Are plane lines a special case of spherical circles?

Question

The equation of plane straight line is: $$y=Kx+b$$

In spherical geometry,the spherical circle equation is:

$$ \beta=\arctan ((1+(\tan \theta )^2) K\sin \alpha + \tan \theta \sqrt {(1+(1+(\tan \theta )^2)(K \sin \alpha)^2} )$$

This is the circle circle equation on the sphere. The alpha is longitude, and the beta is latitudes. $K$ is the slope of a circle, the tangent of angle between plane of a circle and plane of equator. $\theta$ is the distance between point of circle and equator when alpha equals zero. When alpha equals zero, theta equals the value of beta.

When diameter of a sphere tends to infinity, the equation of spherical circle can be changed to:

$$ \beta=K\alpha+ \theta $$

So the plane circles are only a special case of spherical circles. The spherical circles are essentially straight lines.

Answer 1

You can say that lines in the plane is a limiting case of great circles on spheres, but they are not a special case of great circles, because they are not actually great circles.

Answer 2

Using the stereographic projection, we can identify the plane $\mathbb{R}^2$ as the subset $S^2 \setminus \{N\}$ of the sphere $S^2$, where $N$ is the north pole. This is actually a diffeomorphism of smooth manifolds, if that means anything to you. If not, don't worry about it. All that really matters here is that the stereographic projection preserves angles and circles.

Under the sterographic projection, lines in the plane are identified with great circles on $S^2$ that pass through the north pole. Other great circles on $S^2$ get identified with circles in the plane. So in this sense, yes, lines in the plane are a special case of great circles.