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  <div class="section" id="module-spatialmath.geom3d.Plucker">
<span id="plucker-line"></span><h1>Plucker line<a class="headerlink" href="#module-spatialmath.geom3d.Plucker" title="Permalink to this headline">¶</a></h1>
<p>Plucker coordinate class</p>
<p>Concrete class to represent a 3D line using Plucker coordinates.</p>
<p>Methods:</p>
<p>Plucker            Contructor from points
Plucker.planes     Constructor from planes
Plucker.pointdir   Constructor from point and direction</p>
<p>Information and test methods::
closest            closest point on line
commonperp         common perpendicular for two lines
contains           test if point is on line
distance           minimum distance between two lines
intersects         intersection point for two lines
intersect_plane    intersection points with a plane
intersect_volume   intersection points with a volume
pp                 principal point
ppd                principal point distance from origin
point              generate point on line</p>
<p>Conversion methods::
char               convert to human readable string
double             convert to 6-vector
skew               convert to 4x4 skew symmetric matrix</p>
<p>Display and print methods::
display            display in human readable form
plot               plot line</p>
<p>Operators:
*                  multiply Plucker matrix by a general matrix
|                  test if lines are parallel
^                  test if lines intersect
==                 test if two lines are equivalent
~=                 test if lines are not equivalent</p>
<p>Notes:</p>
<blockquote>
<div><ul class="simple">
<li><p>This is reference (handle) class object</p></li>
<li><p>Plucker objects can be used in vectors and arrays</p></li>
</div></blockquote>
<p>References:</p>
<blockquote>
<div><ul class="simple">
<li><p>Ken Shoemake, “Ray Tracing News”, Volume 11, Number 1
<a class="reference external" href="http://www.realtimerendering.com/resources/RTNews/html/rtnv11n1.html#art3">http://www.realtimerendering.com/resources/RTNews/html/rtnv11n1.html#art3</a></p></li>
<li><p>Matt Mason lecture notes <a class="reference external" href="http://www.cs.cmu.edu/afs/cs/academic/class/16741-s07/www/lectures/lecture9.pdf">http://www.cs.cmu.edu/afs/cs/academic/class/16741-s07/www/lectures/lecture9.pdf</a></p></li>
<li><p>Robotics, Vision &amp; Control: Second Edition, P. Corke, Springer 2016; p596-7.</p></li>
</div></blockquote>
<p>Implementation notes:</p>
<blockquote>
<div><ul class="simple">
<li><p>The internal representation is a 6-vector [v, w] where v (moment), w (direction).</p></li>
<li><p>There is a huge variety of notation used across the literature, as well as the ordering
of the direction and moment components in the 6-vector.</p></li>
</div></blockquote>
<p>Copyright (C) 1993-2019 Peter I. Corke</p>
<dl class="py attribute">
<dt class="sig sig-object py" id="spatialmath.geom3d.Plucker.A">
<span class="sig-name descname"><span class="pre">A</span></span><a class="headerlink" href="#spatialmath.geom3d.Plucker.A" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="py attribute">
<dt class="sig sig-object py" id="spatialmath.geom3d.Plucker.pp">
<span class="sig-name descname"><span class="pre">pp</span></span><a class="headerlink" href="#spatialmath.geom3d.Plucker.pp" title="Permalink to this definition">¶</a></dt>
<dd><p>Principal point of the line</p>
<p><code class="docutils literal notranslate"><span class="pre">line.pp</span></code> is the point on the line that is closest to the origin.</p>
<p>Notes:</p>
<blockquote>
<div><ul class="simple">
<li><p>Same as Plucker.point(0)</p></li>
</div></blockquote>
<dl class="field-list simple">
<dt class="field-odd">Seealso</dt>
<dd class="field-odd"><p>Plucker.ppd, Plucker.point</p>
<dl class="py attribute">
<dt class="sig sig-object py" id="spatialmath.geom3d.Plucker.ppd">
<span class="sig-name descname"><span class="pre">ppd</span></span><a class="headerlink" href="#spatialmath.geom3d.Plucker.ppd" title="Permalink to this definition">¶</a></dt>
<dd><p>Distance from principal point to the origin</p>
<dl class="field-list simple">
<dt class="field-odd">Returns</dt>
<dd class="field-odd"><p>Distance from principal point to the origin</p>
<dt class="field-even">Return type</dt>
<dd class="field-even"><p>float</p>
<p><code class="docutils literal notranslate"><span class="pre">line.ppd</span></code> is the distance from the principal point to the origin.
