k000homogeneous.fodt

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{bold nitalic bar r}_ApA
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left lbrace
matrix
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   <text:p text:style-name="Standard"/>
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=
left[{bold nitalic bar M} right]_Ap
{bold nitalic bar h}_pmp
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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">To find the homogeneous matrix to transform from <draw:frame draw:style-name="fr2" draw:name="Object4" text:anchor-type="as-char" svg:width="0.1898in" svg:height="0.2138in" draw:z-index="4"><draw:object>
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       <semantics>
        <msub>
         <mi>t</mi>
         <mn>1</mn>
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        <annotation encoding="StarMath 5.0">t_1</annotation>
       </semantics>
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     </draw:object><draw:image>
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       AP8DAAAAAAAAAAAA/wMAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIA
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     </draw:image>
    </draw:frame><text:s/>to <draw:frame draw:style-name="fr2" draw:name="Object5" text:anchor-type="as-char" svg:width="0.1925in" svg:height="0.2138in" draw:z-index="5"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
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   <text:p text:style-name="Standard"/>
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       <semantics>
        <mrow>
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          <mrow>
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            </mstyle>
           </mrow>
           <mo fence="true" form="postfix" stretchy="true">]</mo>
          </mrow>
          <mi mathvariant="italic">Ap</mi>
          <mi>T</mi>
         </msubsup>
         <mrow>
          <mrow>
           <mo fence="true" form="prefix" stretchy="true">(</mo>
           <mrow>
            <mrow>
             <msub>
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              </mstyle>
              <mi mathvariant="italic">AmA</mi>
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              <mstyle mathvariant="bold">
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              </mstyle>
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          <msub>
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          <msubsup>
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            <mrow>
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         <mrow>
          <mo fence="true" form="prefix" stretchy="true">(</mo>
          <mrow>
           <mrow>
            <msub>
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              <mover accent="true">
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              </mover>
             </mstyle>
             <mi mathvariant="italic">AmA</mi>
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            <msub>
             <mstyle mathvariant="bold">
              <mover accent="true">
               <mi mathvariant="normal">r</mi>
               <mo stretchy="false">¯</mo>
              </mover>
             </mstyle>
             <mi mathvariant="italic">ApA</mi>
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           </mrow>
          </mrow>
          <mo fence="true" form="postfix" stretchy="true">)</mo>
         </mrow>
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   <text:p text:style-name="P1">Finding inverses</text:p>
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=
{bold nitalic r}_ApA
+
left[{bold nitalic A} right]_Ap
{bold nitalic r}_pmp
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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="P1">Pure translation</text:p>
   <text:p text:style-name="Standard"/>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr1" draw:name="Object21" text:anchor-type="as-char" svg:width="1.5283in" svg:height="0.2307in" draw:z-index="21"><draw:object>
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       <semantics>
        <mrow>
         <mrow>
          <msub>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">r</mi>
           </mstyle>
           <mi mathvariant="italic">AmA</mi>
          </msub>
          <mo stretchy="false">=</mo>
          <mrow>
           <msub>
            <mstyle mathvariant="bold">
             <mi mathvariant="normal">r</mi>
            </mstyle>
            <mi mathvariant="italic">ApA</mi>
           </msub>
           <mo stretchy="false">+</mo>
           <msub>
            <mrow>
             <mo fence="true" form="prefix" stretchy="true">[</mo>
             <mrow>
              <mstyle mathvariant="bold">
               <mi mathvariant="normal">A</mi>
              </mstyle>
             </mrow>
             <mo fence="true" form="postfix" stretchy="true">]</mo>
            </mrow>
            <mi mathvariant="italic">Ap</mi>
           </msub>
          </mrow>
         </mrow>
         <msub>
          <mstyle mathvariant="bold">
           <mi mathvariant="normal">r</mi>
          </mstyle>
          <mi mathvariant="italic">pmp</mi>
         </msub>
        </mrow>
