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The title of this question is a bit awkward, as adding two points together on a manifold is usually not considered possible, but in this case there appears to be a nice little hack.

Consider the Riemann surface defined by the complex log function. One can (loosely) think of this as the set of complex numbers with "unwrapped" phases, meaning points of the form $R \angle \theta$, except that $R \neq 0$, and $\theta$ is now allowed to take any value in $\Bbb R$.

Much like with ordinary complex numbers, we can multiply together two points on this surface by multiplying their radii and adding together their angles. Unfortunately, this does not seem to extend to addition in any ordinary way.

However, there is an interesting little hack here. Given any two points on this surface, we can first obtain the average of the two, and then double the amplitude of the result while keeping the angle the same. We can find this average by using the Riemannian metric to find the point on the surface which is equidistant to both of the points in question, and which minimizes this distance.

I think this definition might fail for certain pairs of points of the form $R\angle \theta_1$ and $R \angle \theta_2$, where $\theta_1 = \theta_2+k*2\pi$ for $k \in \Bbb Z$." For these points, the average would intuitively be something like "zero," however there is no zero here. But, leaving that undefined for the moment, it seems as though this would yield a useful notion of addition for almost all points on the surface.

Questions:

• Has this construction been studied, and does it yield any sort of nice algebraic structure?
• Is there an explicit representation for this in terms of the polar coordinates of the two points?

Originally asked here, but seems too advanced for MSE.

The title of this question is a bit awkward, as adding two points together on a manifold is usually not considered possible, but in this case there appears to be a nice little hack.

Consider the Riemann surface defined by the complex log function. One can (loosely) think of this as the set of complex numbers with "unwrapped" phases, meaning points of the form $R \angle \theta$, except that $R \neq 0$, and $\theta$ is now allowed to take any value in $\Bbb R$.

Much like with ordinary complex numbers, we can multiply together two points on this surface by multiplying their radii and adding together their angles. Unfortunately, this does not seem to extend to addition in any ordinary way.

However, there is an interesting little hack here. Given any two points on this surface, we can first obtain the average of the two, and then double the amplitude of the result while keeping the angle the same. We can find this average by using the Riemannian metric to find the point on the surface which is equidistant to both of the points in question, and which minimizes this distance.

I think this definition might fail for certain pairs of points of the form $R\angle \theta_1$ and $R \angle \theta_2$, where $\theta_1 = \theta_2+k*2\pi$ for $k \in \Bbb Z$. For these points, the average would intuitively be something like "zero," however there is no zero here. But, leaving that undefined for the moment, it seems as though this would yield a useful notion of addition for almost all points on the surface.

Questions:

• Has this construction been studied, and does it yield any sort of nice algebraic structure?
• Is there an explicit representation for this in terms of the polar coordinates of the two points?

Originally asked here, but seems too advanced for MSE.

The title of this question is a bit awkward, as adding two points together on a manifold is usually not considered possible, but in this case there appears to be a nice little hack.

Consider the Riemann surface defined by the complex log function. One can (loosely) think of this as the set of complex numbers with "unwrapped" phases, meaning points of the form $R \angle \theta$, except that $R \neq 0$, and $\theta$ is now allowed to take any value in $\Bbb R$.

Much like with ordinary complex numbers, we can multiply together two points on this surface by multiplying their radii and adding together their angles. Unfortunately, this does not seem to extend to addition in any ordinary way.

However, there is an interesting little hack here. Given any two points on this surface, we can first obtain the average of the two, and then double the amplitude of the result while keeping the angle the same. We can find this average by using the Riemannian metric to find the point on the surface which is equidistant to both of the points in question, and which minimizes this distance.

I think this definition might fail for certain pairs of points of the form $R\angle \theta_1$ and $R \angle \theta_2$, where $\theta_1 \neq \theta_2$." For these points, the average would intuitively be something like "zero," however there is no zero here. But, leaving that undefined for the moment, it seems as though this would yield a useful notion of addition for almost all points on the surface.

Has this construction been studied, and does it yield any sort of nice algebraic structure? Also, is there any simple expression for this in terms of the polar coordinates of the two points?

