Saturday, November 23, 2013

Rotation of Planar Regions: Polyhedron in AutoCAD

In a previous post about the angle between intersecting planes, I noted the importance of measuring that angle perpendicular to the line of intersection.  Now I want to apply a similar notion to the rotation of planar regions.  In CAD programs that support solid and surface modelling, it is common to support planar regions. These programs do much more complicated things as well, of course, but they will certainly allow you to make, say, a hexagonal region.  A hexagonal region is not only the outline of the hexagon, but the area enclosed by that outline, all of which lies within the same plane as the outline.  Making a planar region in AutoCAD is straightforward:
  1. Change to one of the 3D workspaces.
  2. Create any closed sequence of line work (sometimes called a closed path), all of which should be coplanar.  (While you can create a closed path that is not coplanar, it won't suit our purpose here and you can't make it into a region.)
  3. Type Region, select the line work and press .
  4. To see the region, change the visual style as necessary:
    1. Click the View tab.
    2. Click the Visual Styles button on the Palettes pane.
    3. In the Visual Styles Manager (which should pop up) choose a style that shows the faces (if/when that is desirable).  I'm partial to Shaded with edges, but other styles may be suitable.
      1. Notice also the settings under Face Settings that can affect the way the faces appear (or not appear).
But I don't just want to make planar regions, we want to do something with them.  I want to rotate them about a line of intersection/adjacency and make multiple regions "fit" together.  I'm going to make, well, you'll see...(at the end of the video in this post)...

Description of the Problem

Suppose we are given three planar regions A, B, and C, such that A is adjacent to B and B is adjacent to C, with all of the regions being initially co-planar with each other.  We want to rotate A and C about their "line of adjacency" with B so that A and C are adjacent to each other.

Informally, we might say that A and C are the box flaps and B is the base.  You may intuit the analogy I'm making.  We are pretending we have a sheet of cardboard cut out in an unusual shape with two creases in it which will serve as the fold lines – our lines of adjacency.  Note also that the proximal lines of A and C (the ones we want to be coincident) need not be of equal length to do the kind of folding/rotating we want to do, but if they are not equal we may not get the result we desire.

We are currently looking at our box flaps in plan view.  Suppose we continue observing in plan view and fold the box flaps (about B).  B is unchanged in appearance but A and C appear "squished" in one direction.  Or, another way to think of it, every point on A will appear to move perpendicularly toward its line of adjacency with B.  Likewise for C.  Thus,

If for some reason the proximal lines of A and C were not of equal length (as they are above), for purposes of our construction, we would need to establish points of equal distance from their common vertex.  A circle centered on their common vertex would do nicely.  In AutoCAD we can work in 3D to do this rotation, but first let's consider how we would do this in 2D.  This will be informative as to how to proceed in 3D.

First observe how the arrows at the end of the proximal lines of A and C cross each other.  These arrows are in plan view with z = 0, but the point where these arrows intersect gives us the xy-coordinates of the point where the non-common vertices of the proximal lines will meet.  (If that doesn't make sense, just go with the picture.)  Now to make our 2D plan view work, we've got to do some single axis scaling.  Here's what we're looking at:

We need to scale region A along the y-axis by a factor of A2/A1 and scale C along an axis perpendicular to the line of adjacency with B by a factor of C2/C1.  Since we are working with only a small number of points here, we can simply draw lines of lengths A3 and C3 and scale them as needed.  We move and copy the lines as necessary so that we can move the vertices of the polygons to their new positions based on the newly scaled lines.  (To scale A in the y direction you could turn A into a block and scale the y-axis as desired.  To scale C you could use the "block"  and scale method too, but you'd have to adjust your UCS line up with the line adjacent to B – use the ucs command with the OBject option and click the line you want to line up with.)  Here's the scaled result:

I used the scale command with the reference option so that I didn't have to do the indicated math myself, but AutoCAD did in fact compute A3(A2/A1) and likewise for C.  (Incidentally, this doesn't work so nice with regions, though it works fine with polylines; it's a simple matter of moving vertices along a line perpendicular to the line of adjacency.)  As for making the other views work for you in 2D – you're on your own, because right now I'm posting how to do this in AutoCAD (full version).

Rotating Planar Regions to Make a Polyhedron in AutoCAD

Making of a Polyhedron in AutoCAD from Darren Irvine on Vimeo.

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