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{bold nitalic r}_IeJeO
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   <text:h text:style-name="Heading_20_3" text:outline-level="3">Coarse Global Search</text:h>
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                </mfrac>
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                 <mrow>
                  <mrow>
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                   <mo stretchy="false">+</mo>
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                  <msub>
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        <annotation encoding="StarMath 5.0">int from CI
sin^-1
left(
left(
{-{bold nitalic r}_IeJeO} 
over 
{abs {-{bold nitalic r}_IeJeO}}
times 
{
{-{bold nitalic r}_IeJeO + d{bold nitalic r}_ImIeIm} 
over 
{abs {-{bold nitalic r}_IeJeO + d{bold nitalic r}_ImIeIm}}
}
right)
cdot
{bold nitalic k}
right)
=
2%pi</annotation>
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       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
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       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
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       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
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   <text:p text:style-name="Standard"/>
   <text:p text:style-name="Standard">For 3D, Discretize surfaces I and J. For every J point, test whether it is outside of surface I. The algorithm is yet to be derived.</text:p>
   <text:p text:style-name="Standard"/>
   <text:p text:style-name="Standard">While testing above, record the minimum<draw:frame draw:style-name="fr1" draw:name="Object6" text:anchor-type="as-char" svg:width="0.3189in" svg:height="0.2102in" draw:z-index="13"><draw:object>
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    </draw:frame>found for the J point. If J point is outside of curve I, record I point and J point if<draw:frame draw:style-name="fr1" draw:name="Object8" text:anchor-type="as-char" svg:width="0.3189in" svg:height="0.2102in" draw:z-index="14"><draw:object>
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    </draw:frame>is the minimum to date. If J point has penetrated curve I, record I and J points of minimum<draw:frame draw:style-name="fr1" draw:name="Object7" text:anchor-type="as-char" svg:width="0.3189in" svg:height="0.2102in" draw:z-index="15"><draw:object>
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       <semantics>
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        <annotation encoding="StarMath 5.0">l_IeJe
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    </draw:frame>. Thus all J points that penetrate curve I are recorded with their I points for minimum <draw:frame draw:style-name="fr1" draw:name="Object9" text:anchor-type="as-char" svg:width="0.3189in" svg:height="0.2102in" draw:z-index="16"><draw:object>
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    </draw:frame>. These I and J points will be the initial guesses for accurate search of the minimum separation or the maximum penetration.</text:p>
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   <text:h text:style-name="Heading_20_3" text:outline-level="3">Minimize<draw:frame draw:style-name="fr1" draw:name="Object10" text:anchor-type="as-char" svg:width="0.3189in" svg:height="0.2102in" draw:z-index="17"><draw:object>
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    </draw:frame>and Parallel Normal Constraint</text:h>
   <text:p text:style-name="Standard">Minimizing<draw:frame draw:style-name="fr1" draw:name="Object11" text:anchor-type="as-char" svg:width="0.3189in" svg:height="0.2102in" draw:z-index="19"><draw:object>
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    </draw:frame>alone has problems when the curves intersect. In such cases we want the maximum penetration and not the intersection. Therefore we require the parallel normal constraint:</text:p>
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   <text:p text:style-name="equation">Applying Lagrange multiplier, the new function to min max is:</text:p>
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        <mrow>
         <mrow>
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           <mi mathvariant="normal">λ</mi>
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         </mrow>
         <msub>
          <mi>A</mi>
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           <mi mathvariant="italic">Je</mi>
