blue duck's explanation is good (though I could wish for a paragraph break to make it easier to read). she talks about really watching the eyes and ears to see if the horse is thinking flee or follow. great advice.
I have to catch my lease horse on 30 acres. he is usually at or near a feeding station, with the rest of the horses. he is dominant, so he will be at the feeder if he wants to be, while theothers wait their turn.
I approach and whistle until he turns to look at me with curiousity. I stop, call and see if he will come. almost never will her at the first encounter.
so, I walk a few steps closer. he will usually turn away and back to the food, becuase of course , that is more interesting. when I see that he has "turned his back on me", so to speak, I pick up a stick or a rock and throw it toward him but not AT him, just in his direction but off to the side. the sound it makes, and the motion, is just enough for him to raise his head and look. at that time, I make a small motion (I am standing) with my hand so that he looks at ME.
See, the rock was to interrupt his outward going attention and once it was up for grabs, I do something to grab it. just the little thing possible. when he's looking at me, I turn my core away from him a bit, strike that relaxed inviting position and call him sweetly. he will look at me, then, maybe look back at the hay. If his movement away from me is minor, I give him a sec, then try to interrupt his outward though with first a shuffle /movement of my body, and if that doesn't work, walk forward a step or two and throw the rock/stick and repeat the above.
I don't do it instantly because I may need to give him time to think about his decision. and this is evident when the hrose turns to look at you, then away a bit , then back to you. he's "searching". when a horse is "searching", do NOT interrupt that. let him find his own choice. HOWEVER, if he gets stuck in the "searching" place, you can help him by adding just a bit if pressure when he turns away, so that turning away feel less comfy than turning toward you. then he'll choose. and he'll start toward you. wait where you are and let him come to you, and if you have a treat, offer it when he stops at arms distance (you don't let him walk up ONto you) and halter up, and go off to your good day of riding or whatever.