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Posted: Sat Jul 05, 2008 2:32 pm
Something that has nagged at me since I first viewed V3.4 is the use of the term “mitre angle” as seen when viewing “Nets\Show Edge Data\Mitre Angles”. I am not a mathematician but I am a carpenter. What you give as a mitre angle, to me is actually a bevel angle. In carpentry the mitre is the angle needed to create the face angle and the bevel is the angle cut on the edge of a polygon so they can be joined. Using the octahedron as an example, the triangles have a face angle of 60 degrees so I need to set the miter to 30 degrees on my saw. The bevel angle is 35.2644 degrees which is derived from the dihedral angle and cut on the edges of the triangles.
Re: Mitre Angle
Posted: Sun Jul 06, 2008 6:10 am
You may well be right. I added that feature at the request of another carpenter (but can't remember who now). I'm no carpenter, so I relied on his terminology. I can change it if you think "bevel angle" is more appropriate.
Any other carpenters here wish to comment?
Posted: Sun Jul 06, 2008 7:19 am
see, Wiktionary at miter
This gives mitre as a cut, usually 45 degrees, designed to fit tightly into another cut. One might use a "mitre box" (typically a pair of parallel boards, with precut grooves for 45, 90 degrees, to achieve this end.
see Wiktionary at bevel
A bevel is a non-90 degree finish to the surface of a thing. For example, one might apply a bevel to the edges of a box to remove the sharp edge.
So, in practice, a mitre is where there is intended a second board would be placed, while a bevel is something done to the finish. In polytope terms, the polytope represents a single peice of wood, and therefore, mitre and bevel mean much the same thing.
Posted: Sun Jul 06, 2008 7:33 am
I read "mitre" and "bevel" somewhat differently.
Roughly, "mitre" is set to half the required angle at the corner, so to get an octagon, (angle 135°), one might mitre the boards at 67° 30'.
"Bevel" is a finishing to the finished product. So a 45-degree bevel applied to a square board might turn it to an octagon cross-section. One does this to cut off sharp corners.
35.2644 seems to be half the supplement of the octahedron's margin-angle of 109.47122 deg. One might suggest 35.2644 is half of 70.528779, which is the exterior of the figure at question.
I prefer 'margin-angle', since margin is used for all dimensions as the boundary between 'faces' according to the PG.
Posted: Mon Jul 07, 2008 7:35 pm
Here's how I have always understood it:
When making a picture frame, you mitre the corner joints at 45 degrees, and may also wish to bevel the edges to make a more pleasant profile than a plain rectangle.
So when making a hollow cube from plywood, you would mitre the edges at 45 deg. and join them along the mitred cuts.
But when making a morph between a cube and a rhombic dodecahedron from a solid cube, you would bevel the edges of the cube at 45 deg. to create the 12 new faces.