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what is the requirement for the sewing to not create curvature at that seam?

In Ted's answer to my question, he wrote:
"In general, you only have curvature if you have to "crumple" or "stretch" the paper. When you form a cone, you don't have to do that, anywhere except at the vertex."

I don't understand why the vertex counts andas crumpling or stretching, but the comment makes intuitive sense. So one condition is that you need to sew the pieces together without stretching. If sewing is mapping a point on one edge to a point on the other edge, then once one pair of points are equated there will be curvature if the other points aren't equated with equal path length along the sewing seam.

I'm not sure how to go further in general, so I'll simplify the problem some. Let's cut the following out of flat space (in polar coordinates): $0<r<\infty, -f(r)<\theta<f(r)$

Sewing it up we can change coordinates
$k \phi=\theta, k=\frac{2\pi - 2f(r)}{2\pi} = 1- f(r)/\pi$
giving the line element in these coordinates as
$$d\theta = \phi \frac{\partial k}{\partial r} dr + k\ d\phi = \phi F(r) dr + k\ d\phi$$ $$ds^2 = dr^2 + r^2 (\phi^2 F^2\ dr^2 + 2\phi F k\ dr\ d\phi + k^2\ d\phi^2)$$ and the coordinates go from $0 \le r < \infty$ and $0 \le \phi \le 2\pi$

Oh god this is going to be messy. But you can in principle get the Riemann curvature and a condition on f(r) for the curvature to be zero. (Does anyone know how to make Mathematica or something calculate the answer here?)

what is the requirement for the sewing to not create curvature at that seam?

In Ted's answer to my question, he wrote:
"In general, you only have curvature if you have to "crumple" or "stretch" the paper. When you form a cone, you don't have to do that, anywhere except at the vertex."

I don't understand why the vertex counts and crumpling or stretching, but the comment makes intuitive sense. So one condition is that you need to sew the pieces together without stretching. If sewing is mapping a point on one edge to a point on the other edge, then once one pair of points are equated there will be curvature if the other points aren't equated with equal path length along the sewing seam.

I'm not sure how to go further in general, so I'll simplify the problem some. Let's cut the following out of flat space (in polar coordinates): $0<r<\infty, -f(r)<\theta<f(r)$

Sewing it up we can change coordinates
$k \phi=\theta, k=\frac{2\pi - 2f(r)}{2\pi} = 1- f(r)/\pi$
giving the line element in these coordinates as
$$d\theta = \phi \frac{\partial k}{\partial r} dr + k\ d\phi = \phi F(r) dr + k\ d\phi$$ $$ds^2 = dr^2 + r^2 (\phi^2 F^2\ dr^2 + 2\phi F k\ dr\ d\phi + k^2\ d\phi^2)$$ and the coordinates go from $0 \le r < \infty$ and $0 \le \phi \le 2\pi$

Oh god this is going to be messy. But you can in principle get the Riemann curvature and a condition on f(r) for the curvature to be zero. (Does anyone know how to make Mathematica or something calculate the answer here?)

what is the requirement for the sewing to not create curvature at that seam?

In Ted's answer to my question, he wrote:
"In general, you only have curvature if you have to "crumple" or "stretch" the paper. When you form a cone, you don't have to do that, anywhere except at the vertex."

I don't understand why the vertex counts as crumpling or stretching, but the comment makes intuitive sense. So one condition is that you need to sew the pieces together without stretching. If sewing is mapping a point on one edge to a point on the other edge, then once one pair of points are equated there will be curvature if the other points aren't equated with equal path length along the sewing seam.

I'm not sure how to go further in general, so I'll simplify the problem some. Let's cut the following out of flat space (in polar coordinates): $0<r<\infty, -f(r)<\theta<f(r)$

Sewing it up we can change coordinates
$k \phi=\theta, k=\frac{2\pi - 2f(r)}{2\pi} = 1- f(r)/\pi$
giving the line element in these coordinates as
$$d\theta = \phi \frac{\partial k}{\partial r} dr + k\ d\phi = \phi F(r) dr + k\ d\phi$$ $$ds^2 = dr^2 + r^2 (\phi^2 F^2\ dr^2 + 2\phi F k\ dr\ d\phi + k^2\ d\phi^2)$$ and the coordinates go from $0 \le r < \infty$ and $0 \le \phi \le 2\pi$

Oh god this is going to be messy. But you can in principle get the Riemann curvature and a condition on f(r) for the curvature to be zero. (Does anyone know how to make Mathematica or something calculate the answer here?)

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source | link

what is the requirement for the sewing to not create curvature at that seam?

In Ted's answer to my question, he wrote:
"In general, you only have curvature if you have to "crumple" or "stretch" the paper. When you form a cone, you don't have to do that, anywhere except at the vertex."

I don't understand why the vertex counts and crumpling or stretching, but the comment makes intuitive sense. So one condition is that you need to sew the pieces together without stretching. If sewing is mapping a point on one edge to a point on the other edge, then once one pair of points are equated there will be curvature if the other points aren't equated with equal path length along the sewing seam.

I'm not sure how to go further in general, so I'll simplify the problem some. Let's cut the following out of flat space (in polar coordinates): $0<r<\infty, -f(r)<\theta<f(r)$

Sewing it up we can change coordinates
$k \phi=\theta, k=\frac{2\pi - 2f(r)}{2\pi} = 1- f(r)/\pi$
giving the line element in these coordinates as
$$d\theta = \phi \frac{\partial k}{\partial r} dr + k\ d\phi = \phi F(r) dr + k\ d\phi$$ $$ds^2 = dr^2 + r^2 (\phi^2 F^2\ dr^2 + 2\phi F k\ dr\ d\phi + k^2\ d\phi^2)$$ and the coordinates go from $0 \le r < \infty$ and $0 \le \phi \le 2\pi$

Oh god this is going to be messy. But you can in principle get the Riemann curvature and a condition on f(r) for the curvature to be zero. (Does anyone know how to make Mathematica or something calculate the answer here?)