little_helpers/tikz/sketch-0.2.161/Doc/sketch/A-hierarchical-model.html

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  56 <h3 class="section">4.3 A hierarchical model</h3>
  57
  58 <p><a name="index-hierarchical-model-472"></a>While <code>sketch</code> was never meant to be a geometric modeling
  59 language, it comes fairly close.  The following example puts all we
  60 have seen to work in a very simple model of the human hand.  Start by
  61 sweeping a line to make a truncated cone, which will be copied over
  62 and over again to make the segments of fingers.
  63 <a name="index-def-473"></a><a name="index-sweep-474"></a><a name="index-rotate-475"></a><a name="index-line-476"></a>
  64 <pre class="verbatim">
  65   def O (0,0,0) % origin
  66   def I [1,0,0] def J [0,1,0] def K [0,0,1] % canonical unit vectors
  67   def segment {
  68     def n_faces 8
  69     sweep { n_faces&lt;>, rotate(360 / n_faces, [J]) }
  70       line(proximal_rad, 0)(distal_rad, distal_len)
  71   }
  72 </pre>
  73 In hand anatomy, <em>distal</em> is &ldquo;at the tip&rdquo; and <em>proximal</em>
  74 is &ldquo;in the area of the palm.&rdquo;  We have omitted all the scalar
  75 constants.  You can find them in <span class="file">hand.sk</span>, which is provided
  76 in the <code>sketch</code> distribution.
  77
  78    <p>We also need a prototypical sphere to use for the joints themselves.
  79 <a name="index-def-477"></a><a name="index-sweep-478"></a><a name="index-rotate-479"></a>
  80 <pre class="verbatim">
  81   def joint_sphere {
  82     def n_joint_faces 8
  83     sweep [fillcolor=red] { n_joint_faces, rotate(360 / n_joint_faces, [J]) }
  84       sweep { n_joint_faces, rotate(180 / n_joint_faces) }
  85         (0, -joint_rad)
  86   }
  87 </pre>
  88
  89    <p>We'll now design the index finger (number&nbsp;<!-- /@w -->1 in our notational
  90 convention; finger&nbsp;<!-- /@w -->0 is the thumb).  The distal rotation for the
  91 finger applies only to the tip, so we define the following.
  92 <a name="index-def-480"></a><a name="index-put-481"></a><a name="index-translate-482"></a><a name="index-rotate-483"></a><a name="index-scale-484"></a>
  93 <pre class="verbatim">
  94   def distal_1 {
  95     put { translate(joint_gap * joint_rad * [J])
  96           then rotate(distal_1_rot, [I])
  97           then translate((distal_len + joint_gap * joint_rad) * [J]) }
  98       {segment}
  99     put { rotate(distal_1_rot / 2, [I])
 100           then translate((distal_len + joint_gap * joint_rad) * [J]) }
 101       {joint_sphere}
 102     put { scale( [J] + proximal_distal_ratio * ([I]+[K]) ) }
 103       {segment}
 104   }
 105 </pre>
 106 The identifiers here are for size and location constants. The
 107 exception is <code>distal_rot_1</code>.  This rotation parameter models the
 108 flexing of the finger tip.  The first <code>put</code> makes a copy of the
 109 finger segment that is translated upward
 110 <a name="index-translation-transform-485"></a><a name="index-transform_002c-translation-486"></a>just far enough to make room
 111 for the spherical joint.  Then it applies the distal rotation.
 112 <a name="index-rotation-transform-487"></a><a name="index-transform_002c-rotation-488"></a>Finally it translates the whole assembly upward again to make room for
 113 the middle phlanges (the next bone toward the palm).  The second
 114 <code>put</code> positions the sphere.  There is a rotation to place the
 115 grid on the sphere surface at an nice angle, then a translation to the
 116 base of the distal phlanges, which is also center of its rotation.
 117 Finally, the last <code>put</code> positions the middle segment itself.
 118
 119    <p>The middle joint is the next one down, with rotation angle
 120 <code>middle_rot_1</code>.  When this angle changes, we need all the objects
 121 in <code>distal_1</code> to rotate as a unit.
 122 <a name="index-rotation-transform-489"></a><a name="index-transform_002c-rotation-490"></a>This is the reasoning behind
 123 the next definition.
