Projection of a plane onto a surface of sphere

What is the effect on the plane equation of projecting a plane onto the surface of a sphere.

Assume we are starting with the standard point-normal plane equation:

$$ax + by + cz + d = 0$$

or alternatively:

$$\hat{n}x + d = 0$$

where $\hat{n}$ is the plane normal, $x$ is a point on the plane, and $d$ is the distance from the origin.

Mathematically, how does one project this plane onto a sphere?

• How do you define projection of a plane on a spherical surface? By the way, what is a spherical surface? Just a sphere? – mathcounterexamples.net Jun 29 '18 at 19:06
• @mathcounterexamples.net: Yeah I had trouble articulating this. Onto a the surface of a sphere is a better way to say it. And projection of a plane is basically warping the plane onto the surface of a sphere. – marcman Jun 30 '18 at 17:27
• @marcman "wrapping the plane onto the surface of a sphere" doesn't make sense either though: there is no isometric map from a region of the plane to a region of the sphere. Intuitively you've likely discovered this yourself, when trying to gift-wrap a round object, and noticing that no matter what you do the wrapping paper "puckers" – user7530 Jun 30 '18 at 18:36
• @user7530: That's not entirely true. Sure, for physical objects perhaps. But in the case I'm referring to it's entirely plausible that the plane should deform. Think of panoramas and image stitching where planes on images are projected onto a sphere. – marcman Jul 11 '18 at 17:04