What causes the mirror image of my finger in a spoon to flip when I put it closer?

Question

When my finger is far away, its image is flipped:

But when I move it closer, it's not flipped:

What causes the flip? According to the illustration of this answer, it flips when the object moves closer than the focal point, is that right?

Answer 1

We can use ray optics to explain what's happening. To do so we can imagine the inner surface of the spoon as a spherical mirror having it's focus at some distance from its surface on its symmetry axis (as the red dot in the figures below). Of course the spoon is not so simple but this is enough to capture this behavior.

Now remember that a mirror like this has the property of making the parallel rays (the ones which would be produced by a very far source) converge in its focus and vice versa (light rays from the focus will be reflected parallel to the optic axis, this would be half of the radius o curvature of the mirror). Another property we need is that, by symmetry, a light ray incident on the very center of the mirror will be reflected as the mirror was flat, so with the same angle of incidence. We can use these properties to understand what light rays are doing and where images are formed.

Now let's look at the figures

To understand where light "is going" we only need to look at two light rays coming from the same point $P$ on the object. There are of course infinite rays but we can imagine, by continuity, that they will all converge roughly in the same points. When this doesn't happen we talk about aberration, and this will make our images blurred.

Now, if the rays converge in some point $Q$ (different from the point $P$ on the object itself), to us it will look as if $P$ was actually in $Q$ and there will be an image formed in that point although there is no object. This is what happens in the first image, matching your first image. We call this image a real image because light rays actually converge in that point.

Something different happens in the second image. In this case, using the same properties explained at the beginning, we can see that light rays coming from the tip of the object never really converge. However we see that they converge "going backwards" at some point behind the mirror, therefore it looks as if the object was behind the mirror and we call this a virtual image.

As you can see, in the first case the image of the object is upside down and we keep seeing it until the object is placed closer to the surface than the focus, at which point we start seeing the object upright and magnified. When the object is placed exactly in the focus there is no image formed and we don't see any image.