) has at least two distinct real roots \
(see [17]).  Call these two real roots \[Alpha] and \[Beta] , and the other \
two roots \[Gamma] and \[Delta].  The equation (1.8) implies that the \
discriminant ",
  Cell[BoxData[
      \(TraditionalForm\`D\)]],
  " of the quartic (1.7) is given by\n \n                                     \
",
  Cell[BoxData[
      \(TraditionalForm\`\(D = \)\)]],
  Cell[BoxData[
      \(TraditionalForm
      \`\((b\ y\_0)\)\^6\ \(\((\[Alpha] - \[Beta])\)\^2\) 
        \(\((\[Alpha] - r)\)\^2\) \(\((\[Alpha] - \[Delta])\)\^2\) 
        \(\((\[Beta] - \[Gamma])\)\^2\) \(\((\[Beta] - \[Delta])\)\^2\) 
        \((\[Gamma] - \[Delta])\)\^2\)]],
  "\t\t\t                (1.10)\n \nCase (a):  Suppose the equation (1.7) has \
exactly two distinct real roots.  Then \n                (i)  \[Gamma] and \
\[Delta] are real \[DoubleRightArrow] ",
  Cell[BoxData[
      \(TraditionalForm\`D = 0\)]],
  "                           (ii) \[Gamma]  and  \[Delta] are non real \
\[DoubleRightArrow] ",
  Cell[BoxData[
      \(TraditionalForm\`D < 0\)]],
  ".\n\nCase (b):  Suppose the equation (1.7) has exactly three distinct real \
roots.  Then clearly ",
  Cell[BoxData[
      \(TraditionalForm\`D = 0\)]],
  ".  \n                \nCase (c):  Suppose that the equation (1.7) has \
exactly four distinct real roots.  Then clearly ",
  Cell[BoxData[
      \(TraditionalForm\`D > 0\)]],
  ".\n \nLet us now calculate ",
  Cell[BoxData[
      \(TraditionalForm\`D\)]],
  " in terms of the coefficients of (1.7).  By comparing the polynomial ",
  Cell[BoxData[
      \(TraditionalForm
      \`\[Phi](x) = 
        \(a\_0\) x\^4 + 4 \( a\_1\) x\^3 + 6 \( a\_2\) x\^2 + 4 \( a\_3\) x + 
          a\_4\)]],
  "  with the left-hand side of equation of (1.7), one finds that ",
  Cell[BoxData[
      \(TraditionalForm\`a\_0 = b\ y\_0, \ a\_1 = \(a(x\_0 + a\)\)]],
  Cell[BoxData[
      \(TraditionalForm\`\(e\^2)\)/2, \ a\_2 = 0, \ 
      a\_3 = \(a(x\_0 - a\ e\^2)\)/2\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`a\_4 = \(-b\)\ \(y\_0 . \)\)]],
  " Therefore, equation (1.9) implies that\n\n                  ",
  Cell[BoxData[
      \(TraditionalForm
      \`D = \(D(x\_0, y\_0) = 
          256\ {\ \([\ 
                    \((a\^2 - b\^2)\)\^2 - \(a\^2\) x\_0\^2 - 
                      \(b\^2\) y\_0\^2]\)\^3 - 
                27 \( a\^2\) \(\(b\^2\)(a\^2 - b\^2)\) \(x\_0\^2\) y\_0\^2}
              \)\)]],
  "        \t                (1.11)\n\nOne can use ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " for simplification purposes to arrive at the above equation.  Depending \
on the values of ",
  Cell[BoxData[
      \(TraditionalForm\`x\_0\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`y\_0\)]],
  ", ",
  Cell[BoxData[
      \(TraditionalForm\`D(x\_0, y\_0)\)]],
  " could be positive (i.e. 4 distinct normals), negative (i.e. 2 distinct \
normals) , or zero (i. e. 2 or 3 normals).  To find out when does this \
happen, one can draw the graph of the following equation:\n\n                 \
                        ",
  Cell[BoxData[
      \(TraditionalForm
      \`f(x, y) = \ 
        \(\([\ \((a\^2 - b\^2)\)\^2 - \(a\^2\) x\^2 - \(b\^2\) y\^2]\)\^3 - 
            27 \( a\^2\) \(\(b\^2\)(a\^2 - b\^2)\) \(x\^2\) y\^2 = 0\)\)]],
  "                         (1.12)\n                       \nFor given \
specific values of  ",
  Cell[BoxData[
      \(TraditionalForm\`a\)]],
  " and  ",
  Cell[BoxData[
      \(TraditionalForm\`b\)]],
  " one can use the ",
  StyleBox["\"ImplicitPlot\"",
    FontWeight->"Bold"],
  " command of ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " to see the shape of the graph given by equation (1.12).  In general, one \
