1200kine.fodt

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   <text:h text:style-name="Heading_20_3" text:outline-level="3">Radius of Curvature,<draw:frame draw:style-name="fr1" draw:name="Object71" text:anchor-type="as-char" svg:width="0.5839in" svg:height="0.2165in" draw:z-index="57"><draw:object>
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   <text:p text:style-name="equation">Refer to figure2.sxd.</text:p>
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    </draw:frame>is just a parameter. It is not arc length.</text:p>
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=
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over 
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=
abs{{partial {bold nitalic r}_mem(t,s_1)} 
over 
{partial s_1}} 
over 
{{d %theta} 
over 
{d s_1}}</annotation>
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   <text:p text:style-name="Standard">The direction cosine between the two unit tangent vectors is</text:p>
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        <annotation encoding="StarMath 5.0">cos(d%theta) 
=
{partial {bold nitalic r}_mem} 
over 
{partial s_1}  
over 
abs{
{partial {bold nitalic r}_mem} 
over 
{partial s_1}
}   
cdot 
{
{
{partial {bold nitalic r}_mem
} 
over 
{partial s_1}  
	+
	{partial^2 {bold nitalic r}_mem}
	over 
	{partial s_1 partial s_1}
	ds_1
}
over 
abs{
{partial {bold nitalic r}_mem} 
over 
{partial s_1}
	+
	{partial^2 {bold nitalic r}_mem}
	over 
	{partial s_1 partial s_1}
	ds_1
}   
}
</annotation>
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                  <mo stretchy="false">∂</mo>
                  <msub>
                   <mi>s</mi>
                   <mn>1</mn>
                  </msub>
                  <mo stretchy="false">∂</mo>
                  <msub>
                   <mi>s</mi>
                   <mn>1</mn>
                  </msub>
                 </mrow>
                </mfrac>
                <msub>
                 <mi mathvariant="italic">ds</mi>
                 <mn>1</mn>
                </msub>
               </mrow>
              </mrow>
              <mo stretchy="false">+</mo>
              <msup>
               <mrow>
                <mo fence="true" form="prefix" stretchy="true">|</mo>
                <mrow>
                 <mrow>
                  <mfrac>
                   <mrow>
                    <msup>
                     <mo stretchy="false">∂</mo>
                     <mn>2</mn>
                    </msup>
                    <msub>
                     <mstyle mathvariant="bold">
                      <mi mathvariant="normal">r</mi>
                     </mstyle>
                     <mi mathvariant="italic">mem</mi>
                    </msub>
                   </mrow>
                   <mrow>
                    <mo stretchy="false">∂</mo>
                    <msub>
                     <mi>s</mi>
                     <mn>1</mn>
                    </msub>
                    <mo stretchy="false">∂</mo>
                    <msub>
                     <mi>s</mi>
                     <mn>1</mn>
                    </msub>
                   </mrow>
                  </mfrac>
                  <msub>
                   <mi mathvariant="italic">ds</mi>
                   <mn>1</mn>
                  </msub>
                 </mrow>
                </mrow>
                <mo fence="true" form="postfix" stretchy="true">|</mo>
               </mrow>
               <mn>2</mn>
              </msup>
             </mrow>
            </mrow>
           </msqrt>
          </mrow>
         </mfrac>
        </mrow>
        <annotation encoding="StarMath 5.0">1 - (d%theta)^2 + dotsaxis 
=
{
	{
		{partial {bold nitalic r}_mem} 
		over 
		{partial s_1} 
	} 
	cdot 
	{
		{partial {bold nitalic r}_mem} 
		over 
		{partial s_1} 
	} 
	+
	{
		{partial {bold nitalic r}_mem} 
		over 
		{partial s_1} 
	} 
	cdot 
	{
		{partial^2 {bold nitalic r}_mem} 
		over 
		{partial s_1 partial s_1} 
		ds_1
	} 
} 
over 
{
	abs {
		{partial {bold nitalic r}_mem} 
		over 
		{partial s_1} 
	}
	sqrt{
		{abs {
			{partial {bold nitalic r}_mem} 