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          </mrow>
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        </mrow>
        <annotation encoding="StarMath 5.0">l^&apos;_IeJe
=
l_IeJe
+
%lambda
A_Ie3Je1
</annotation>
       </semantics>
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     </draw:object><draw:image>
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      </office:binary-data>
     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="equation">Differentiating wrt<draw:frame draw:style-name="fr1" draw:name="Object14" text:anchor-type="as-char" svg:width="0.6098in" svg:height="0.2102in" draw:z-index="22"><draw:object>
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       <semantics>
        <mrow>
         <msub>
          <mstyle mathvariant="bold">
           <mi mathvariant="normal">s</mi>
          </mstyle>
          <mi>I</mi>
         </msub>
         <mi>,</mi>
         <msub>
          <mstyle mathvariant="bold">
           <mi mathvariant="normal">s</mi>
          </mstyle>
          <mi>J</mi>
         </msub>
         <mi>,</mi>
         <mi mathvariant="normal">λ</mi>
        </mrow>
        <annotation encoding="StarMath 5.0">{bold nitalic s}_I,
{bold nitalic s}_J,
%lambda</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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       AAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAHsC
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       AQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABcAAAA0AwAA3AEAAAEASnIA
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       AAEAAAAAAIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJvbAAAAAAAAKcBAAD/
       /wAAAAAFAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8A
       hgABAAQAAAAAAAAAcgACABoAAACnBAAAcgEAAAEAAAC7A+UAAAAAAP//AQC7A4wAAQAAAAAA
       lQABAAQAAAAAAAAAlgABAAIAAAAJAIwAAQAAAAAA
      </office:binary-data>
     </draw:image>
    </draw:frame>and setting to zeros</text:p>
   <text:p text:style-name="equation"><draw:frame draw:style-name="fr1" draw:name="Object15" text:anchor-type="as-char" svg:width="1.4764in" svg:height="0.4457in" draw:z-index="23"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <mfrac>
           <mrow>
            <mo stretchy="false">∂</mo>
            <msub>
             <mi>l</mi>
             <mi mathvariant="italic">IeJe</mi>
            </msub>
           </mrow>
           <mrow>
            <mo stretchy="false">∂</mo>
            <msub>
             <mstyle mathvariant="bold">
              <mi mathvariant="normal">s</mi>
             </mstyle>
             <mi>I</mi>
            </msub>
           </mrow>
          </mfrac>
          <mo stretchy="false">+</mo>
          <mi mathvariant="normal">λ</mi>
         </mrow>
         <mrow>
          <mfrac>
           <mrow>
            <mo stretchy="false">∂</mo>
            <msub>
             <mi>A</mi>
             <mrow>
              <mi mathvariant="italic">Ie</mi>
              <mn>3</mn>
              <mi mathvariant="italic">Je</mi>
              <mn>1</mn>
             </mrow>
            </msub>
           </mrow>
           <mrow>
            <mo stretchy="false">∂</mo>
            <msub>
             <mstyle mathvariant="bold">
              <mi mathvariant="normal">s</mi>
             </mstyle>
             <mi>I</mi>
            </msub>
           </mrow>
          </mfrac>
          <mo stretchy="false">=</mo>
          <mstyle mathvariant="bold">
           <mn>0</mn>
          </mstyle>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">{partial l_IeJe}
over
{partial {bold nitalic s}_I}
+
%lambda
{partial A_Ie3Je1}
over
{partial {bold nitalic s}_I}

=
{bold nitalic 0}</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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   <text:p text:style-name="equation"><draw:frame draw:style-name="fr1" draw:name="Object16" text:anchor-type="as-char" svg:width="1.4764in" svg:height="0.4457in" draw:z-index="24"><draw:object>
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              <mi mathvariant="normal">s</mi>
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            <msub>
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              <mi mathvariant="normal">s</mi>
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             <mi>J</mi>
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           </mrow>
          </mfrac>
