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In version 10.0, Mathematica introduced a new function about region plotting: ImplicitRegion. But I'm wondered that haven't this function been already existed in the older version? That is, RegionPlot. I truly can't see the difference.

Another question is, in version 10, Mathematica also introduced a function called ParametricRegion, my question is the same again: isn't it equivalent to the third usage of ParametricPlot? PS: I found a little bit difference is that ParametricRegion(new function in ver. 10) have the ability to constrain the parameters in more specific way(not just required to be {u,u_min,u_max}, {v,v_min,v_max}, but also {1 <= u <= 5, 3 <= v <= 10, u+v < 7.5}), but it can seemingly be done by option RegionFunction in the older version!

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There's a lot in Mma that, strictly speaking, isn't necessary. But look at the NDSolve-FEM stuff, as well as integration, and you'll see that there are things that were difficult, perhaps impossible, to do before that can be done now. The region functionality is part of the development of those capabilities. –  Michael E2 Aug 17 at 14:30
    
For instance, you can't Integrate over RegionPlot, but you can do it over an ImplicitRegion. New computational geometry functionality with regions is more symbolic than old, which was mostly related to visualization. Sadly enough naming can be a bit confusing and prevent seeing the difference... –  kirma Aug 17 at 14:32
    
I would suggest that this Q is primarily opinion based or requires advice from Wolfram support, except I feel the answer is clearly yes. –  Michael E2 Aug 17 at 14:33

1 Answer 1

ImplicitRegion (and ParametricRegion) represent a region. They are not for plotting. Thus RegionPlot is not even remotely an alternative.

You can do many operations on regions that you can look up in the documentation centre. Just a few examples: you can compute their size, decide if a point is within, compute the distance between them, find their boundary, intersection, union, etc. You can also integrate over regions numerically or symbolically, use them as a domain in several symbolic and numerical functions (e.g. optimization, PDE solving) or plotting functions.

All of these uses are completely unrelated to what RegionPlot can do.

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So in older version, RegionPlot and ParametricPlot are just for plotting something, then can not be put altogether with Reduce, Solve, such like Solve >> Scope >> Geometric Region –  Eric Aug 17 at 15:49
    
And RegionPlotand ParametricPlot can have a lot of options to set its colors, aspect ratios, mesh styles and plot points, while ImplicitRegion and ParametricRegion cannot, since they are more likely to be treated as domain specification. However, they can be involved in equation/differential equation/inequalities solving, which is impossible to be done (directly) in the previous versions? –  Eric Aug 17 at 15:55
    
@Eric RegionPlot is for visualizing a region. ImplicitRegion is for representing a region. These are two entirely different purposes, and it makes little sense to compare the two functions. You can use RegionPlot to plot any region object, e.g. RegionPlot[ImplicitRegion[x^2+y^2<1, {x,y}]] or RegionPlot[Disk[]]. –  Szabolcs Aug 17 at 16:19
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@Eric, and you can have regions in 5D –  Rojo Aug 17 at 23:19
    
I want to know whether Integrate(in version 10) can integrate on an arbitrary region such like triangles or ploygons(ie. 6-gon)? If possible, what's the algorithm inside kernel? It's pretty miraculous! –  Eric Aug 17 at 23:38

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