How can I get the texture to perfectly match a portion of the sphere (360 image)?

In Three.js, I’m new. I’m having trouble matching as I work on a spherical panorama with three.js. I want to obtain a distinct texture that matches the original spherical. I’m now attempting to fix my issue by manually adjusting the positions of the PlaneGeometry objects while they have texture applied, however I think this is a poor idea. The sphere curves away from the plane.

Is it possible to use PlaneGeometry to correctly match if I locate the middle point P and the corners P1, P2, P3, and P4 as shown in the picture below?

I also discovered that it is possible to use Theta and Phi to make a segment of a sphere, but I’m not handle all for corner of the new plane(perspective view).