          <mo stretchy="false">=</mo>
          <mstyle mathvariant="bold">
           <mn>0</mn>
          </mstyle>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">{partial l_IeJe}
over
{partial {bold nitalic s}_J}
+
%lambda
{partial A_Ie3Je1}
over
{partial {bold nitalic s}_J}

=
{bold nitalic 0}</annotation>
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   <text:h text:style-name="Heading_20_4" text:outline-level="4">Newton Raphson Iteration</text:h>
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              <mrow>
               <mrow>
                <mrow>
                 <mfrac>
                  <mrow>
                   <msup>
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                   </msup>
                   <msub>
                    <mi>l</mi>
                    <mi mathvariant="italic">IeJe</mi>
                   </msub>
                  </mrow>
                  <mrow>
                   <mo stretchy="false">∂</mo>
                   <msub>
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                    </mstyle>
                    <mi>J</mi>
                   </msub>
                   <mo stretchy="false">∂</mo>
                   <msub>
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                     <mi mathvariant="normal">s</mi>
                    </mstyle>
                    <mi>J</mi>
                   </msub>
                  </mrow>
                 </mfrac>
                 <mo stretchy="false">+</mo>
                 <mi mathvariant="normal">λ</mi>
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                <mfrac>
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                  <msup>
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                  </msup>
                  <msub>
                   <mi>A</mi>
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                    <mn>3</mn>
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                    <mn>1</mn>
                   </mrow>
                  </msub>
                 </mrow>
                 <mrow>
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                   </mstyle>
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                </mfrac>
               </mrow>
              </mrow>
              <mo fence="true" form="postfix" stretchy="true">)</mo>
             </mrow>
            </mrow>
            <mi>d</mi>
            <mrow>
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              </mstyle>
              <mi>J</mi>
             </msub>
             <mo stretchy="false">+</mo>
             <mfrac>
              <mrow>
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               <msub>
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                </mrow>
               </msub>
              </mrow>
              <mrow>
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               <msub>
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               </msub>
              </mrow>
             </mfrac>
            </mrow>
            <mi>d</mi>
            <mrow>
             <mi mathvariant="normal">λ</mi>
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             <mstyle mathvariant="bold">
              <mn>0</mn>
             </mstyle>
            </mrow>
           </mrow>
          </mtd>
         </mtr>
        </mtable>
        <annotation encoding="StarMath 5.0">left(
{partial l_IeJe}
over
{partial {bold nitalic s}_J}
+
%lambda
{partial A_Ie3Je1}
over
{partial {bold nitalic s}_J}
right)
+
left(
{partial^2 l_IeJe}
over
{partial {bold nitalic s}_J partial {bold nitalic s}_I}
+
%lambda
{partial^2 A_Ie3Je1}
over
{partial {bold nitalic s}_J partial {bold nitalic s}_I}
right)
d{bold nitalic s}_I

newline
&quot;&quot;
+
left(
{partial^2 l_IeJe}
over
{partial {bold nitalic s}_J partial {bold nitalic s}_J}
+
%lambda
{partial^2 A_Ie3Je1}
over
{partial {bold nitalic s}_J partial {bold nitalic s}_J}
right)
d{bold nitalic s}_J
+
{partial A_Ie3Je1}
over
{partial {bold nitalic s}_J}
d%lambda
=
{bold nitalic 0}</annotation>
       </semantics>
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matrix
{
left(
{partial^2 l_IeJe}
over
{partial {bold nitalic s}_I partial {bold nitalic s}_I}
+
%lambda
{partial^2 A_Ie3Je1}
over
{partial {bold nitalic s}_I partial {bold nitalic s}_I}
right)
# 
left(
{partial^2 l_IeJe}
over
{partial {bold nitalic s}_I partial {bold nitalic s}_J}
+
%lambda
{partial^2 A_Ie3Je1}
over
