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      <div class="title_topic4" id="xps10_pagetitle">Usage Notes</div>
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      <p class="para_topic">The displayed color-coded face analysis output provides a qualitative check on the shape of the face. If the face is smooth, then the display pattern will also look smooth, with no sharp breaks or oscillations.</p>
      <div class="title_division">Sectioning Plane</div>
      <p class="para_division">To understand curvature analysis functions, consider a particular point P on a face, and let N be the face normal vector at P. Any plane containing the point P and the vector N will intersect the face in some curve through P. This plane is called the &quot;sectioning plane.&quot; As the sectioning plane rotates about N, a family of intersection curves will be produced as shown below. These curves can be used to analyze the curvature of the face at the point P.</p>
      <div class="figure">
         <p align="center"><img align="bottom" src="graphics/ana_infoat36.gif"></p>
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      <div class="title_figure">Family of Intersection Curves</div>
      <div class="title_division">Minimum Maximum Radius Curvature</div>
      <p class="para_division">In general, each section curve in the family will have a different radius of curvature at the point P. The smallest of these radii is called the Minimum radius of curvature of the face at the point P, and the largest is called the Maximum radius of curvature of the face at the point P.</p>
      <p class="para_division">The minimum radius tells you how large a sphere can be placed in contact with the face at the point P without gouging. This can be useful in choosing NC cutting tools.</p>
      <p class="para_division">The Mean radius of curvature is the reciprocal of the mean curvature, which is the average of maximum and minimum curvature. If we let &quot;r&quot; and &quot;R&quot; denotes the Minimum and Maximum radii of curvature respectively, then the Mean radius of curvature is (2rR)  (R+r), and the Gaussian radius of curvature is  <img align="bottom" src="graphics/ana_infoeq22.gif" border="0">. (Remember, the sign of r and R is dependent on the direction of the face normal vector N.)
      </p>
      <p class="para_division">The radius of curvature values will be positive or negative depending on whether the corresponding planar section is concave or convex. In particular, this means that Gaussian radius will only be negative at &quot;saddle&quot; shaped points (where one section is concave and the other is convex).</p>
      <p class="para_division">The sign of the numbers that are output tells about the convexity or concavity of the face. For example, if the face is convex, the Minimum radius of curvature will have the same sign at all points on the face. The Gaussian radius will only be negative at &quot;saddle-shaped&quot; points where the Minimum and Maximum radii of curvature have opposite signs (i.e., rR&lt;0).</p>
      <p class="para_division">The face analysis results can be used qualitatively and quantitatively. For quantitative analysis, only the Minimum Radius results can be used to decide the maximum tool radius that can be used for machining purposes.</p>
      <div class="title_division">Qualitative Analysis</div>
      <p class="para_division">Part of the attractiveness of a product lie in the smooth blending of its highlights and shadows and this, in turn, is determined by the shape of its faces. Thus a major part of the computer-aided design of outer faces is construction of mathematically smooth, aesthetically pleasing faces from given data. A key element of this process is surface curvature analysis and its correct interpretation.</p>
      <p class="para_division">At a local protrusion or bump, both Minimum Radius and Maximum Radius of curvature will have negative values when the face normal is pointing out from the face; e.g., when it is on the `same side' as the bump. In this case, the Mean Radius will also be negative.</p>
      <p class="para_division">At a local hollow or indentation, both Minimum Radius and Maximum Radius of curvature will have positive values. In this case, the Mean Radius will also be positive. But if Minimum Radius and Maximum Radius differ in sign or are near &quot;infinity,&quot; there will neither be a local bump or a hollow.</p>
      <p class="para_division">Gaussian Radius, on the other hand, characterizes local curvature into three cases.</p>
      <ol type="1" start="1">
         <li>
            <p class="para_item">In the case of a bump or a hollow, Minimum Radius and Maximum Radius have the same sign, so Gaussian Radius is positive.</p>
         </li>
         <li>
            <p class="para_item">If Minimum Radius and Maximum Radius have opposite signs, then Gaussian Radius is negative - indicating a saddle point.</p>
         </li>
         <li>
            <p class="para_item">If one or both Minimum Radius and Maximum Radius is &quot;infinity,&quot; then Gaussian Radius is &quot;infinity&quot; - indicating a locally cylindrical ridge or a cylindrical hollow or a planar face.</p>
         </li>
      </ol>
      <p class="para_division">Thus, Gaussian Radius can be used to distinguish between elliptical, hyperbolic and cylindrical cases. Subsequently, Minimum Radius and Maximum Radius can be used to separate bumps and hollows, ridges and hollows, etc. Overall, if a face is `smooth', the face analysis results will look smooth - with no sharp breaks or oscillations.</p>
      <p class="para_division">Following are examples of elliptical, hyperbolic and cylindrical shapes:</p>
      <div class="figure">
         <p align="center"><img align="bottom" src="graphics/ana_infofi12.gif"></p>
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      <div class="title_figure">Examples of Elliptical, Hyperbolic and Cylindrical Shapes</div>
      <div class="title_division">Quantitative Analysis</div>
      <p class="para_division">Minimum Radius of curvature can be used for finding the approximate maximum tool radius that can be used in a machining operation.</p>
      <p class="para_division">For example, a ball-end cutter can safely be used to mill a face, with no danger of gouging, provided that the tool radius is smaller than the minimum radius of curvature at all points on the face. The determination of maximum tool radius should also take into account (besides local Face Analysis) some kind of obstacle avoidance checking to avoid problems such as restrictive part dimensions, the presence of the tool shank and other moving parts, etc.</p>
      <p class="para_division">To determine the maximum tool radius:</p>
      <ul>
         <li>
            <p class="para_item">Edit the face normal to point away from the material.</p>
         </li>
         <li>
            <p class="para_item">Next, plot the Minimum Radius of curvature.</p>
         </li>
         <li>
            <p class="para_item">Ignore all negative values. The minimum of all positive values gives the approximate maximum tool radius.</p>
         </li>
         <li>
            <p class="para_item">To get more accurate results, specify a smaller UV rectangle and a large number of display points in the region where the least positive minimum radius was observed.</p>
         </li>
      </ul>
      <p class="para_division">A maximum tool radius of &quot; Infinity&quot; signifies that a cylindrical ridge or plane is being machined, and any size tool can be used. If the Minimum Radius is negative everywhere on the face, a convex face is being machined - and then again a tool of any size can be used. </p>
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