			over 
			{partial s_1} 
		}}^2
		+2
		{
			{partial {bold nitalic r}_mem} 
			over 
			{partial s_1} 
		} 
		cdot 
		{
			{partial^2 {bold nitalic r}_mem} 
			over 
			{partial s_1 partial s_1} 
			ds_1
		} 
		+
		{abs {
			{partial^2 {bold nitalic r}_mem} 
			over 
			{partial s_1 partial s_1} 
			ds_1
		}}^2
	} 
} 
</annotation>
       </semantics>
      </math>
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			{partial s_1 partial s_1} 
			ds_1
		} 
	}
	over
	{
		{abs {
		{partial {bold nitalic r}_mem} 
				over 
		{partial s_1} 
		}}^2
	}
right)
left(
	1+
	{2	
		{
			{partial {bold nitalic r}_mem} 
			over 
			{partial s_1} 
		} 
		cdot 
		{
			{partial^2 {bold nitalic r}_mem} 
			over 
			{partial s_1 partial s_1} 
			ds_1
		} 
	}
	over
	{
		{abs {
		{partial {bold nitalic r}_mem} 
				over 
		{partial s_1} 
		}}^2
	}
	+
	{
		{abs {
		{partial^2 {bold nitalic r}_mem} 
				over 
		{partial s_1 partial s_1} 
		ds_1
		}}^2
	} 
	over 
	{
		{abs {
		{partial {bold nitalic r}_mem} 
				over 
		{partial s_1} 
		}}^2
	}
right)^-{1 over 2}
</annotation>
       </semantics>
      </math>
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		}}^2
	}
right)
left(
	1-
	{
		{
			{partial {bold nitalic r}_mem} 
			over 
			{partial s_1} 
		} 
		cdot 
		{
			{partial^2 {bold nitalic r}_mem} 
			over 
			{partial s_1 partial s_1} 
			ds_1
		} 
	}
	over
	{
		{abs {
		{partial {bold nitalic r}_mem} 
				over 
		{partial s_1} 
		}}^2
	}
	-
	{
		{abs {
		{partial^2 {bold nitalic r}_mem} 
				over 
		{partial s_1 partial s_1} 
		ds_1
		}}^2
	} 
	over 
	{
		{2 abs {
		{partial {bold nitalic r}_mem} 
				over 
		{partial s_1} 
		}}^2
	}
	+
	dotsaxis 
right)
</annotation>
       </semantics>
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        </mrow>
        <annotation encoding="StarMath 5.0">left(
{d%theta} 
over
{ds_1}
right)^2
=
	{
		{abs {
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				over 
		{partial s_1 partial s_1}
		}}^2
	} 
	over 
	{
		{2 abs {
		{partial {bold nitalic r}_mem} 
				over 
		{partial s_1} 
		}}^2
	}
	+
	left (
	{	
		{
			{partial {bold nitalic r}_mem} 
			over 
			{partial s_1} 
		} 
		cdot 
		{
			{partial^2 {bold nitalic r}_mem} 
			over 
			{partial s_1 partial s_1}
		} 
	}
	over
	{
		{abs {
		{partial {bold nitalic r}_mem} 
				over 
		{partial s_1} 
		}}^2
	}
	right )^2</annotation>
       </semantics>
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{{bold nitalic vec k}_e times {bold nitalic vec i}_e}
=
{{bold nitalic vec n}_{2e} over n_{2e}}</annotation>
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    </draw:frame>is binormal to the curve.</text:p>
   <text:p text:style-name="Standard">Problem:<draw:frame draw:style-name="fr1" draw:name="Object12" text:anchor-type="as-char" svg:width="0.2228in" svg:height="0.2402in" draw:z-index="72"><draw:object>
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     </draw:object><draw:image>
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       <semantics>
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          <mover accent="true">
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        </msub>
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     </draw:object><draw:image>
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    </draw:frame>that is never parallel to the curve. Then</text:p>
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        <annotation encoding="StarMath 5.0">{bold nitalic vec i}_e
=
{{bold nitalic vec t}_{1e} over t_{1e}}</annotation>
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     </draw:image>
    </draw:frame>is tangent to the curve in increasing<draw:frame draw:style-name="fr1" draw:name="Object351" text:anchor-type="as-char" svg:width="0.2047in" svg:height="0.2138in" draw:z-index="76"><draw:object>
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       <semantics>
        <msub>
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        </msub>
        <annotation encoding="StarMath 5.0">s_1</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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