        <annotation encoding="StarMath 5.0">{bold nitalic r}_AmA
=
{bold nitalic r}_ApA
+
left[{bold nitalic A} right]_Ap
{bold nitalic r}_pmp
</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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     </draw:image>
     <svg:desc>formula</svg:desc>
    </draw:frame><draw:frame draw:style-name="fr1" draw:name="Object23" text:anchor-type="as-char" svg:width="1.2075in" svg:height="0.2307in" draw:z-index="23"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <msub>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">h</mi>
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           <mi mathvariant="italic">AmA</mi>
          </msub>
          <mo stretchy="false">=</mo>
          <msub>
           <mrow>
            <mo fence="true" form="prefix" stretchy="true">[</mo>
            <mrow>
             <mstyle mathvariant="bold">
              <mi mathvariant="normal">M</mi>
             </mstyle>
            </mrow>
            <mo fence="true" form="postfix" stretchy="true">]</mo>
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           <mi mathvariant="italic">Ap</mi>
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         </mrow>
         <msub>
          <mstyle mathvariant="bold">
           <mi mathvariant="normal">h</mi>
          </mstyle>
          <mi mathvariant="italic">pmp</mi>
         </msub>
        </mrow>
        <annotation encoding="StarMath 5.0">{bold nitalic h}_AmA
=
left[{bold nitalic M} right]_Ap
{bold nitalic h}_pmp
</annotation>
       </semantics>
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard"/>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr1" draw:name="Object22" text:anchor-type="as-char" svg:width="2.0256in" svg:height="0.2307in" draw:z-index="22"><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">AmA</mi>
          </msub>
          <mo stretchy="false">=</mo>
          <mrow>
           <msub>
            <mstyle mathvariant="bold">
             <mi mathvariant="normal">r</mi>
            </mstyle>
            <mi mathvariant="italic">ApA</mi>
           </msub>
           <mo stretchy="false">+</mo>
           <mi mathvariant="normal">δ</mi>
          </mrow>
         </mrow>
         <mrow>
          <msub>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">r</mi>
           </mstyle>
           <mi mathvariant="italic">ApA</mi>
          </msub>
          <mo stretchy="false">+</mo>
          <msub>
           <mrow>
            <mo fence="true" form="prefix" stretchy="true">[</mo>
            <mrow>
             <mstyle mathvariant="bold">
              <mi mathvariant="normal">A</mi>
             </mstyle>
            </mrow>
            <mo fence="true" form="postfix" stretchy="true">]</mo>
           </mrow>
           <mi mathvariant="italic">Ap</mi>
          </msub>
         </mrow>
         <msub>
          <mstyle mathvariant="bold">
           <mi mathvariant="normal">r</mi>
          </mstyle>
          <mi mathvariant="italic">pmp</mi>
         </msub>
        </mrow>
        <annotation encoding="StarMath 5.0">{bold nitalic r}_AmA
=
{bold nitalic r}_ApA
+
%delta{bold nitalic r}_ApA
+
left[{bold nitalic A} right]_Ap
{bold nitalic r}_pmp
</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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     </draw:image>
     <svg:desc>formula</svg:desc>
    </draw:frame><draw:frame draw:style-name="fr1" draw:name="Object24" text:anchor-type="as-char" svg:width="1.7299in" svg:height="0.2307in" draw:z-index="24"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <msub>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">h</mi>
           </mstyle>
           <mi mathvariant="italic">AmA</mi>
          </msub>
          <mo stretchy="false">=</mo>
          <msub>
           <mrow>
            <mo fence="true" form="prefix" stretchy="true">[</mo>
            <mrow>
             <mrow>
              <mi>d</mi>
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               <mi mathvariant="normal">M</mi>
              </mstyle>
             </mrow>
            </mrow>
            <mo fence="true" form="postfix" stretchy="true">]</mo>
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           <mi mathvariant="italic">Ap</mi>
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         </mrow>
         <msub>
          <mrow>
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           </mrow>
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          </mrow>
          <mi mathvariant="italic">Ap</mi>
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         <msub>
          <mstyle mathvariant="bold">
           <mi mathvariant="normal">h</mi>
          </mstyle>
          <mi mathvariant="italic">pmp</mi>
         </msub>
        </mrow>
        <annotation encoding="StarMath 5.0">{bold nitalic h}_AmA
=
left[{d bold nitalic M} right]_Ap
left[{bold nitalic M} right]_Ap
{bold nitalic h}_pmp
</annotation>
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      </math>
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   <text:p text:style-name="Standard"/>
   <text:p text:style-name="Standard"/>
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   <text:p text:style-name="Standard"/>
   <text:p text:style-name="Standard"/>
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</office:document>