best1plane.fodt

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right rbrace 
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left lbrace 
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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">Euler parameters</text:p>
   <text:p text:style-name="Standard"/>
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            <mtr>
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E_3 ## 
E_4
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right rbrace 
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{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
} 
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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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        <annotation encoding="StarMath 5.0">G_E = E^2_1 + E^2_2 + E^2_3 + E^2_4 - 1 = 0</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">First partial derivative</text:p>
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        </mrow>
        <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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over
{partial {bold nitalic q}_E partial {bold nitalic q}_E}
=
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   <text:p text:style-name="Standard">Rotation matrix</text:p>
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=
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left(
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over
{partial {bold nitalic q}_E partial {bold nitalic q}_E}
=
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over
{partial {bold nitalic q}_E partial {bold nitalic q}_E}
left(
{bold nitalic r}_OpO - {bold nitalic r}_OoO
right)



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   <text:p text:style-name="Standard"/>
   <text:p text:style-name="Standard">Least squares</text:p>
   <text:p text:style-name="Standard"/>
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         <mrow>
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              </mrow>
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               <msubsup>
                <mi>y</mi>
                <mi mathvariant="italic">opo</mi>
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               <mo stretchy="false">+</mo>
               <msubsup>
                <mi>c</mi>
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        </mrow>
        <annotation encoding="StarMath 5.0">s
=
1 over 2

sum from i=1 to n 
left(
x^2_opo
+
c_y 
y^2_opo
+
c_z
z^2_opo
right)
+ %lambda G_E
</annotation>
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     </draw:image>
     <svg:desc>formula</svg:desc>
    </draw:frame></text:p>
   <text:p text:style-name="Standard">z in normal, x attract more points.</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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       <semantics>
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          <mrow>
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             <mi mathvariant="normal">q</mi>
            </mstyle>
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           </msub>
          </mrow>
         </mfrac>
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         <mrow>
          <munderover>
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               <mrow>
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        <annotation encoding="StarMath 5.0">{partial s}
over
{partial {bold nitalic q}_X}
=
sum from i=1 to n 
left(
x_opo
{partial x_opo}
over
{partial {bold nitalic q}_X}
+
c_y
y_opo
{partial y_opo}
over
{partial {bold nitalic q}_X}
+
c_z
z_opo
{partial x_opo}
over
{partial {bold nitalic q}_X}
right)

={bold 0}

</annotation>
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{partial {bold nitalic q}_E}
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over
{partial {bold nitalic q}_E}
+
c_y
y_opo
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over
{partial {bold nitalic q}_E}
+
c_z
z_opo
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over
{partial {bold nitalic q}_E}
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+ %lambda 
{partial G_E}
over
{partial {bold nitalic q}_E}
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   <text:p text:style-name="Standard"/>
   <text:p text:style-name="Standard">Second partial derivatives</text:p>
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over
{partial {bold nitalic q}_X partial {bold nitalic q}_X}
=
sum from i=1 to n
left(
{partial x_opo}
over
{partial {bold nitalic q}_X}
{partial x_opo}
over
{partial {bold nitalic q}_X}
+
c_y
{partial y_opo}
over
{partial {bold nitalic q}_X}
{partial y_opo}
over
{partial {bold nitalic q}_X}
+
c_z
{partial z_opo}
over
{partial {bold nitalic q}_X}
{partial z_opo}
over
{partial {bold nitalic q}_X}
right)


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{partial {bold nitalic q}_X partial {bold nitalic q}_E}
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{partial {bold nitalic q}_X}
{partial x_opo}
over
{partial {bold nitalic q}_E}
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over
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c_y
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over
{dotsaxis}
dotsaxis right)
+
c_z
left(
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over
{dotsaxis}
dotsaxis right)
right)


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{partial {bold nitalic q}_X partial %lambda}

= {bold 0}

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        <annotation encoding="StarMath 5.0">{partial^2 s}
over
{partial {bold nitalic q}_E partial {bold nitalic q}_E}
=
sum from i=1 to n
left(
left(
{partial x_opo}
over
{partial {bold nitalic q}_E}
{partial x_opo}
over
{partial {bold nitalic q}_E}
+
x_opo
{partial^2 x_opo}
over
{partial {bold nitalic q}_E partial {bold nitalic q}_E}
right)+
c_y
left(
{partial y_opo}
over
{dotsaxis}
dotsaxis right)
+
c_z
left(
{partial z_opo}
over
{dotsaxis}
dotsaxis right)
right)
+ 
%lambda 
{partial^2 G_E}
over
{partial {bold nitalic q}_E partial {bold nitalic q}_E}

</annotation>
       </semantics>
      </math>
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{partial %lambda partial %lambda}

= 0

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   <text:p text:style-name="Standard">Simplified expressions</text:p>
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     </draw:image>
     <svg:desc>formula</svg:desc>
    </draw:frame></text:p>
  </office:text>
 </office:body>
</office:document>