Originally asked here, but seems too advanced for MSE.

The title of this question is a bit awkward, as adding two points together on a manifold is usually not considered possible, but in this case there appears to be a nice little hack.

Consider the Riemann surface defined by the complex log function. One can (loosely) think of this as the set of complex numbers with "unwrapped" phases, meaning points of the form $R \angle \theta$, except that $R \neq 0$, and $\theta$ is now allowed to take any value in $\Bbb R$.

Much like with ordinary complex numbers, we can multiply together two points on this surface by multiplying their radii and adding together their angles. Unfortunately, this does not seem to extend to addition in any ordinary way.

However, there is an interesting little hack here. Given any two points on this surface, we can first obtain the average of the two, and then double the amplitude of the result while keeping the angle the same. We can find this average by using the Riemannian metric to find the point on the surface which is equidistant to both of the points in question, and which minimizes this distance.

I think this definition might fail for certain pairs of points of the form $R\angle \theta_1$ and $R \angle \theta_2$, where $\theta_1 = \theta_2+k*2\pi$ for $k \in \Bbb Z$." For these points, the average would intuitively be something like "zero," however there is no zero here. But, leaving that undefined for the moment, it seems as though this would yield a useful notion of addition for almost all points on the surface.

Questions:

• Has this construction been studied, and does it yield any sort of nice algebraic structure?
• Is there an explicit representation for this in terms of the polar coordinates of the two points?

Originally asked here, but seems too advanced for MSE.

The title of this question is a bit awkward, as adding two points together on a manifold is usually not considered possible, but in this case there appears to be a nice little hack.

Consider the Riemann surface defined by the complex log function. One can (loosely) think of this as the set of complex numbers with "unwrapped" phases, meaning points of the form $R \angle \theta$, except that $R \neq 0$, and $\theta$ is now allowed to take any value in $\Bbb R$.

Much like with ordinary complex numbers, we can multiply together two points on this surface by multiplying their radii and adding together their angles. Unfortunately, this does not seem to extend to addition in any ordinary way.

However, there is an interesting little hack here. Given any two points on this surface, we can first obtain the average of the two, and then double the amplitude of the result while keeping the angle the same. We can find this average by using the Riemannian metric to find the point on the surface which is equidistant to both of the points in question, and which minimizes this distance.

I think this definition might fail for certain pairs of points which are, intuitively, "opposite" one another on the Riemann surface, where the average would (intuitively) be zero, except there is no zero here. However, leaving that undefined for the moment, it seems as though this would yield a useful notion of addition for almost all points on the surface.

Has this construction been studied, and does it yield any sort of nice algebraic structure? Also, is there any simple expression for this in terms of the polar coordinates of the two points?

Originally asked here, but seems too advanced for MSE.

The title of this question is a bit awkward, as adding two points together on a manifold is usually not considered possible, but in this case there appears to be a nice little hack.

Consider the Riemann surface defined by the complex log function. One can (loosely) think of this as the set of complex numbers with "unwrapped" phases, meaning points of the form $R \angle \theta$, except that $R \neq 0$, and $\theta$ is now allowed to take any value in $\Bbb R$.

Much like with ordinary complex numbers, we can multiply together two points on this surface by multiplying their radii and adding together their angles. Unfortunately, this does not seem to extend to addition in any ordinary way.

However, there is an interesting little hack here. Given any two points on this surface, we can first obtain the average of the two, and then double the amplitude of the result while keeping the angle the same. We can find this average by using the Riemannian metric to find the point on the surface which is equidistant to both of the points in question, and which minimizes this distance.

I think this definition might fail for certain pairs of points of the form $R\angle \theta_1$ and $R \angle \theta_2$, where $\theta_1 \neq \theta_2$." For these points, the average would intuitively be something like "zero," however there is no zero here. But, leaving that undefined for the moment, it seems as though this would yield a useful notion of addition for almost all points on the surface.

Has this construction been studied, and does it yield any sort of nice algebraic structure? Also, is there any simple expression for this in terms of the polar coordinates of the two points?

Originally asked here, but seems too advanced for MSE.