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Oh good gosh I hated this process. Not because it this was a hard drawing to sketch, but because I did not know some very simple commands/processes like I do for AutoCAD. For the following drawing I came up with a weaving pattern for the 1 mm dia. wire.

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Its actually just one wire pattern that winds up getting mirrored and rotated.

I started out by drawing up sketch1 being the profile of the wire's centerline.

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I then followed this by creating a perpendicular work plane placed at the start of the wire. On this plane I projected the start point of the wire and drew a 1 mm dia. circle. I then proceeded to sweep the circle along the centerline profile. All pretty straight forward.

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Then the head banging on the desk started as I tried to figure out how to copy the solid. I realized I could array it but I wanted copy in case I wanted to do this in the future and the copy position was not lined up in an arrayable pattern. Busted out Google and did some searching. The best I could come up with was "copy paste" of the item from the command history on the left, selecting a new work plane, and then pasting the sketch on the new plane. This brought over a all the previous constraints and dimensions. This seemed like potentially a lot of information to be potentially repeating depending on the nature of the sketch being copied.

Then when I went to mirror the profile vertically in the sketch there was not option to delete the source information like there is in AutoCad. I wound up having to delete portions of the source profile then re dimension everything.

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When It came time to create the transverse wire, things got ugly for me. First time I attempted it, I created a sketch in the transverse plan to the original wire where the two wires would cross. Easy enough! I then copied my sketch1 and proceeded to paste it into the transverse plane...&^#@(@!!!! The thing went from being horizontal to vertical. I tried rotating it to be horizontal and despite relaxing and ignoring constraints, the prg tossed an error and would not rotate the sketch.

The second time I tired it with a redefined profile with new constraints, it worked with relaxed constraints but then all the dimension went wonky, though technically worked. What was really annoying was all the vertical dimensions became horizontal and vice versa. Even when I redimensioned the sketch and selected horizontal as the type of dimension I wanted, the dimension displayed vertical. Very annoying thing I have noticed when switching to the right view. I then had to repeat the mirroring step on a different plane.

What I really wanted to do prior to this stage was simply select all the solid wires copy and rotate them about a central point and I would have been done.

One step I did realize 1/4 of the way through was that instead of copying the profile sketch to another plane and sweeping each individual wire, I could instead sweep all wires with matching profiles at the same time. Very helpful when things are lined up and you have more than 1 copy of the wire to make.

Is there an easier way to copy, mirror, rotate a solid part that I am missing?

My model below just feels like a mess to me. It works, but I would have expected something cleaner or more simple. Admittedly I don't think I would actually model a wire mesh, I would just call out a surface or simplified grid as being a wire mesh. This was a learning exercise telling me that some of the AutoCAD feature I would have used are not the same in Inventor and I don't know the best work arounds yet.

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Forward Ed
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    if all you are after is an auto-cad like non-parameterized approach, you can literally manipulate bodies without datums usual method derives parts to do this to form a quasi-assembly layer (derived with just bodies can be faster than an assembly). For this specific case however, you should be able to get away with cleverly placing the origin in your first sketch, and use just the origin planes, axes followed by pattern, mirror features – Abel Nov 11 '21 at 16:38

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