geometry-doc.sgml

机器人开源项目orocos的源代码

</para>
</listitem>

<listitem>
<para>
  <parameter>vectorVectorCoordinates</parameter>: &hellip;
</para>
</listitem>

</itemizedlist>
</para>

</section>


<section id="plane">
<title>Plane</title>
<para>
A Plane has the following
<link linkend="symbolic-class">symbolic properties</link>:
<variablelist>
 <varlistentry>
 <term>
  <anchor id="plane-coordinate-type">
  <parameter>coordinateType</parameter>:
 </term>
  <listitem>
  <para>
  the type of the plane coordinates:
  <parameter>vectorVectorPlane</parameter>, 
  <parameter>abcdParameterPlane</parameter>, 
  <parameter>lineLinePlane</parameter>, 
  <parameter>pointNormalPlane</parameter>, etc.
  </para>
  </listitem>
 </varlistentry>
</variablelist>

The corresponding numeric coordinate representations are:
<itemizedlist>

<listitem>
<para>
  <parameter>vectorVectorPlane</parameter>: &hellip;
</para>
</listitem>

<listitem>
<para>
  <parameter>abcdParameterPlane</parameter>: &hellip;
</para>
</listitem>

<listitem>
<para>
  <parameter>lineLinePlane</parameter>: &hellip;
</para>
</listitem>

<listitem>
<para>
  <parameter>pointNormalPlane</parameter>: &hellip;
</para>
</listitem>

</itemizedlist>
</para>

</section>


<section id="polyhedron">
<title>Polyhedron</title>
<para>
This is a planar geometric primitive, consisting of an ordered list of
<emphasis>vertices</emphasis>. Since the vertices all lie in the same
plane, an obvious representation consists of a plane, with 2D coordinates
of the vertices in that plane.
</para>

</section>


<section id="mesh">
<title>Mesh</title>
<para>
This is a non-planar geometric primitive, consisting of a connected
set of <emphasis>vertices</emphasis>. Since the vertices form a
surface the interconnection graph is <emphasis>planar</emphasis>.
(TODO Is this true? Is this useful information?)
</para>

</section>


<section id="screw">
<title>Screw</title>
<para>
A screw is the geometric object that combines 
<emphasis role="strong">two parallel vectors</emphasis>,
(the first one is bound to a line, the second one is a free vector), with a 
<emphasis role="strong">scalar number</emphasis> (the
&ldquo;pitch&rdquo;) which is the ratio of the two vectors.
Screws are the geometric primitives behind rigid body velocity and
force.
<variablelist>
<title>Symbolic properties</title>
 <varlistentry>
 <term>
  <anchor id="screw-coordinate-type">
  <parameter>coordinateType</parameter>:
 </term>
  <listitem>
  <para>
  the type of the screw
  <link linkend="screw-coordinates">coordinates</link>:
  <parameter>PluckerAxisCoordinates</parameter>,
  <parameter>PluckerRayCoordinates</parameter>,
  <parameter>pointVectorCoordinates</parameter>, 
  <parameter>vectorVectorCoordinates</parameter>, etc.
  </para>
  </listitem>
 </varlistentry>
 <varlistentry>
 <term>
  <anchor id="pitch-units">
  <parameter>pitchPhysicalUnits</parameter>
  </term>
  <listitem>
   <para>
A pitch has the physical units of <emphasis>distance</emphasis>.
   </para>
  </listitem>
 </varlistentry>
</variablelist>

<variablelist>
<title>Physical properties</title>
 <varlistentry>
 <term><parameter>inversion</parameter></term>
  <listitem>
   <para>
   reversing the direction of both vectors.
   </para>
  </listitem>
 </varlistentry>
 <varlistentry>
 <term>
  <anchor id="screw-product">
  <parameter>product</parameter>
  </term>
  <listitem>
   <para>
  The (&ldquo;vector&rdquo;, &ldquo;screw&rdquo;) product of two
  screws is again a screw.
   </para>
  </listitem>
 </varlistentry>
</variablelist>

<variablelist>
<title>Numeric coordinate represenations:</title>
 <varlistentry>
 <term>
  <anchor id="pitch-coordinate">
  <parameter>pitch</parameter>
  </term>
  <listitem>
   <para>
numeric value of the pitch (which can have the value
&ldquo;infinity&rdquo;!).
   </para>
  </listitem>
 </varlistentry>
 <varlistentry>
 <term>
  <anchor id="screw-coordinates">
  <parameter>coordinates</parameter>: 
 </term>
  <listitem>
   <para>
Very similar to the six-dimensional coordinates of a
<link linkend="line">Line</link>, but in this case of a Screw,
<emphasis>five</emphasis> of the six numbers are independent
coordinates. (A <link linkend="line">Line</link> only has four
independent coordinates.)
   </para>
  </listitem>
 </varlistentry>
</variablelist>
</para>

</section>


<section id="orientation">
<title>Orientation</title>
<para>
An orientation is an (choice of) <emphasis>sign</emphasis>
attached to some geometric objects.
</para>
<para>
<variablelist>
<title>Symbolic properties</title>
 <varlistentry>
 <term>
  <anchor id="orientation-type">
  <parameter>type</parameter>:
  </term>
  <listitem>
  <para>
   In 1D, orientation is <parameter>left</parameter> or
   <parameter>right</parameter>; in 2D,
   orientation is <parameter>clockwise</parameter> or
   <parameter>counter-clockwise</parameter>; in 3D, orientation is
   <parameter>positive</parameter> or <parameter>negative</parameter>
	(or, <parameter>right-handed</parameter> or
   <parameter>left-handed</parameter>).
  </para>
  </listitem>
 </varlistentry>
</variablelist>