This is the smallest distance of any point on the line
to the origin.</p>
<dl class="field-list simple">
<dt class="field-odd">Seealso</dt>
<dd class="field-odd"><p>Plucker.pp</p>
<dl class="py attribute">
<dt class="sig sig-object py" id="spatialmath.geom3d.Plucker.shape">
<span class="sig-name descname"><span class="pre">shape</span></span><a class="headerlink" href="#spatialmath.geom3d.Plucker.shape" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="py attribute">
<dt class="sig sig-object py" id="spatialmath.geom3d.Plucker.skew">
<span class="sig-name descname"><span class="pre">skew</span></span><a class="headerlink" href="#spatialmath.geom3d.Plucker.skew" title="Permalink to this definition">¶</a></dt>
<dd><p>Line as a Plucker skew-matrix</p>
<dl class="field-list simple">
<dt class="field-odd">Returns</dt>
<dd class="field-odd"><p>Skew-symmetric matrix form of Plucker coordinates</p>
<dt class="field-even">Return type</dt>
<dd class="field-even"><p>numpy.ndarray, shape=(4,4)</p>
<p><code class="docutils literal notranslate"><span class="pre">M</span> <span class="pre">=</span> <span class="pre">line.skew()</span></code> is the Plucker matrix, a 4x4 skew-symmetric matrix
representation of the line.</p>
<p>Notes:</p>
<blockquote>
<div><ul class="simple">
<li><p>For two homogeneous points P and Q on the line, <span class="math notranslate nohighlight">\(PQ^T-QP^T\)</span> is also skew
symmetric.</p></li>
<li><p>The projection of Plucker line by a perspective camera is a homogeneous line (3x1)
given by <span class="math notranslate nohighlight">\(\vee C M C^T\)</span> where <span class="math notranslate nohighlight">\(C \in \mathbf{R}^{3 \times 4}\)</span> is the camera matrix.</p></li>
</div></blockquote>
<dl class="py attribute">
<dt class="sig sig-object py" id="spatialmath.geom3d.Plucker.uw">
<span class="sig-name descname"><span class="pre">uw</span></span><a class="headerlink" href="#spatialmath.geom3d.Plucker.uw" title="Permalink to this definition">¶</a></dt>
<dd><p>Line direction as a unit vector</p>
<dl class="field-list simple">
<dt class="field-odd">Returns</dt>
<dd class="field-odd"><p>Line direction</p>
<dt class="field-even">Return type</dt>
<dd class="field-even"><p>numpy.ndarray, shape=(3,)</p>
<p><code class="docutils literal notranslate"><span class="pre">line.uw</span></code> is a unit-vector parallel to the line.</p>
<dl class="py attribute">
<dt class="sig sig-object py" id="spatialmath.geom3d.Plucker.v">
<span class="sig-name descname"><span class="pre">v</span></span><a class="headerlink" href="#spatialmath.geom3d.Plucker.v" title="Permalink to this definition">¶</a></dt>
<dd><p>Moment vector</p>
<dl class="field-list simple">
<dt class="field-odd">Returns</dt>
<dd class="field-odd"><p>the moment vector</p>
<dt class="field-even">Return type</dt>
<dd class="field-even"><p>numpy.ndarray, shape=(3,)</p>
<dl class="py attribute">
<dt class="sig sig-object py" id="spatialmath.geom3d.Plucker.vec">
<span class="sig-name descname"><span class="pre">vec</span></span><a class="headerlink" href="#spatialmath.geom3d.Plucker.vec" title="Permalink to this definition">¶</a></dt>
<dd><p>Line as a Plucker coordinate vector</p>
<dl class="field-list simple">
<dt class="field-odd">Returns</dt>
<dd class="field-odd"><p>Coordinate vector</p>
<dt class="field-even">Return type</dt>
<dd class="field-even"><p>numpy.ndarray, shape=(6,)</p>
<p><code class="docutils literal notranslate"><span class="pre">line.vec</span></code> is the  Plucker coordinate vector <code class="docutils literal notranslate"><span class="pre">X</span></code> = [V,W] where V (3-vector)
is the moment and W (3-vector) is the line direction.</p>
<dl class="py attribute">
<dt class="sig sig-object py" id="spatialmath.geom3d.Plucker.w">
<span class="sig-name descname"><span class="pre">w</span></span><a class="headerlink" href="#spatialmath.geom3d.Plucker.w" title="Permalink to this definition">¶</a></dt>
<dd><p>Direction vector</p>
<dl class="field-list simple">
<dt class="field-odd">Returns</dt>
<dd class="field-odd"><p>the direction vector</p>
<dt class="field-even">Return type</dt>
<dd class="field-even"><p>numpy.ndarray, shape=(3,)</p>
<dt class="field-odd">Seealso</dt>
<dd class="field-odd"><p>Plucker.uw</p>
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