A little before noon each day, when the sun has warmed the muddy watering hole in the pasture, pairs of dragonflies lay eggs. They have had acrobatic sex before they arrive, and now they need to leave their eggs in the warm mud. The one in front, the male, is bright red. He holds the duller female by the head in his anal appendages...
But you must know more.
He, the flashy red one, makes sperm down in the usual spot near the tip of his tail, but his penis and a sperm reservoir are located farther up the body, closer to his head than to the tip of his tail. When he's ready to mate, he swings his tail up to transfer a spermatophore (a big package of sperm and fixings) to the sperm reservoir.
When he meets a willing female, he uses his anal appendages (pincers at the tip of his tail) to grasp the female in flight behind her head. As they fly on, she curls her tail down and around and up to meet his penis. He scoops out any previous lover's sperm with a ticklish looking penile appendage, and then he releases his own sperm. They form a wheel, eggs meet sperm, and her belly fills with the first sparks of life.
After copulation she uncurls, stretches out, probably needs a smoke, but he keeps a grip on her head � he doesn't want her meeting any other males until she has deposited her eggs � and they fly off, eight wings beating together, in search of a good place to lay eggs. This is when I meet them in the mornings, down by the mud hole.
They fly in low, up to two dozen pairs at a time, always along the edge closest to the sun and always facing into the wind. They hover within three or four inches of the surface and then the laying begins. His head remains relatively motionless in the air, the pivot on which the rest of his body and all of hers swings down once every couple of seconds or so. With each swing, her tails slaps the surface and she releases eggs.
To try this at home, the man might strip naked and use his feet to grip the woman, also naked, by her neck so that they form a long chain of nakedness. Vertically, he would be standing on her shoulders. She would then curl up to copulate with him (while he continues to hold her head with his feet) and then uncurl into the original position.
Now comes the tricky part. He grasps a tree branch growing above a natural body of water and hangs there with her suspended by her neck from his feet. The two would then swing as one (like tandem gymnasts on a bar) while her abdomen slaps repeatedly into the mud.
It's an unusual dance to us, but dragonflies have made it work for three hundred million years. They are older than the dinosaurs, which copulated much like you do and laid eggs like domestic chickens brooding, and which nonetheless are gone.
This fine site
has video of a pair laying their eggs. The species is different but the laying behavior is similar.