Without photoshop, extensive sets or anything not commonly found in a normal household...
What's the best way to make a figure look like they're flying through the air to do a flying kick?

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RIP Minouche, 1994-2009.

My kid always suggests using fishing wire. At the same time, when I was shooting on the beach down at, uh, Long Beach, Wa, I had kids tell me the same thing. That I could probably do some really cool pictures with figures suspended with fishing wire. In theory, you'd just spray paint the wires out.

It might be on account that the Saint-Lawrence is so damn polluted, but very few people in Montreal have fishing wires as common household items.

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RIP Minouche, 1994-2009.

One trick I've used is to stick a small piece of blu-tac in a figure's screw hole, and then insert a short length of coat hanger (or similar) wire into it. It will support the figure, and if you get the angle right, you can hide it behind the figure so that it doesn't show in the photo.

Green floral wire, 22 gauge. It is flexible enough to bend in anyway you need and at the same time is thin enough to hide or disguise in the shot. If you have Photoshop, the green can be removed and background colors blended in.