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I'm trying to calculate the area of intersection of a cone and a disk which is a circular slice of the cone rotated through a particular angle about the apex of the cone. I've defined the cone using a set of inequalities with ImplicitRegion and have found a parametrization for the circle. I'm confident I have found the right equations because plotting RegionUnion produces what I expect to see, for example for the disk rotated through 1 degree:

R = 1.4; d = 30; T = Pi/180; cone = ImplicitRegion[{x^2 + y^2 <= (z Tan[R/d])^2,0 <= z < 1.2*d}, {x, y, z}]; disk = ParametricRegion[{d Sin[T] - r Cos[T] Cos[t], r Sin[t],d Cos[T] - r Cos[t] Sin[T]}, {{r, 0, 1.42}, {t, 0, 2 Pi}}]; Region[cone] Region[disk] Region[RegionUnion[cone, disk]]

cone

disk

union of cone and disk

However, I can't get it to plot the intersection or calculate the area of the intersection:

Region[RegionIntersection[cone, disk]]

produces the output

RegionIntersection output

and asking it for the area of intersection,

Area[RegionIntersection[cone, disk]]

gives the output: Area output

Can anyone help me figure out why this isn't working and how to calculate the area of intersection?

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Is this what you need?

In[7]:= RegionMeasure[RegionIntersection[cone, disk]] Out[7]= 4.78199
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    $\begingroup$ Cannot reproduce it, obtaing Throw::sysexc: Uncaught SystemException returned to top level. Can be caught with Catch[[Ellipsis], _SystemException]. SystemException["MemoryAllocationFailure"] in MMA 11.3. $\endgroup$ Commented Apr 8, 2018 at 6:12
  • $\begingroup$ I tried this in version 11.2. I'll try 11.3 tomorrow and file a bug report if I get the same error. $\endgroup$ Commented Apr 8, 2018 at 11:42
  • $\begingroup$ Don't trouble. I already submitted that. $\endgroup$ Commented Apr 8, 2018 at 14:49

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