can show that the equation (1.12) has the following parametrization:\n\n      \
                            ",
  Cell[BoxData[
      \(TraditionalForm
      \`x = \([\((a\^2 - b\^2)\)\ \(Cos\^3\) \[Alpha]]\)/a\)]],
  "  and  ",
  Cell[BoxData[
      \(TraditionalForm
      \`y = \([\((a\^2 - b\^2)\) \(Sin\^3\) \[Alpha]]\)/b\)]],
  " ;  0 \[LessEqual]  \[Alpha] < 2\[Pi]                              (1.13) \
\n  \nFor  ",
  Cell[BoxData[
      \(TraditionalForm\`a = 6\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`b = 4\)]],
  " , the following ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " command ",
  StyleBox["\"ParametricPlot\"",
    FontWeight->"Bold"],
  " produces the graph of equation (1.13):"
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Cell[TextData[{
  StyleBox["                                                Fig. 1.2.",
    FontWeight->"Bold"],
  " The Graph of Equation (1.13): Generalized Astroid"
}], "Text",
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Cell[TextData[{
  "Observe that the above graph is quite similar to the astroid ",
  Cell[BoxData[
      \(TraditionalForm\`x\^\(2/3\) + y\^\(2/3\) = \ c\^\(2/3\)\)]],
  " where ",
  Cell[BoxData[
      \(TraditionalForm\`\(\ c\)\)]],
  " is a constant (see [20]).  Motivated by this, it is not hard to show that \
any of the equations (1.12) or (1.13) is equivalent to the following \
equation, which we will refer to as the \"generalized astroid\":\n\n          \
                                             ",
  Cell[BoxData[
      \(TraditionalForm
      \`\((x\ a)\)\^\(2/3\) + \((y\ b)\)\^\(2/3\) = 
        \((a\^2 - b\^2)\)\^\(2/3\)\)]],
  "                                                                 (1.14)\n\n\
Using the above graph, one sees that if  (",
  Cell[BoxData[
      \(TraditionalForm\`\(x\_0, y\_0)\)\)]],
  "  is inside the curve, then  ",
  Cell[BoxData[
      \(TraditionalForm\`D(x\_0, y\_0)\)]],
  " is positive; If (",
  Cell[BoxData[
      \(TraditionalForm\`\(x\_0, y\_0)\)\)]],
  "  is on the curve, then ",
  Cell[BoxData[
      \(TraditionalForm\`D(x\_0, y\_0)\)]],
  "  is equal to zero; If  (",
  Cell[BoxData[
      \(TraditionalForm\`\(x\_0, y\_0)\)\)]],
  "  is outside the curve, then  ",
  Cell[BoxData[
      \(TraditionalForm\`D(x\_0, y\_0)\)]],
  " is negative.  \n\nOne can now combine the outcomes of  cases 1 and 2 in \
this section to arrive at the following conclusion: If ",
  Cell[BoxData[
      \(TraditionalForm\`P\_0\)]],
  " is inside the generalized astroid, then one can draw four distinct normal \
lines from to the ellipse; If ",
  Cell[BoxData[
      \(TraditionalForm\`P\_0\)]],
  " is outside the generalized astroid, then one can draw two distinct normal \
lines from ",
  Cell[BoxData[
      \(TraditionalForm\`P\_0\)]],
  " to the ellipse; If ",
  Cell[BoxData[
      \(TraditionalForm\`P\_0\)]],
  " is on the generalized astroid, but not a cusp point, then one can draw \
three distinct normal lines from ",
  Cell[BoxData[
      \(TraditionalForm\`P\_0\)]],
  " to the ellipse; If ",
  Cell[BoxData[
      \(TraditionalForm\`P\_0\)]],
  " is one of the four cusp points on the generalized astroid, then one can \
draw two distinct normal lines from ",
  Cell[BoxData[
      \(TraditionalForm\`P\_0\)]],
  " to the ellipse.  This is illustrated by the following figure.             \
                   "
}], "Text",
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  "One can use ",
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following programs draws these normals, along with the graphs of the ellipse \
(1.1) and the modified astroid (1.13):\n\n",
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