{partial {bold nitalic s}_I partial {bold nitalic s}_J}
right)
# 
{partial A_Ie3Je1}
over
{partial {bold nitalic s}_I}
##
left(
{partial^2 l_IeJe}
over
{partial {bold nitalic s}_J partial {bold nitalic s}_I}
+
%lambda
{partial^2 A_Ie3Je1}
over
{partial {bold nitalic s}_J partial {bold nitalic s}_I}
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# left(
{partial^2 l_IeJe}
over
{partial {bold nitalic s}_J partial {bold nitalic s}_J}
+
%lambda
{partial^2 A_Ie3Je1}
over
{partial {bold nitalic s}_J partial {bold nitalic s}_J}
right)
# 
{partial A_Ie3Je1}
over
{partial {bold nitalic s}_J}
##
{partial A_Ie3Je1}
over
{partial {bold nitalic s}_I}
# 
{partial A_Ie3Je1}
over
{partial {bold nitalic s}_J}
# 0 
}
right]
left lbrace
matrix
{
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d{bold nitalic s}_J ##
%lambda
}
right rbrace
=
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left lbrace
matrix
{
{partial l_IeJe}
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{partial {bold nitalic s}_I}
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%lambda
{partial A_Ie3Je1}
over
{partial {bold nitalic s}_I}
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{partial l_IeJe}
over
{partial {bold nitalic s}_J}
+
%lambda
{partial A_Ie3Je1}
over
{partial {bold nitalic s}_J}
 ##
A_Ie3Je1 
}
right rbrace


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   <text:h text:style-name="Heading_20_4" text:outline-level="4">Newton Raphson Iteration</text:h>
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   <text:h text:style-name="Heading_20_4" text:outline-level="4">Method of Steepest Descent</text:h>
   <text:p text:style-name="Standard">This method was tried and abandoned.</text:p>
   <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. Try method of steepest descent.</text:p>
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    </draw:frame>are good. Also the roots do not converge to high accuracy.</text:p>
   <text:h text:style-name="Heading_20_4" text:outline-level="4">Search for Minimum</text:h>
   <text:h text:style-name="Heading_20_5" text:outline-level="5">Ad Hoc method</text:h>
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   <text:p text:style-name="Standard">Need to use a combination of steepest descent, bisection and NewtonRaphson to find mininum separation.</text:p>
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   <text:p text:style-name="Standard">The vector down the valley should be the average of the unit vectors of the two sides.</text:p>
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   <text:p text:style-name="Standard">The ad hoc method was not reliable.</text:p>
   <text:h text:style-name="Heading_20_5" text:outline-level="5">Very General Method</text:h>
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   <text:p text:style-name="Standard">Now minimize <draw:frame draw:style-name="fr3" draw:name="Object98" text:anchor-type="as-char" svg:width="0.3335in" svg:height="0.2217in" draw:z-index="3"><draw:object-ole>
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       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ///////////////////////////////////9/////v////7///8EAAAABQAAAP7/////////
       //////////8KAAAAAQAAAP//////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
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       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
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       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ///////////////////////////////////9/////v////7///8EAAAABQAAAP7/////////
       //////////8KAAAAAQAAAP//////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       /////////////////////////////////////////////////////////////////////0UA
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       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       //////////////////////////////////////////////////////////////////8BAAAC
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       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ///////////////////////////////////9/////v////7///8EAAAABQAAAP7/////////
       //////////8KAAAAAQAAAP//////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       /////////////////////////////////////////////////////////////////////0UA
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       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       ////////////////////////////////////////////////////////////////////////
       //////////////////////////////////////////////////////////////////8BAAAC
