From our POV, the stars of the Little Dipper ( or Ursa Minor) seem on the same plane. Have astronomers created a model of how the stars are in relation to each other in reality? In other words, I’m looking at them straight ahead, but how they look when I look at them from side?
You wouldn’t recognize them from another angle. They’re not in any way isolated except from Earth. Unless you had a very reliable 3D star chart, they’d just be lost among billions of other stars.
What panache45 said. And the same is pretty much true of all the constellations. The distances between the stars of a constellation can easily be greater than the distance of any of those stars from our solar system.
I wonder why OP asks about Little Dipper specifically, rather than any-old-constellation.
There is such a model for the Big Dipper. I saw it at a museum, some years ago. (Sorry, don’t remember where.) It had a box with the relevant stars inside, illuminated as pinpoints of light, in their proper positions.
The front of the box had a peep-hole. Look in, and you see the Big Dipper!
The side of the box had another peep-hole. Look in there, and you see an unrecognizable hodge-podge of stars.
Lift the cover, and you see the whole inside of the box, so you can see the arrangement in living 3-D!
That depends on what you mean by model. I’ve seen 3D models of some asterisms showing how they would change if you had a magical ship that let you change the viewing angle significantly, but that’s something people have made for fun. It’s not useful for astronomers in any way.
They have, however, done the ground work and determined the distances to each star. The stars that make up the actual Little Dipper are 431, 126, 480, 346, 376, 183, 97 and 832 ly away, according to wikipedia Ursa Minor - Wikipedia . How we would “see” the asterism would of course also depend on the changes in magnitude as the angle changed, and what other stars would become more visible.
This raises an interesting possibility. Given a 3D map of the stars in our Galaxy and some pattern-recognition software, I wonder if a computer could “fly around” and find a vantage point where the stars form a passable portrait of, say, Justin Bieber? 
I now need a new keyboard.
:dubious: Bieber? Really?
Those are estimates, though, in some cases with large margins of error relative to the distance. There are many stars whose distances we are not sure about, so any 3D map would be a bit of a guess.
Out to a few hundred lightyears, though, our distance measurements are pretty good.
Did you not know? The prophecy states that if we can find his likeness in the stars, the world will end.
I can’t find a map for the Little Dipper specifically, but this sideways view of Orion should give you a good idea of how arbitrary our point of view is. The stars that make up Orion have no actual relation to each other, and are hundreds of light years apart. It’s only when projected on a sphere centred on the Earth that they assume the familiar shape of the constellation.
IIRC, National Geographic had an article about Orion some years ago that had a better version of this 3D map, but I can’t seem to find it.
That would be an interesting idea. Do you mind if I float the idea to a friend of mine? He is an astronomy professor who did a sabbatical with Google which partially led to Google Sky. Google is apparently always looking for ways to use the data- it might be an interesting idea to float by them.
Instead of your beloved Justin Bieber, I could imagine taking dot-to-dot images and using those as a search and match template before building up to his awesomeness.
A display like this used to be one of my favorite exhibits at the Hayden Planetarium in New York when I was a kid. I forget whether the constellation was the Big Dipper or Orion.
This can be done with the Big Dipper because it’s unique. Unlike most asterisms, most of the stars that make up the Big Dipper asterism are gravitationally bound together. Unlike almost every other asterism, most of Big Dipper’s stars are close enough together that they’re moving through space together as a single cluster. (They’re also close enough to us to make them look further apart than they really are.) There’s some models that show you how the asterism will look in thousands of years - it’s mostly recognizable even without the couple of stars that aren’t part of the cluster.
Actually, I need to correct myself (the dangers of posting without verifying!). Big Dipper’s stars are not gravitationally bound - that’s my mistake. Instead, five of the seven stars are part of the same moving group and are moving generally at the same speed in generally the same direction. (Apparently those in the know postulate that these five used to form an open cluster that was gravitationally bound, but they aren’t any longer - they’re moving together in mostly the same direction, but they’re not gravitationally bound anymore.)
Because these five are moving on the same path at the same speed, the Big Dipper will look mostly the same (to us) even thousands of years from now. Minus those other two.
The software Space Engine will allow you to “fly around” the universe, and thus see stuff from different angles. You want to be a little careful, because although it uses database values when such are available, it will proceduraly generate content when they are not. For relatively nearby stars such as you can see individually, it should work fine.
This has almost nothing directly to do with the OP question, but when my kids were little, grammar school age, I tried to impress them with the relative sizes and distances of the Solar System by converting things to the scale of our house being taken as the size of the Sun and then the planets to things like a basketball, a softball, a BB, and such at distances from our house that they could relate to. The nearby planets were on our block somewhere and the distant ones miles from home.
Once that conversion was done it was difficult, even for me, to see how Gravity could bind such tiny things at such distances together as a “system.” To add to the fun, the nearest star (at that scale) would have been in Australia.
Here’s a series of images of the Big Dipper from 100,000 years ago to 100,000 years in the future. The central 5 stars move less in relation to each other than the other two.
Yeah, that’s about the threshold where the parallax is relatively easy to detect, right?
Beyond the threshold, what are the estimates based on? Trying to get a handle on a parralax that’s really too small for our current estimates? Figuring absolute magnitude based on the star’s spectrum and comparing that with the star’s apparent brightness? A combination of both?
For “clustered” stars a long way away, you can use a combination of the above plus Cepheid (variable) stars to estimate distance. The variable stars follow a known luminosity curve and their distance can be estimated fairly precisely. Once you’ve got those to “center” your cluster, you can use magnitudes and spectrums to locate nearby stars relative to it (with much higher error).