How to distribute points on a Sphere[] cap from a normal distribution?

Question

I was wondering if one could distribute points on a "cap" of a sphere, following a normal distribution of the points instead of a uniform distribution. This normal could be centered at the cap. Maybe one can use SpherePoints[]?

By "cap" I mean that we select only these points within a radius of some reference point. For instance, using the case for uniform points (from Carl Woll's answer here):

SeedRandom[1]
numberofPoints = 1000;
radiusofCap = .8;
ctr = RandomPoint[Sphere[]];
pts = RandomPoint[
   RegionIntersection[Ball[ctr, radiusofCap], Sphere[]], 
   numberofPoints];
Graphics3D[{Red, Point@pts, White, Opacity[.5], Sphere[]}]

We get:

How could we extend this so that the points are normal-distributed? A solution might be in the form:

myNormalCapPoints[sphereRadius_, capRadius_, numberofPoints_, std_] :=

Where we take the desired sphere and cap radius, then select a point on the sphere (randomly perhaps?), and generate the normal-distributed points around it with some standard deviation from the reference point.

Thanks!

Note/update 1: It would be like a 2D gaussian cloud of points on the surface of the sphere (at the "cap").

Answer 1

Here I randomly rotate points away from the pole by a distance along the surface that is drawn from a truncated normal distribution. I hope this is close to what you need:

n = 1000;
capc = 0.4; (* Pi/2: cover hemisphere, Pi: cover the whole thing *)
dist = TruncatedDistribution[{-capc, capc}, NormalDistribution[0, 2]];
cvals = RandomVariate[dist, n];
dirs = Append[#, 0] & /@ RandomPoint[Circle[], n];
newpts = MapThread[RotationTransform[#1, #2][{0, 0, 1}] &, {cvals, dirs}];
Graphics3D[{Opacity[.5], Sphere[], Opacity[1], Point[newpts]}]