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       BgAAcgEAAAEAAABlIssAAAAAAP//AQBlIowAAQAAAAAAiwABAAIAAAAfAIoAAQA7AAAAAwA1
       AAAACQBUaG9ybmRhbGUAAAAAAACnAQAAAAAAAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAA
       AACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAXAAAA2wcAAHIBAAAB
       ADDYAAAAAAD//wEAMACMAAEAAAAAAJUAAQAEAAAAAAAAAJYAAQACAAAACQCMAAEAAAAAAA==
      </office:binary-data>
     </draw:image>
    </draw:frame></text:p>
   <text:h text:style-name="Heading_20_2" text:outline-level="2">Check for lift off</text:h>
   <text:p text:style-name="Standard">For no lift off,</text:p>
   <text:p text:style-name="equation"><draw:frame draw:style-name="fr1" draw:name="Object48" text:anchor-type="as-char" svg:width="1.1134in" svg:height="0.2102in" draw:z-index="55"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <msub>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">F</mi>
           </mstyle>
           <mi mathvariant="italic">IcontactO</mi>
          </msub>
          <mo stretchy="false">⋅</mo>
          <msub>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">k</mi>
           </mstyle>
           <mi mathvariant="italic">IeO</mi>
          </msub>
         </mrow>
         <mo stretchy="false">≤</mo>
         <mn>0</mn>
        </mrow>
        <annotation encoding="StarMath 5.0">{bold nitalic F}_IcontactO 
cdot
{bold nitalic k}_IeO
&lt;=
0</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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       hgABAAQAAAAAAAAAcgACAC8AAACNAQAA3AEAAAkASWNvbnRhY3RPBQQAAAAA//8JAEkAYwBv
       AG4AdABhAGMAdABPAIwAAQAAAAAAiwABAAIAAAAfAIoAAQA8AAAAAwA2AAAACgBPcGVuU3lt
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       AP////8AhgABAAQAAAAAAAAAcgACABcAAAD/BQAAcgEAAAEAa9gAAAAAAP//AQBrAIwAAQAA
       AAAAiwABAAIAAAAfAIoAAQA7AAAAAwA1AAAACQBUaG9ybmRhbGUAAAAAAAD9AAAAAAAAAAAA
       BQAAAAAAAgD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAE
       AAAAAAAAAHIAAgAdAAAABwcAANwBAAADAEllT30BAAAAAP//AwBJAGUATwCMAAEAAAAAAIsA
       AQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJvbAAAAAAAAKcBAAD//wAAAAAFAAAA
       AAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAA
       AAAAcgACABoAAACUCAAAcgEAAAEAAABkIssAAAAAAP//AQBkIowAAQAAAAAAiwABAAIAAAAf
       AIoAAQA7AAAAAwA1AAAACQBUaG9ybmRhbGUAAAAAAACnAQAAAAAAAAAABQAAAAAAAAD/AwAA
       AAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAX
       AAAA0gkAAHIBAAABADDYAAAAAAD//wEAMACMAAEAAAAAAJUAAQAEAAAAAAAAAJYAAQACAAAA
       CQCMAAEAAAAAAA==
      </office:binary-data>
     </draw:image>
    </draw:frame></text:p>
   <text:h text:style-name="Heading_20_1" text:outline-level="1">No Bounce Contact</text:h>
   <text:p text:style-name="Standard">2D first.</text:p>
   <text:p text:style-name="Standard">Define curve in xy plane.</text:p>
   <text:h text:style-name="Heading_20_2" text:outline-level="2">CurveI-PointJ contact</text:h>
   <text:h text:style-name="Heading_20_2" text:outline-level="2">CurveI-CurveJ contact</text:h>
   <text:p text:style-name="Standard">Monitor<draw:frame draw:style-name="fr1" draw:name="Object49" text:anchor-type="as-char" svg:width="1.1134in" svg:height="0.2102in" draw:z-index="56"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <msub>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">F</mi>
           </mstyle>
           <mi mathvariant="italic">IcontactO</mi>
          </msub>
          <mo stretchy="false">⋅</mo>
          <msub>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">k</mi>
           </mstyle>
           <mi mathvariant="italic">IeO</mi>
          </msub>
         </mrow>
         <mo stretchy="false">≤</mo>
         <mn>0</mn>
        </mrow>
        <annotation encoding="StarMath 5.0">{bold nitalic F}_IcontactO 
cdot
{bold nitalic k}_IeO
&lt;=
0</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAQsLAAAX
       AgAAOQAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAAAoLAAAWAgAAlQAB
       AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAOwAAAAMANQAAAAkAVGhvcm5kYWxl
       AAAAAAAApwEAAAAAAAAAAAgAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcA
       AQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAGoAAAByAQAAAQBG/gAAAAAA//8BAEYA
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       hgABAAQAAAAAAAAAcgACAC8AAACNAQAA3AEAAAkASWNvbnRhY3RPBQQAAAAA//8JAEkAYwBv
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       AIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAGgAAAGAFAAByAQAAAQAAAMUiZgAAAAAA
