best1plane2.fodt

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                <mi>o</mi>
               </msub>
               <mi>sin</mi>
               <mfrac>
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                <mn>2</mn>
               </mfrac>
              </mrow>
             </mtd>
            </mtr>
            <mtr>
             <mtd>
              <mrow>
               <mi>cos</mi>
               <mfrac>
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                <mn>2</mn>
               </mfrac>
              </mrow>
             </mtd>
            </mtr>
           </mtable>
          </mrow>
          <mo fence="true" form="postfix" stretchy="true">}</mo>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">{bold nitalic q}_E =
left lbrace 
matrix{
E_1 ## 
E_2 ## 
E_3 ## 
E_4
} 
right rbrace 
= left lbrace 
matrix{
{bold nitalic vec n cdot bold nitalic vec i}_o sin %theta over 2 ## 
{bold nitalic vec n cdot bold nitalic vec j}_o sin %theta over 2 ## 
{bold nitalic vec n cdot bold nitalic vec k}_o sin %theta over 2 ## 
cos %theta over 2
} 
right rbrace </annotation>
       </semantics>
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     </draw:image>
     <svg:desc>formula</svg:desc>
    </draw:frame></text:p>
   <text:p text:style-name="Standard"/>
   <text:p text:style-name="Standard">Euler parameters constraint</text:p>
   <text:p text:style-name="Standard"/>
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       <semantics>
        <mrow>
         <msub>
          <mi>G</mi>
          <mi>E</mi>
         </msub>
         <mo stretchy="false">=</mo>
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          <mo stretchy="false">+</mo>
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          </msubsup>
          <mo stretchy="false">+</mo>
          <msubsup>
           <mi>E</mi>
           <mn>3</mn>
           <mn>2</mn>
          </msubsup>
          <mo stretchy="false">+</mo>
          <msubsup>
           <mi>E</mi>
           <mn>4</mn>
           <mn>2</mn>
          </msubsup>
          <mo stretchy="false">−</mo>
          <mn>1</mn>
         </mrow>
         <mo stretchy="false">=</mo>
         <mn>0</mn>
        </mrow>
        <annotation encoding="StarMath 5.0">G_E = E^2_1 + E^2_2 + E^2_3 + E^2_4 - 1 = 0</annotation>
       </semantics>
      </math>
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     <svg:desc>formula</svg:desc>
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   <text:p text:style-name="Standard"/>
   <text:p text:style-name="Standard">First partial derivative</text:p>
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            <mtr>
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        <annotation encoding="StarMath 5.0">{partial G_E}
over
{partial {bold nitalic q}_E}
=
left lbrace 
matrix{
2E_1 ## 
2E_2 ## 
2E_3 ## 
2E_4
} 
right rbrace 
</annotation>
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     </draw:image>
     <svg:desc>formula</svg:desc>
    </draw:frame></text:p>
   <text:p text:style-name="Standard"/>
   <text:p text:style-name="Standard">Second partial derivative</text:p>
   <text:p text:style-name="Standard"/>
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       <semantics>
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         <mrow>
          <mfrac>
           <mrow>
            <msup>
             <mo stretchy="false">∂</mo>
             <mn>2</mn>
            </msup>
            <msub>
             <mi>G</mi>
             <mi>E</mi>
            </msub>
           </mrow>
           <mrow>
            <mo stretchy="false">∂</mo>
            <msub>
             <mstyle mathvariant="bold">
              <mi mathvariant="normal">q</mi>
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             <mi>E</mi>
            </msub>
            <mo stretchy="false">∂</mo>
            <msub>
             <mstyle mathvariant="bold">
              <mi mathvariant="normal">q</mi>
             </mstyle>
             <mi>E</mi>
            </msub>
           </mrow>
          </mfrac>
          <mo stretchy="false">=</mo>
          <mn>2</mn>
         </mrow>
         <mrow>
          <mo fence="true" form="prefix" stretchy="true">[</mo>
          <mrow>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">I</mi>
           </mstyle>
          </mrow>
          <mo fence="true" form="postfix" stretchy="true">]</mo>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">{partial^2 G_E}
over
{partial {bold nitalic q}_E partial {bold nitalic q}_E}
=
2 left[{bold nitalic I} right]
</annotation>
       </semantics>
      </math>
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     </draw:image>
     <svg:desc>formula</svg:desc>
    </draw:frame></text:p>
   <text:p text:style-name="Standard"/>
   <text:p text:style-name="Standard">Rotation matrix</text:p>
   <text:p text:style-name="Standard"/>
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               <mtd>
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               <mtd>
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              <mtr>
               <mtd>
                <msub>
                 <mi>E</mi>
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=
left[{bold nitalic A} right]^T_Oo
left(
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right)