<variablelist>
<title>Physical properties</title>
 <varlistentry>
 <term><parameter>inversion</parameter></term>
  <listitem>
   <para>
	Reversing the <link linkend="orientation-type">type</link> of the
   orientation.
   </para>
  </listitem>
 </varlistentry>
 <varlistentry>
 <term>
  <anchor id="orientation-product">
  <parameter>product</parameter>
  </term>
  <listitem>
   <para>
The product of two orientations with opposite type is
&ldquo;negative&rdquo;, the product of orientations of the same type
is &ldquo;positive&rdquo;.
   </para>
  </listitem>
 </varlistentry>
</variablelist>
</para>

</section>


<section id="rotation">
<title>Rotation</title>
<para>
Consider the same rigid body in two different configurations, in which
(at least) one (&ldquo;fixed&rdquo;) point connected to the body
remains in the same position in space. Then, one can prove that there
exists a <link linkend="line">Line</link> through that point such that
the first body can be rotated about the line and moved to coincide
with the second body. In 1D, this Rotation is trivial: the rigid body
is a subset of the 1D line, and it is &ldquo;mirrored&rdquo; on the
line about the point, or it remains in the same place.
In 2D (<emphasis>i.e.</emphasis> a plane), the rotation line is
perpendicular to the plane, and goes through the fixed point of the
body. In 3D, the line can have any direction in space. Note also that,
in 2D and 3D, there are in general always two of these rotations
(&ldquo;clockwise&rdquo; and &ldquo;counter-clockwise&rdquo;, about
the same line), whose rotation angles add up (in absolute value) to
360 degrees.
</para>
<para>
Note that the fixed point must not necessarily be physically located
on the rigid body itself: it can be a point outside of the body, that
moves rigidly together with the body.
</para>
<para>
A Rotation can be given two meanings:
<itemizedlist>

<listitem>
<para>
<emphasis role="strong">Active</emphasis>:
the <emphasis role="strong">motion</emphasis> of a rigid body
from a first configuration to a second configuration. (With the
constraint that at least one point remains &ldquo;motionless&rdquo;!)
</para>
</listitem>

<listitem>
<para>
<emphasis role="strong">Passive</emphasis>:
the position of a <emphasis>reference frame</emphasis> on
the rigid body with respect to a
<emphasis>fixed world reference frame</emphasis> at the origin of the
world.
</para>
</listitem>

</itemizedlist>
Only the passive interpretation
<emphasis role="strong">requires</emphasis> the choice of one
reference frame on the object, and another reference frame in the
world; and the given orientation depends on this choice. 
So, passive Rotation is not a 
<link linkend="physical-properties">physical property</link> but a
property of the choice of coordinates.  The active interpretation,
however, <emphasis>is</emphasis> physically well defined,
independently of any arbitrary choices.
</para>
<para>
So, the following properties are required to define a Rotation:
</para>
<variablelist>
<title>Symbolic property</title>
 <varlistentry>
 <term><parameter>active</parameter>, <parameter>passive</parameter></term>
  <listitem>
   <para>
Only in the case of a <parameter>passive</parameter> Orientation, the
generic <link linkend="symbolic-class">symbolic information</link>
about the <link linkend="geometric-port">geometric Port</link> to
the reference <link linkend="frame">frame</link> is relevant.
   </para>
  </listitem>
 </varlistentry>
</variablelist>
<variablelist>
<title>Physical properties</title>
 <varlistentry>
 <term><parameter>inversion</parameter></term>
  <listitem>
   <para>
The inversion of a given Rotation is another Rotation. The rotation
line is the same, and the rotation angle is the inverse of the
original angle. Or, equivalently, the direction fo the rotation line
is reversed.
   </para>
  </listitem>
 </varlistentry>
 <varlistentry>
 <term>
  <anchor id="orientation-composition">
  <parameter>composition</parameter>
  </term>
  <listitem>
   <para>
The composition of two Rotations is another Rotation.
<note>
<para>
The composition of Rotation is a
<emphasis role="strong">multiplicative</emphasis> operation, and 
<emphasis role="strong">not additive</emphasis>,
<emphasis>i.e.</emphasis> no coordinate representation exists in which
any composition corresponds to the addition of the rotation
coordinates of the two Rotations involved in the composition; while
there do exist coordinate representations in which the composition
corresponds to the multiplication of the coordinates. Note that the
opposite is not true: not in every coordinate representation,
composition corresponds to multiplication!
</para>
</note>
   </para>
  </listitem>
 </varlistentry>
</variablelist>
<para>
Many different
<emphasis role="strong">coordinate respresentations</emphasis> exist
for Rotations. For each of them, the composition operation is called
<parameter>multiplication</parameter>, in accordance with the physical
properties of Rotations. All coordinate representations depend on the
<emphasis>choice</emphasis> of a reference frame on the object and in
the fixed world. Usually, right-handed and orthogonal reference frames
are used, but this is not necessary; of course, the numerical values
of the coordinates depend on this choice, so the frame objects should
contain this information among their symbolic properties.
<variablelist>
 <varlistentry>
 <term>
   <anchor id="rotation-matrix">
   <parameter>rotationMatrix</parameter></term>
  <listitem>