       //8BAMUijAABAAAAAACLAAEAAgAAAB8AigABADsAAAADADUAAAAJAFRob3JuZGFsZQAAAAAA
       AKcBAAAAAAAAAAAIAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAA
       AP////8AhgABAAQAAAAAAAAAcgACABcAAAD/BQAAcgEAAAEAa9gAAAAAAP//AQBrAIwAAQAA
       AAAAiwABAAIAAAAfAIoAAQA7AAAAAwA1AAAACQBUaG9ybmRhbGUAAAAAAAD9AAAAAAAAAAAA
       BQAAAAAAAgD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAE
       AAAAAAAAAHIAAgAdAAAABwcAANwBAAADAEllT30BAAAAAP//AwBJAGUATwCMAAEAAAAAAIsA
       AQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJvbAAAAAAAAKcBAAD//wAAAAAFAAAA
       AAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAA
       AAAAcgACABoAAACUCAAAcgEAAAEAAABkIssAAAAAAP//AQBkIowAAQAAAAAAiwABAAIAAAAf
       AIoAAQA7AAAAAwA1AAAACQBUaG9ybmRhbGUAAAAAAACnAQAAAAAAAAAABQAAAAAAAAD/AwAA
       AAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAX
       AAAA0gkAAHIBAAABADDYAAAAAAD//wEAMACMAAEAAAAAAJUAAQAEAAAAAAAAAJYAAQACAAAA
       CQCMAAEAAAAAAA==
      </office:binary-data>
     </draw:image>
    </draw:frame>for valid contact.</text:p>
   <text:p text:style-name="Standard">Find lift off<draw:frame draw:style-name="fr1" draw:name="Object50" text:anchor-type="as-char" svg:width="1.1283in" svg:height="0.2102in" draw:z-index="57"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <msub>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">F</mi>
           </mstyle>
           <mi mathvariant="italic">IcontactO</mi>
          </msub>
          <mo stretchy="false">⋅</mo>
          <msub>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">k</mi>
           </mstyle>
           <mi mathvariant="italic">IeO</mi>
          </msub>
         </mrow>
         <mo stretchy="false">=</mo>
         <mn>0</mn>
        </mrow>
        <annotation encoding="StarMath 5.0">{bold nitalic F}_IcontactO 
cdot
{bold nitalic k}_IeO
=
0</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAATILAAAX
       AgAAOQAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAADELAAAWAgAAlQAB
       AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAOwAAAAMANQAAAAkAVGhvcm5kYWxl
       AAAAAAAApwEAAAAAAAAAAAgAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcA
       AQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAGoAAAByAQAAAQBG/gAAAAAA//8BAEYA
       jAABAAAAAACLAAEAAgAAAB8AigABADsAAAADADUAAAAJAFRob3JuZGFsZQAAAAAAAP0AAAAA
       AAAAAAAFAAAAAAACAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8A
       hgABAAQAAAAAAAAAcgACAC8AAACNAQAA3AEAAAkASWNvbnRhY3RPBQQAAAAA//8JAEkAYwBv
       AG4AdABhAGMAdABPAIwAAQAAAAAAiwABAAIAAAAfAIoAAQA8AAAAAwA2AAAACgBPcGVuU3lt
       Ym9sAAAAAAAApwEAAP//AAAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAAB
       AIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAGgAAAGAFAAByAQAAAQAAAMUiZgAAAAAA
       //8BAMUijAABAAAAAACLAAEAAgAAAB8AigABADsAAAADADUAAAAJAFRob3JuZGFsZQAAAAAA
       AKcBAAAAAAAAAAAIAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAA
       AP////8AhgABAAQAAAAAAAAAcgACABcAAAD/BQAAcgEAAAEAa9gAAAAAAP//AQBrAIwAAQAA
       AAAAiwABAAIAAAAfAIoAAQA7AAAAAwA1AAAACQBUaG9ybmRhbGUAAAAAAAD9AAAAAAAAAAAA
       BQAAAAAAAgD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAE
       AAAAAAAAAHIAAgAdAAAABwcAANwBAAADAEllT30BAAAAAP//AwBJAGUATwCMAAEAAAAAAIsA
       AQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJvbAAAAAAAAKcBAAD//wAAAAAFAAAA
       AAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAA
       AAAAcgACABoAAACUCAAAcgEAAAEAAAA9AEoBAAAAAP//AQA9AIwAAQAAAAAAiwABAAIAAAAf
       AIoAAQA7AAAAAwA1AAAACQBUaG9ybmRhbGUAAAAAAACnAQAAAAAAAAAABQAAAAAAAAD/AwAA
       AAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAX
       AAAA7AkAAHIBAAABADDYAAAAAAD//wEAMACMAAEAAAAAAJUAAQAEAAAAAAAAAJYAAQACAAAA
       CQCMAAEAAAAAAA==
      </office:binary-data>
     </draw:image>
    </draw:frame>.</text:p>
   <text:p text:style-name="Standard">Remove curve-curve contact constraints.</text:p>
   <text:p text:style-name="Standard">Restart simulation.</text:p>
   <text:p text:style-name="Standard">Monitor separation distance between curves. <draw:frame draw:style-name="fr1" draw:name="Object51" text:anchor-type="as-char" svg:width="0.922in" svg:height="0.2102in" draw:z-index="58"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <msub>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">r</mi>
           </mstyle>
           <mi mathvariant="italic">IeJeO</mi>
          </msub>
          <mo stretchy="false">⋅</mo>
          <msub>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">k</mi>