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=
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left(
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right)


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   <text:p text:style-name="Standard">First partial derivatives</text:p>
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        <annotation encoding="StarMath 5.0">{partial {bold nitalic r}_opo}
over
{partial {bold nitalic q}_X}
=
-
left[{bold nitalic A} right]^T_Oo



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=
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over
{partial {bold nitalic q}_E}

left(
{bold nitalic r}_OpO - {bold nitalic r}_OoO
right)


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               </mstyle>
              </mrow>
              <mo fence="true" form="postfix" stretchy="true">]</mo>
             </mrow>
             <mi mathvariant="italic">Oo</mi>
             <mi>T</mi>
            </msubsup>
           </mrow>
           <mrow>
            <mo stretchy="false">∂</mo>
            <msub>
             <mstyle mathvariant="bold">
              <mi mathvariant="normal">q</mi>
             </mstyle>
             <mi>E</mi>
            </msub>
            <mo stretchy="false">∂</mo>
            <msub>
             <mstyle mathvariant="bold">
              <mi mathvariant="normal">q</mi>
             </mstyle>
             <mi>E</mi>
            </msub>
           </mrow>
          </mfrac>
         </mrow>
         <mrow>
          <mo fence="true" form="prefix" stretchy="true">(</mo>
          <mrow>
           <mrow>
            <msub>
             <mstyle mathvariant="bold">
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             </mstyle>
             <mi mathvariant="italic">OpO</mi>
            </msub>
            <mo stretchy="false">−</mo>
            <msub>
             <mstyle mathvariant="bold">
              <mi mathvariant="normal">r</mi>
             </mstyle>
             <mi mathvariant="italic">OoO</mi>
            </msub>
           </mrow>
          </mrow>
          <mo fence="true" form="postfix" stretchy="true">)</mo>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">{partial^2 {bold nitalic r}_opo}
over
{partial {bold nitalic q}_E partial {bold nitalic q}_E}
=
{partial^2 left[{bold nitalic A} right]^T_Oo}
over
{partial {bold nitalic q}_E partial {bold nitalic q}_E}
left(
{bold nitalic r}_OpO - {bold nitalic r}_OoO
right)



</annotation>
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      </office:binary-data>
     </draw:image>
     <svg:desc>formula</svg:desc>
    </draw:frame></text:p>
   <text:p text:style-name="Standard"/>
   <text:p text:style-name="Standard">Constrain first point to z axis.</text:p>
   <text:p text:style-name="Standard"/>
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      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <msub>
          <mi>G</mi>
          <mn>11</mn>
         </msub>
         <mo stretchy="false">=</mo>
         <msub>
          <mi>x</mi>
          <mrow>
           <mi mathvariant="italic">op</mi>
           <mn>1</mn>
           <mi>o</mi>
          </mrow>
         </msub>
         <mo stretchy="false">=</mo>
         <msub>
          <mi>r</mi>
          <mrow>
           <mi mathvariant="italic">op</mi>
           <mn>1</mn>
           <mi>o</mi>
           <mn>1</mn>
          </mrow>
         </msub>
         <mo stretchy="false">=</mo>
         <mn>0</mn>
        </mrow>
        <annotation encoding="StarMath 5.0">G_11 
=
x_op1o
=
r_op1o1
=
0


</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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      </office:binary-data>
     </draw:image>
     <svg:desc>formula</svg:desc>
    </draw:frame></text:p>
   <text:p text:style-name="Standard"/>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr1" draw:name="Object59" text:anchor-type="as-char" svg:width="1.4535in" svg:height="0.2098in" draw:z-index="33"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <msub>
          <mi>G</mi>
          <mn>12</mn>
         </msub>
         <mo stretchy="false">=</mo>
         <msub>
          <mi>y</mi>
          <mrow>
           <mi mathvariant="italic">op</mi>
           <mn>1</mn>
           <mi>o</mi>
          </mrow>
         </msub>
         <mo stretchy="false">=</mo>
         <msub>
          <mi>r</mi>
          <mrow>
           <mi mathvariant="italic">op</mi>
           <mn>1</mn>
           <mi>o</mi>
           <mn>2</mn>
          </mrow>
         </msub>
         <mo stretchy="false">=</mo>
         <mn>0</mn>
        </mrow>
        <annotation encoding="StarMath 5.0">G_12
=
y_op1o
=
r_op1o2
=
0


