How to calculate the intersect of a point-vector and a cone in 3D space.

I have a 3D modeling problem where I am trying to modify a mesh to make it 3d printable without supports. I have been able to identify “unsupported” vertices in the mesh. What I want to do it move the vertex along its normal vector until it intersects a cone. This cone will originate from a neighboring vertex in the mesh, will open upwards, and will always have the axis be the vertical (z-axis). The relative locations of the vertices and the slope of the cone can vary.

I need to calculate the coordinates of the intersecting point given the coordinates of the original point, a vector from the original point, the coordinates of the tip of the cone, and the angle of the cone.

I’m at a complete loss for how to do this. Probably something I learned 25 years ago in college and forgot because I haven’t used it since.

vector intersecting a cone