           </mstyle>
           <mi mathvariant="italic">IeO</mi>
          </msub>
         </mrow>
         <mo stretchy="false">≥</mo>
         <mn>0</mn>
        </mrow>
        <annotation encoding="StarMath 5.0">{bold nitalic r}_IeJeO 
cdot
{bold nitalic k}_IeO
&gt;=
0</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAScJAAAX
       AgAAOQAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAACYJAAAWAgAAlQAB
       AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAOwAAAAMANQAAAAkAVGhvcm5kYWxl
       AAAAAAAApwEAAAAAAAAAAAgAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcA
       AQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAGoAAAByAQAAAQByvwAAAAAA//8BAHIA
       jAABAAAAAACLAAEAAgAAAB8AigABADsAAAADADUAAAAJAFRob3JuZGFsZQAAAAAAAP0AAAAA
       AAAAAAAFAAAAAAACAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8A
       hgABAAQAAAAAAAAAcgACACMAAAA+AQAA3AEAAAUASWVKZU9iAgAAAAD//wUASQBlAEoAZQBP
       AIwAAQAAAAAAiwABAAIAAAAfAIoAAQA8AAAAAwA2AAAACgBPcGVuU3ltYm9sAAAAAAAApwEA
       AP//AAAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA////
       /wCGAAEABAAAAAAAAAByAAIAGgAAAGkDAAByAQAAAQAAAMUiZgAAAAAA//8BAMUijAABAAAA
       AACLAAEAAgAAAB8AigABADsAAAADADUAAAAJAFRob3JuZGFsZQAAAAAAAKcBAAAAAAAAAAAI
       AAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQA
       AAAAAAAAcgACABcAAAAiBAAAcgEAAAEAa9gAAAAAAP//AQBrAIwAAQAAAAAAiwABAAIAAAAf
       AIoAAQA7AAAAAwA1AAAACQBUaG9ybmRhbGUAAAAAAAD9AAAAAAAAAAAABQAAAAAAAgD/AwAA
       AAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAd
       AAAAKwUAANwBAAADAEllT30BAAAAAP//AwBJAGUATwCMAAEAAAAAAIsAAQACAAAAHwCKAAEA
       PAAAAAMANgAAAAoAT3BlblN5bWJvbAAAAAAAAKcBAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAA
       AAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoAAACd
       BgAAcgEAAAEAAABlIssAAAAAAP//AQBlIowAAQAAAAAAiwABAAIAAAAfAIoAAQA7AAAAAwA1
       AAAACQBUaG9ybmRhbGUAAAAAAACnAQAAAAAAAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAA
       AACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAXAAAA2wcAAHIBAAAB
       ADDYAAAAAAD//wEAMACMAAEAAAAAAJUAAQAEAAAAAAAAAJYAAQACAAAACQCMAAEAAAAAAA==
      </office:binary-data>
     </draw:image>
    </draw:frame>for valid separation.</text:p>
   <text:p text:style-name="Standard">Find contact<draw:frame draw:style-name="fr1" draw:name="Object52" text:anchor-type="as-char" svg:width="0.9374in" svg:height="0.2102in" draw:z-index="59"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <msub>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">r</mi>
           </mstyle>
           <mi mathvariant="italic">IeJeO</mi>
          </msub>
          <mo stretchy="false">⋅</mo>
          <msub>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">k</mi>
           </mstyle>
           <mi mathvariant="italic">IeO</mi>
          </msub>
         </mrow>
         <mo stretchy="false">=</mo>
         <mn>0</mn>
        </mrow>
        <annotation encoding="StarMath 5.0">{bold nitalic r}_IeJeO 
cdot
{bold nitalic k}_IeO
=
0</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAU4JAAAX
       AgAAOQAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAAE0JAAAWAgAAlQAB
       AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAOwAAAAMANQAAAAkAVGhvcm5kYWxl
       AAAAAAAApwEAAAAAAAAAAAgAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcA
       AQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAGoAAAByAQAAAQByvwAAAAAA//8BAHIA
       jAABAAAAAACLAAEAAgAAAB8AigABADsAAAADADUAAAAJAFRob3JuZGFsZQAAAAAAAP0AAAAA
       AAAAAAAFAAAAAAACAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8A
       hgABAAQAAAAAAAAAcgACACMAAAA+AQAA3AEAAAUASWVKZU9iAgAAAAD//wUASQBlAEoAZQBP
       AIwAAQAAAAAAiwABAAIAAAAfAIoAAQA8AAAAAwA2AAAACgBPcGVuU3ltYm9sAAAAAAAApwEA
       AP//AAAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA////
       /wCGAAEABAAAAAAAAAByAAIAGgAAAGkDAAByAQAAAQAAAMUiZgAAAAAA//8BAMUijAABAAAA
       AACLAAEAAgAAAB8AigABADsAAAADADUAAAAJAFRob3JuZGFsZQAAAAAAAKcBAAAAAAAAAAAI
       AAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQA
       AAAAAAAAcgACABcAAAAiBAAAcgEAAAEAa9gAAAAAAP//AQBrAIwAAQAAAAAAiwABAAIAAAAf
       AIoAAQA7AAAAAwA1AAAACQBUaG9ybmRhbGUAAAAAAAD9AAAAAAAAAAAABQAAAAAAAgD/AwAA
       AAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAd
       AAAAKwUAANwBAAADAEllT30BAAAAAP//AwBJAGUATwCMAAEAAAAAAIsAAQACAAAAHwCKAAEA
       PAAAAAMANgAAAAoAT3BlblN5bWJvbAAAAAAAAKcBAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAA
       AAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoAAAC4
