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Consider the following task:

enter image description here

There are two cross-sections: the first one is located in the $yz$ plane (the case 1), and the second one is in the $xy$ plane (the case 2). In the spherical coordinate system, given the known value of the polar angle $\theta$, I want to estimate the domain of azimuthal angle $\phi$ for which the vector $\mathbf{r} = r(\cos(\phi)\sin(\theta),\sin(\phi)\sin(\theta),\cos(\theta))$ intersects the plane.

In case 1, it is simple:

(*Radius R of a circle assuming fixed longitudinal distance \
Subscript[x, long] and the angle th*)
ROffAxisArb[th_, xLong_] = xLong*Tan[th];
(*Circle and a disk defined by the value of Subscript[x, long] and th*)
ParticleAsimuthalCoverage[xLong_, th_] = 
  Circle[{0, 0}, ROffAxisArb[th, xLong]];
x1Perpmin = 0.5;
x2Perpmin = 0.5;
x1Perpmax = 3;
x2PerpMax = 4.2;
(*Cross section located at the distance Subscript[x, long]*)
CrossSection1 = 
 Rectangle[{x1Perpmin, x2Perpmin}, {x1Perpmax, x2PerpMax}]
(*Region of the intersection of the R-circle and the cross section*)
Allowed\[Phi]ValuesZ[xLong_, th_] := 
 RegionIntersection[CrossSection1, 
  ParticleAsimuthalCoverage[xLong, th]]
Show@Graphics@{Black, RegionBoundary[CrossSection1], Blue, 
   ParticleAsimuthalCoverage[3, 0.36], Red, 
   Allowed\[Phi]ValuesZ[3, 0.36]}

enter image description here

But how to evaluate the allowed $\phi$ range for case 2? This task should be similarly 2D, but I am looking for some Mathematica function projecting the circle on the plane.

Could you please help me?

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    $\begingroup$ Change Circle to Disk or there will be no intersection. Consider two circles: reg1 = Circle[{0, 0}, 1]; and reg2 = Circle[{0.5, 0}, 1]; and find: RegionIntersection[reg1, reg2] $\endgroup$
    – Syed
    Commented Feb 4, 2022 at 12:22
  • $\begingroup$ @Syed: What do you mean by "no intersection"? I have a rectangle (2D) and a circle (1D). In case 1, the intersection is non-zero (see the figure). In case 2, I should project the circle on the plane with the rectangle's plane, find the intersection of the rectangle and the obtained section, and finally derive the allowed $\phi$ values. $\endgroup$
    – John Taylor
    Commented Feb 4, 2022 at 12:47
  • $\begingroup$ I mean that the area of the intersection would be zero as the Line has no area. With a disk it will be as shown in red. $\endgroup$
    – Syed
    Commented Feb 4, 2022 at 12:49
  • $\begingroup$ @Syed: I need not the area but rather RegionMeasure (i.e. length of the ark). $\endgroup$
    – John Taylor
    Commented Feb 4, 2022 at 12:50
  • $\begingroup$ @Syed : and apart from that I need the domain of $\phi$, not just the length. $\endgroup$
    – John Taylor
    Commented Feb 4, 2022 at 12:51

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