# Volume projections to plane, cannot run demo code

I'm copying the code from "New in Mathematica 10 › Basic and Formula Regions"

ℛ =
ImplicitRegion[
x^6 - 5 x^4 y z + 3 x^4 y^2 + 10 x^2 y^3 z + 3 x^2 y^4 - y^5 z +
y^6 + z^6 <= 1, {x, y, z}];

RegionPlot3D[ℛ,
PlotRange -> {{-1.5, 1.5}, {-1.5, 1.5}, {-1.6, 1.6}},
PlotPoints -> 50]

{
RegionPlot[
Resolve[Exists[z, {x,y,z} ∈ ℛ], Reals]
, {x, -1.5, 1.5}
, {y, -1.5, 1.5}
],
RegionPlot[
Resolve[Exists[y, {x,y,z} ∈ ℛ], Reals]
, {x, -1.5, 1.5}
, {z, -1.5, 1.5}
],
RegionPlot[
Resolve[Exists[x, {x,y,z} ∈ ℛ], Reals]
, {y, -1.5, 1.5}
, {z, -1.5, 1.5}
]
}


It fails (I'm using version 11.3). Also I've heard that using Resolve is slow.

If I want to project a volume on to an arbitrary infinite plane, and observe the result area, what should I do instead?

• Hi seilgu, welcome to Mma.SE. Start by taking the tour now and learning about asking and what's on-topic. Always edit if improvable, show due diligence, give brief context, include minimal working example of code and data in formatted form. By doing all this you help us to help you and likely you will inspire great answers. I can reproduce your problem, it's not obvious to me what is the case. – rhermans Nov 1 '18 at 16:14

It's not clear to me whether this ever worked in any version of Mathematica. At any rate, the issue is that the Resolve command needs to be evaluated before values for x and y are inserted. So, the following produces the output given in the documentation (albeit with some error messages related to underflow that I quieted).

Quiet @ {
RegionPlot[
Evaluate @ Resolve[Exists[z, {x,y,z} ∈ ℛ], Reals],
{x, -1.5, 1.5},
{y, -1.5, 1.5}
],
RegionPlot[
Evaluate @ Resolve[Exists[y, {x,y,z} ∈ ℛ], Reals],
{x, -1.5, 1.5},
{z, -1.5, 1.5}
],
RegionPlot[
Evaluate @ Resolve[Exists[x, {x,y,z} ∈, Reals],
{y, -1.5, 1.5},
{z, -1.5, 1.5}
]
}


• Thanks. Is there anyway to find the projected region faster? – seilgu Nov 2 '18 at 9:13