Is there a notion of similarity for shapes on the sphere?

Question

There are similar shapes in the plane, such as similar triangles. So is there a similar shape on the sphere? For example, is a great circle similar to a small circle on a sphere? I think great circles and small circles are similar, so there are similar shapes on the sphere. So as shown in the figure, are there similar two sides on the sphere?

Here is my opinion:

Can two rectangles of different sizes be similar on a plane? Obviously, rectangles can be similar. Under what conditions are they similar?

1. The proportion of their corresponding sides is equal;
2. Their corresponding angles are equal.

These two conditions must be met at the same time.

Squares are also similar because they meet both conditions 1 and 2.

The conditions for the similarity of rectangle and triangle are not exactly the same. Although, we can think of each rectangle as consisting of two triangles.

In the plane, two regular polygons with the same number of sides are similar because they meet the requirements of conditions 1 and 2 at the same time.

We can think that any two circles are regular polygons with the same number of sides, so any two circles are similar. Because they meet both conditions 1 and 2. (their corresponding angles are all π).

If we take two points on each circle, we get some shapes with two sides. Can these shapes be similar? Obviously they can be similar. As long as they meet the requirements of condition 1 (their corresponding angles are π).

Can two shapes formed by closed curves be similar? I think they can be similar, and the conditions of judgment are similar, but the judgment is more complicated.

Are two circles similar on a sphere? Obviously they are similar. Because they meet both conditions 1 and 2.

On a sphere, are the two shapes shown in the figure similar (they are composed of arcs)? Obviously they can be similar as long as they meet the requirements of conditions 1 and 2 at the same time.

All is exploration. I'm not sure what I said is right. I hope you can talk about your knowledge.

Answer 1

Similarity can be tricky business.

Instead of the broad category of "shapes", first consider triangles.

In Euclidean geometry, "similar" triangles have the "same shape", a notion we clarify with the properties

1. Corresponding angles are congruent.
2. Corresponding sides are proportional.

Conveniently, property (1) implies (2), and vice-versa. Perhaps because angle-congruence is easier to check, "angles are congruent" is sometimes taken as the defining characteristic of similar triangles, with "sides are proportional" being a useful consequence. It doesn't seem to matter. Of course, we when consider "similar polygons", the logical equivalence breaks: squares and, say, golden rectangles have congruent angles, but their sides aren't proportional; conversely, squares and non-square rhombi have proportional sides, but their angles aren't congruent. "Same shape"-ness for polygons in general requires both (1) and (2).

Defining "same shape"-ness for curves is a little more nuanced, since we can't compare angles or sides directly. (We can get technical and establish a "similarity transformation" involving a dilation.) Nevertheless, it's "obvious" that all circles have the "same shape", so if the descriptor "similar" applies to any figures, it certainly applies to them.

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On the sphere, triangles already drive a wedge between (1) and (2). If corresponding sides are proportional, but not congruent, the corresponding angles are not congruent. (Specifically, larger sides make for larger angles. For example, consider a tiny-tiny equilateral triangle, whose angles are close to $60^\circ$ each, and the equilateral triangle joining the North, "East", and "West" Poles, whose angles are $90^\circ$.) Contrariwise, if corresponding angles are congruent, then the corresponding sides are congruent; "Angle-Angle-Angle" is a congruence pattern! This phenomenon can be summarized as

"There are no similar triangles in spherical geometry."

which is shorthand for "There are no similar triangles ---in the sense of (1) and (2)--- that aren't fully congruent triangles, so we have no use for the term 'similar'".

But what about circles?

In the vague sense of "same shape"-ness, then all circles on the sphere should be considered "similar". In the technical sense of "similarity transformation"-ability, too: simply align centers with an isometry and apply an appropriate dilation. Case closed!

And yet ...

On the (unit-radius) sphere, a spherical circle with radius $\theta$ as measured along the surface of the sphere is a plane circle with radius $\sin\theta$ as measured on the plane containing the circle; such a circle has circumference $C = 2\pi\sin\theta$. But, then, a circle with (surface) radius $2\theta$ has circumference $2\pi\sin 2\theta \neq 2C$: doubling the radius does not double the circumference; indeed, the circumference could get smaller! Likewise for other scale factors (apart from $1$).

This isn't the kind of behavior we've come to expect from "same shape" figures in the Euclidean plane. In light of this situation, we recognize that our (1) and (2) are actually insufficient to capture the entirety of our expectations. We find that a refinement is in order:

• 2'. Corresponding lengths are proportional.

Here, "length" isn't limited to just sides. It applies to medians, altitudes, arbitrary cevians, midpoint segments, etc, for triangles; diagonals of quadrilaterals and polygons; perimeters for everything; radii and circumferences for circles; and on and on and on. Here again, Euclidean geometry spoils us: figures that satisfy (1) and (2) ---and/or admit a similarity transformation--- automatically satisfy (2'); it's a freebie. Spherical geometry, however, reveals (2') to be a stronger condition of "same shape"-ness; a condition that spherical circles do not satisfy unless they are fully congruent.

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In the context of $(2')$, we find that, apart from segments,

"There are no [non-congruent] similar figures at all in spherical geometry."

If you choose to reject $(2')$ as a requirement, then arbitrary circles might reasonably be called "similar", but in a way that's almost-no help to further investigation and that's almost-certain to cause confusion. It would be better to apply a different descriptor ("quasi-similar"?).

Answer 2

As was mentioned in the comments, it depends on what you are willing to call "similar".

One reasonable option would be to take into account the curvature of the sphere, since it is an intrinsic feature. In the plane, it makes no difference, since it is everywhere flat, but when you compare similar triangles you do take into account the angles, which is a geometric feature as well.

On the sphere, making a triangle "bigger" will also automatically change its angles, because it changes the total curvature of the piece of surface enclosed by the shape. So all in all the natural notion of similarity is much more rigid on the sphere and you end up with geometric figures being similar only if they are actually isometric, in most cases.

Answer 3

It should not be said that the great circle is geodesic, so we think that the great circle and the small circle are not similar. Because we can also ask: are two small circles with different diameters similar? When the diameter increases, the small circle can coincide with the great circle, and the small circle can coincide with the small circle. So I think: on the sphere, any two circles are similar, whether they are great circles and small circles, or small circles and small circles. Only in this way can our geometric theory be unified, otherwise our geometric theory will be split.