       BgAAcgEAAAEAAAA9AEoBAAAAAP//AQA9AIwAAQAAAAAAiwABAAIAAAAfAIoAAQA7AAAAAwA1
       AAAACQBUaG9ybmRhbGUAAAAAAACnAQAAAAAAAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAA
       AACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAXAAAAEAgAAHIBAAAB
       ADDYAAAAAAD//wEAMACMAAEAAAAAAJUAAQAEAAAAAAAAAJYAAQACAAAACQCMAAEAAAAAAA==
      </office:binary-data>
     </draw:image>
    </draw:frame>.</text:p>
   <text:p text:style-name="Standard">Restore curve-curve contact constraints.</text:p>
   <text:p text:style-name="Standard">Restart simulation.</text:p>
   <text:h text:style-name="Heading_20_1" text:outline-level="1">Collision</text:h>
   <text:p text:style-name="Standard">2D first.</text:p>
   <text:p text:style-name="Standard">Define curve in xy plane.</text:p>
   <text:p text:style-name="Standard">Find point of contact.</text:p>
   <text:p text:style-name="Standard"><text:tab/>Curve compose of piecewise curves and corners.</text:p>
   <text:p text:style-name="Standard"><text:tab/>Corner - curve contact.</text:p>
   <text:p text:style-name="Standard"><text:tab/>Curve - curve contact.</text:p>
   <text:p text:style-name="Standard">Introduce bushing at point of contact. Normal stiffness can be different from the tangential stiffness.</text:p>
   <text:p text:style-name="Standard">Or introduce coefficient of restitution at point of contact.</text:p>
   <text:p text:style-name="Standard">Restart simulation.</text:p>
   <text:p text:style-name="Standard">Monitor<draw:frame draw:style-name="fr1" draw:name="Object55" text:anchor-type="as-char" svg:width="1.1134in" svg:height="0.2102in" draw:z-index="60"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <msub>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">F</mi>
           </mstyle>
           <mi mathvariant="italic">IcontactO</mi>
          </msub>
          <mo stretchy="false">⋅</mo>
          <msub>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">k</mi>
           </mstyle>
           <mi mathvariant="italic">IeO</mi>
          </msub>
         </mrow>
         <mo stretchy="false">≤</mo>
         <mn>0</mn>
        </mrow>
        <annotation encoding="StarMath 5.0">{bold nitalic F}_IcontactO 
cdot
{bold nitalic k}_IeO
&lt;=
0</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAQsLAAAX
       AgAAOQAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAAAoLAAAWAgAAlQAB
       AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAOwAAAAMANQAAAAkAVGhvcm5kYWxl
       AAAAAAAApwEAAAAAAAAAAAgAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcA
       AQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAGoAAAByAQAAAQBG/gAAAAAA//8BAEYA
       jAABAAAAAACLAAEAAgAAAB8AigABADsAAAADADUAAAAJAFRob3JuZGFsZQAAAAAAAP0AAAAA
       AAAAAAAFAAAAAAACAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8A
       hgABAAQAAAAAAAAAcgACAC8AAACNAQAA3AEAAAkASWNvbnRhY3RPBQQAAAAA//8JAEkAYwBv
       AG4AdABhAGMAdABPAIwAAQAAAAAAiwABAAIAAAAfAIoAAQA8AAAAAwA2AAAACgBPcGVuU3lt
       Ym9sAAAAAAAApwEAAP//AAAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAAB
       AIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAGgAAAGAFAAByAQAAAQAAAMUiZgAAAAAA
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      </office:binary-data>
     </draw:image>
    </draw:frame>for valid contact.</text:p>
   <text:p text:style-name="Standard">Find lift off<draw:frame draw:style-name="fr1" draw:name="Object53" text:anchor-type="as-char" svg:width="1.1283in" svg:height="0.2102in" draw:z-index="61"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <msub>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">F</mi>
           </mstyle>
           <mi mathvariant="italic">IcontactO</mi>
          </msub>
          <mo stretchy="false">⋅</mo>
          <msub>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">k</mi>
           </mstyle>
           <mi mathvariant="italic">IeO</mi>
          </msub>
         </mrow>
         <mo stretchy="false">=</mo>
         <mn>0</mn>
        </mrow>
        <annotation encoding="StarMath 5.0">{bold nitalic F}_IcontactO 
cdot
{bold nitalic k}_IeO
=
0</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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      </office:binary-data>
     </draw:image>
    </draw:frame>.</text:p>
   <text:p text:style-name="Standard">Remove contact forces.</text:p>
   <text:p text:style-name="Standard">Restart simulation.</text:p>
   <text:p text:style-name="Standard"/>
  </office:text>
 </office:body>
</office:document>