</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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      </office:binary-data>
     </draw:image>
     <svg:desc>formula</svg:desc>
    </draw:frame></text:p>
   <text:p text:style-name="Standard"/>
   <text:p text:style-name="Standard">Define</text:p>
   <text:p text:style-name="Standard"/>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr1" draw:name="Object57" text:anchor-type="as-char" svg:width="1.0264in" svg:height="0.4429in" draw:z-index="34"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mfrac>
          <mrow>
           <mo stretchy="false">∂</mo>
           <msub>
            <mi>G</mi>
            <mi mathvariant="italic">kl</mi>
           </msub>
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{partial {bold nitalic q}_X}
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{partial r_opkol}
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{partial {bold nitalic q}_X}


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over
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=
{partial r_opkol}
over
{partial {bold nitalic q}_E}


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=
{partial^2 r_opkol}
over
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{partial {bold nitalic q}_X partial {bold nitalic q}_E}
=
{partial^2 r_opkol}
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{partial {bold nitalic q}_X partial {bold nitalic q}_E}



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over
{partial {bold nitalic q}_E partial {bold nitalic q}_E}
=
{partial^2 r_opkol}
over
{partial {bold nitalic q}_E partial {bold nitalic q}_E}



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   <text:p text:style-name="Standard"/>
   <text:p text:style-name="Standard">Constrain second point on xz plane.</text:p>
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        <annotation encoding="StarMath 5.0">G_22
=
y_op2o
=
r_op2o2
=
0


</annotation>
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     </draw:image>
     <svg:desc>formula</svg:desc>
    </draw:frame></text:p>
   <text:p text:style-name="Standard"/>
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=
1 over 2

sum from i=1 to n 
left(
r^2_opo3
right)
+ %lambda_E G_E
+ %lambda_11 G_11
+ %lambda_12 G_12
+ %lambda_22 G_22

</annotation>
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     </draw:image>
     <svg:desc>formula</svg:desc>
    </draw:frame></text:p>
   <text:p text:style-name="Standard"/>
   <text:p text:style-name="Standard">z in normal.</text:p>
   <text:p text:style-name="Standard">z-axis passes through first point.</text:p>
   <text:p text:style-name="Standard">xz plane passes through second point.</text:p>
   <text:p text:style-name="Standard"/>
   <text:p text:style-name="Standard">Minimization</text:p>
   <text:p text:style-name="Standard"/>
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      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
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           </mrow>
           <mrow>
            <mo stretchy="false">∂</mo>
            <msub>
             <mstyle mathvariant="bold">
              <mi mathvariant="normal">q</mi>
             </mstyle>
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          </mfrac>
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             <mrow>
              <mrow>
               <msub>
                <mi>r</mi>
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               </msub>
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                  </mrow>
                 </msub>
                </mrow>
                <mrow>
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          </mstyle>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">{partial s}
over
{partial {bold nitalic q}_X}
=
sum from i=1 to n 
left(
r_opo3
{partial r_opo3}
over
{partial {bold nitalic q}_X}
right)
+ %lambda_11 
G_11
over
{partial {bold nitalic q}_X}
+ %lambda_12 
G_12
over
{partial {bold nitalic q}_X}
+ %lambda_22 
G_22
over
{partial {bold nitalic q}_X}
={bold 0}

</annotation>
       </semantics>
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        <annotation encoding="StarMath 5.0">{partial s}
over
{partial {bold nitalic q}_E}
=
sum from i=1 to n 
left(
r_opo3
{partial r_opo3}
over
{partial {bold nitalic q}_E}
right)
+ %lambda_E
{partial G_E}
over
{partial {bold nitalic q}_E}
+ %lambda_11 
G_11
over
{partial {bold nitalic q}_E}
+ %lambda_12 
G_12
over
{partial {bold nitalic q}_E}
+ %lambda_22 
G_22
over
{partial {bold nitalic q}_E}
={bold 0}

