# Commits

committed e493b17

copyediting.

# Documents/Positioning/Tutorial 06.xml

`             meaningful and useful depends on what you are doing.</para>`
`         <para>Successive transformations can be seen as doing successive multiplication operations.`
`             For example, if S is a pure scale matrix, T is a pure translation matrix, and R is a`
`-            pure scale matrix, then the shader can compute the result of a transformation as`
`+            pure rotation matrix, then the shader can compute the result of a transformation as`
`             follows:</para>`
`         <programlisting language="glsl">vec4 temp;`
` temp = T * position;`
`                 <para>The order that successive transforms are applied in matters. Matrix`
`                     multiplication is not commutative, and neither is object transformation.</para>`
`             </listitem>`
`+            <listitem>`
`+                <para>Successive transformations can be used to build hierarchies of objects, each`
`+                    dependent on the accumulated transformations of lower ones. This is done using a`
`+                    matrix stack.</para>`
`+            </listitem>`
`         </itemizedlist>`
`         <section>`
`             <title>Further Study</title>`
`                         drawn.</para>`
`                 </listitem>`
`                 <listitem>`
`-                    <para>Given the generalized Hierarchy code, remove the matrix stack. Use objects`
`-                        created on the C++ stack instead. The node render function would take a`
`-                        const&amp; to a matrix rather than a matrix stack reference.</para>`
`+                    <para>Given the generalized Hierarchy code, remove the matrix stack. Use matrix`
`+                        objects created on the C++ stack instead. The node render function would`
`+                        take a const&amp; to a matrix rather than a matrix stack reference.</para>`
`                 </listitem>`
`             </itemizedlist>`
`         </section>`

# Documents/Positioning/Tutorial 07.xml

`             is enough to measure the size of atoms in inches at more than 60 miles away from the`
`             origin.</para>`
`         <para>However, you would be sacrificing a lot of performance to do this. Even though the`
`-            hardware <emphasis>can</emphasis> do double-precision math, it loses about 50% of its`
`-            speed in doing so. And why bother, when the real solution is much easier?</para>`
`+            hardware <emphasis>can</emphasis> do double-precision math, it loses quite a bit of`
`+            performance in doing so (anywhere between 25% and 75% or more, depending on the GPU).`
`+            And why bother, when the real solution is much easier?</para>`
`         <para>Let's look at our shader again.</para>`
`         <programlisting language="glsl">#version 330`
` `