 124 <pre class="verbatim">
 125   def finger_1 {
 126     put { translate(joint_gap * joint_rad * [J])
 127           then rotate(middle_1_rot, [I])
 128           then translate((middle_ratio * distal_len +
 129                           joint_gap * joint_rad) * [J]) }
 130       {distal_1}
 131     put { scale(proximal_distal_ratio)
 132           then rotate(middle_1_rot / 2, [I])
 133           then translate((middle_ratio * distal_len +
 134                           joint_gap * joint_rad) * [J]) }
 135       {joint_sphere}
 136     put { scale( middle_ratio * [J] +
 137                  proximal_distal_ratio^2 * ([I]+[K]) ) }
 138       {segment}
 139   }
 140 </pre>
 141 This looks very similar to the previous definition, and it is.  The
 142 important difference is that rather than positioning and rotating a
 143 single segment, we position and rotate the entire &ldquo;assembly&rdquo; defined
 144 as <code>distal_1</code>.
 145 <a name="index-rotation-transform-491"></a><a name="index-transform_002c-rotation-492"></a>The rest is just arithmetic to compute sizes and
 146 positions that look nice.  The last <code>put</code> places an appropriately
 147 shaped segment that is the <em>proximal phlanges</em>, the bone that
 148 joins the palm of the hand.  This completes the finger itself.
 149
 150    <p>All the other fingers are described identically to this one.  We
 151 account for the fact that real fingers are different sizes in the next
 152 step, which is to build the entire hand.
 153
 154    <p>The <code>hand</code> definition that follows includes a section for each
 155 finger.  We'll continue with finger&nbsp;<!-- /@w -->1 and omit all the others.
 156 (Of note is that the thumb needs slightly special treatment&mdash;an extra
 157 rotation to account for its opposing angle. This is clear in the full
 158 source code.) Not surprisingly, the <code>hand</code> definition looks very
 159 much like the previous two.  It should be no surprise that when the
 160 rotation parameter <code>meta_1_rot</code> changes, the entire finger
 161 rotates!
 162 <a name="index-rotation-transform-493"></a><a name="index-transform_002c-rotation-494"></a>There is an additional rotation that allows the fingers to spread
 163 laterally.  We say these joints of the proximal phlanges have two
 164 <em>degrees of freedom</em>. The joints higher on the finger have only
 165 one. Finally, each finger is scaled by a factor to lend it proportion.
 166 <pre class="verbatim">
 167   def hand {
 168     % finger 1 [all other fingers omitted]
 169     def scale_1 .85
 170     put { scale(scale_1)
 171           then translate((joint_gap * joint_rad) * [J])
 172           then rotate(meta_1_rot, [I])
 173           then rotate(-spread_rot, [K])
 174           then translate((proximal_1_loc) - (O)) }
 175       {finger_1}
 176     put { scale(scale_1 * proximal_distal_ratio^2)
 177           then rotate(meta_1_rot / 2, [I])
 178           then rotate(-spread_rot, [K])
 179           then translate((proximal_1_loc) - (O)) }
 180       {joint_sphere}
 181
 182     % palm
 183     sweep { 1, rotate(6, (0,15,0), [I]) }
 184       put { rotate(-3, (0,15,0), [I]) } {
 185         polygon(proximal_1_loc)(proximal_2_loc)
 186                (proximal_3_loc)(proximal_4_loc)
 187                (h5)(h6)(h6a)(h9)(h10)
 188         polygon(h6a)(h7)(h8)(h9)
 189    }  }
 190 </pre>
 191 The last section of the definition creates the polytope for the palm
 192 of the hand by <code>sweep</code>ing
 193 <a name="index-swept-polygon-495"></a><a name="index-polygon-sweep-496"></a>a 10-sided polygon through a very short
 194 arc (9&nbsp;<!-- /@w -->degrees).  This provides a wedge-shaped profile when viewed
 195 from the side. The thick end of the wedge is the wrist.  Because the
 196 polygon is concave, it is split into into two convex shapes with nine
 197 and four vertices.
 198
 199    <p>We can now have fun positioning the hand by adjusting the various
 200 rotation angles.  The complete source includes definitions with
 201 alternatives that include the following views and more.
 202
 203 <div align="center"><img src="ex210.png" alt="ex210.png"><img src="ex220.png" alt="ex220.png"><img src="ex230.png" alt="ex230.png"><img src="ex240.png" alt="ex240.png"></div>
 204
 205    </body></html>
 206