</annotation>
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     <svg:desc>formula</svg:desc>
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        <annotation encoding="StarMath 5.0">{partial s}
over
{partial %lambda_E}
=
G_E
=0

</annotation>
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     </draw:image>
     <svg:desc>formula</svg:desc>
    </draw:frame></text:p>
   <text:p text:style-name="Standard"/>
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       <semantics>
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          <mrow>
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          </mrow>
         </mfrac>
         <mo stretchy="false">=</mo>
         <msub>
          <mi>G</mi>
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         </msub>
         <mo stretchy="false">=</mo>
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        </mrow>
        <annotation encoding="StarMath 5.0">{partial s}
over
{partial %lambda_11}
=
G_11
=0

</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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   <text:h text:style-name="Heading_20_1" text:outline-level="1">Newton-Raphson method</text:h>
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{
{partial s}
over
{partial {bold nitalic q}_X}
##
{partial s}
over
{partial {bold nitalic q}_E}
##
{partial s}
over
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##
{partial s}
over
{partial %lambda_11}
##
{partial s}
over
{partial %lambda_12}
##
{partial s}
over
{partial %lambda_22}
}
right rbrace_i
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   <text:p text:style-name="Standard"/>
   <text:p text:style-name="Standard">New guess</text:p>
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   <text:p text:style-name="Standard">Second partial derivatives</text:p>
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{partial {bold nitalic q}_X partial {bold nitalic q}_X}
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over
{partial {bold nitalic q}_X}
{partial r_opo3}
over
{partial {bold nitalic q}_X}
right)


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over
{partial {bold nitalic q}_X partial {bold nitalic q}_E}
=
sum from i=1 to n
left(
left(
{partial r_opo3}
over
{partial {bold nitalic q}_X}
{partial r_opo3}
over
{partial {bold nitalic q}_E}
+
r_opo3
{partial^2 r_opo3}
over
{partial {bold nitalic q}_X partial {bold nitalic q}_E}
right)
+
%lambda_11
{partial^2 G_11}
over
{partial {bold nitalic q}_X partial {bold nitalic q}_E}
+
%lambda_12
{partial^2 G_12}
over
{partial {bold nitalic q}_X partial {bold nitalic q}_E}
+
%lambda_22
{partial^2 G_22}
over
{partial {bold nitalic q}_X partial {bold nitalic q}_E}
right)


</annotation>
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over
{partial {bold nitalic q}_X partial %lambda_22}
= 
{partial G_22}
over
{partial {bold nitalic q}_X}

</annotation>
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over
{partial {bold nitalic q}_E partial {bold nitalic q}_E}
=
sum from i=1 to n
left(
left(
{partial r_opo3}
over
{partial {bold nitalic q}_E}
{partial r_opo3}
over
{partial {bold nitalic q}_E}
+
r_opo3
{partial^2 r_opo3}
over
{partial {bold nitalic q}_E partial {bold nitalic q}_E}
right)
+
%lambda_11
{partial^2 G_11}
over
{partial {bold nitalic q}_E partial {bold nitalic q}_E}
+
%lambda_12
{partial^2 G_12}
over
{partial {bold nitalic q}_E partial {bold nitalic q}_E}
+
%lambda_22
{partial^2 G_22}
over
{partial {bold nitalic q}_E partial {bold nitalic q}_E}
right)


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     <svg:desc>formula</svg:desc>
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        <annotation encoding="StarMath 5.0">{partial^2 s}
over
{partial {bold nitalic q}_E partial %lambda_E}
= 
{partial G_E}
over
{partial {bold nitalic q}_E}

</annotation>
       </semantics>
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over
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= 
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over
{partial {bold nitalic q}_E}

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over
{partial %lambda_E partial %lambda_E}

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over
{partial %lambda_E partial %lambda_11}

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{partial %lambda_E partial %lambda_12}

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{partial %lambda_E partial %lambda_22}

= 0

</annotation>
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over
{partial %lambda_11 partial %lambda_11}

= 0

</annotation>
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     <svg:desc>formula</svg:desc>
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over
{partial %lambda_11 partial %lambda_12}

= 0

</annotation>
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over
{partial %lambda_11 partial %lambda_22}

= 0

</annotation>
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over
{partial %lambda_12 partial %lambda_12}

= 0

</annotation>
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   <text:p text:style-name="Standard">Simplified expressions</text:p>
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