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   <text:h text:style-name="Heading_20_1" text:outline-level="1">Method of Steepest Descent</text:h>
   <text:p text:style-name="Standard">Newton Raphson does not behave well in finding the separation distance between two curves especially when the curves are near contact.</text:p>
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   <text:p text:style-name="Standard">The direction of steepest descent is parallel to</text:p>
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   <text:p text:style-name="Standard">Using Taylor series</text:p>
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        <annotation encoding="StarMath 5.0">l_IJ
left(
{bold nitalic s}_I
+
%alpha
{partial l_IJ}
over
{partial {bold nitalic s}_I}
,
{bold nitalic s}_J
+
%alpha
{partial l_IJ}
over
{partial {bold nitalic s}_J}
right)
=
l_IJ
left(
{bold nitalic s}_I
,
{bold nitalic s}_J
right)
+
%alpha
left(
{partial l_IJ}
over
{partial {bold nitalic s}^T_I}
{partial l_IJ}
over
{partial {bold nitalic s}_I}
+
{partial l_IJ}
over
{partial {bold nitalic s}^T_J}
{partial l_IJ}
over
{partial {bold nitalic s}_J}
right)

newline
&quot;&quot;~~~~~
{%alpha^2}
over
{2}
left(
{partial l_IJ}
over
{partial {bold nitalic s}^T_I}
{partial^2 l_IJ}
over
{partial {bold nitalic s}_I partial {bold nitalic s}_I}
{partial l_IJ}
over
{partial {bold nitalic s}_I}
+
2
{partial l_IJ}
over
{partial {bold nitalic s}^T_I}
{partial^2 l_IJ}
over
{partial {bold nitalic s}_I partial {bold nitalic s}_J}
{partial l_IJ}
over
{partial {bold nitalic s}_J}
+
{partial l_IJ}
over
{partial {bold nitalic s}^T_J}
{partial^2 l_IJ}
over
{partial {bold nitalic s}_J partial {bold nitalic s}_J}
{partial l_IJ}
over
{partial {bold nitalic s}_J}
right)
+
dotsaxis





</annotation>
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=
{
-
left(
{partial l_IJ}
over
{partial {bold nitalic s}^T_I}
{partial l_IJ}
over
{partial {bold nitalic s}_I}
+
{partial l_IJ}
over
{partial {bold nitalic s}^T_J}
{partial l_IJ}
over
{partial {bold nitalic s}_J}
right)
}
over
{
left(
{partial l_IJ}
over
{partial {bold nitalic s}^T_I}
{partial^2 l_IJ}
over
{partial {bold nitalic s}_I partial {bold nitalic s}_I}
{partial l_IJ}
over
{partial {bold nitalic s}_I}
+
2
{partial l_IJ}
over
{partial {bold nitalic s}^T_I}
{partial^2 l_IJ}
over
{partial {bold nitalic s}_I partial {bold nitalic s}_J}
{partial l_IJ}
over
{partial {bold nitalic s}_J}
+
{partial l_IJ}
over
{partial {bold nitalic s}^T_J}
{partial^2 l_IJ}
over
{partial {bold nitalic s}_J partial {bold nitalic s}_J}
{partial l_IJ}
over
{partial {bold nitalic s}_J}
right)
}




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left(
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+
%alpha
{bold nitalic n}_V
right)
=
l_IJ
left(
{bold nitalic s}
right)
+
%alpha
{partial l_IJ}
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{partial {bold nitalic s}_I}
{bold nitalic n}_V
+
{%alpha^2}
over
{2}

{bold nitalic n}^T_V
{partial^2 l_IJ}
over
{partial {bold nitalic s} partial {bold nitalic s}}
{bold nitalic n}_V
+
dotsaxis





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=
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-
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over
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over
{
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over
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