# DiscretizeRegion fails for a triangular region

Bug introduced in 10.2.0 or earlier and fixed in 11.0.0

I define an implicit region like

R = ImplicitRegion[
0 < Sin[u]/Cos[v] < 1 &&
0 < Sin[v]/Cos[u] < 1, {{u, 0, 2}, {v, 0, 2}}]


But when I try to do

DiscretizeRegion[R]


Mathematica 10.4 seems to hang and doesn't return any result. This is strange, because the region is really just a simple triangle

RegionPlot[0 < Sin[u]/Cos[v] < 1 && 0 < Sin[v]/Cos[u] < 1, {u, 0, 2}, {v, 0, 2}] and discretization of a more complex region like

DiscretizeRegion[ImplicitRegion[0 < Sinh[u]/Cosh[v] < 1 && 0 < Sinh[v]/Cosh[u] < 1,
{{u, 0, 2}, {v, 0, 2}}]]


works without problems. What is the problem with my implicit triangle?

• I can confirm that this problem is also present in MM 10.2 running on Mac OS. – Michael Seifert Apr 18 '16 at 22:20
• With version 10.4 on Windows 7 x64 DiscretizeRegion[R] returns the result after 6m34s of work on my system (screenshot). – Alexey Popkov Apr 19 '16 at 0:40
• Could you report this as a bug to Wolfram Research and add information on the case number they give to it? – kirma Apr 19 '16 at 3:58
• I reported a bug to Wolfram and they opened CASE:3585657. They also said, that DiscretizeRegion can be sped up by specifying the Method to use (e.g. Boolean or RegionPlot), e.g. DiscretizeRegion[R, Method -> "RegionPlot"] . – asmaier May 1 '16 at 20:03

Looks like you've uncovered a bug. I can confirm this behavior in 10.3.1 and 10.4. You can still discretize your region using DiscretizeGraphics though:

r = DiscretizeGraphics@
RegionPlot[0 < Sin[u]/Cos[v] < 1 && 0 < Sin[v]/Cos[u] < 1, {u, 0, 2}, {v, 0, 2}] And if you want finer areas, use DiscretizeRegion:

DiscretizeRegion[r, MaxCellMeasure -> 0.001] 