双曲圆的周长、面积

hbghlyj

CG15-0.pdf使用hyperboloid model，把圆/双曲线、球/双曲面作对照，由变换群自然地引出度量

习题
6. Show that the circumference of the disk in $\mathbb{H}^2$ of radius $\theta$ is $2 \pi \sinh \theta$. This circumference is the set of points of distance $\theta$ from a fixed point. (Hint: Choose the center point to be Paris, and think about what kind of circle you have in $\mathbb{H}^2$.) What is the circumference of the disk of radius $\theta$ in $\mathbb{S}^2$?

7. Use Problem 6 to show that the area of the disk of radius $θ$ in $\Bbb H^2$ is $2π(\cosh θ - 1)$. You may use calculus. Think of a geometric way of finding the circumference of a disk by taking a derivative. What is the area of a disk of radius $θ$ in $\Bbb S^2$?

8. Use Problem 7 to calculate the area in $\Bbb H^2$ of the annular region between the circle of radius $θ$ and the circle of radius $θ + 1$. What is the limit as $θ$ goes to infinity of the ratio of this annular area and the total area of the disk of radius $θ+1$? Where is most of the area in a hyperbolic disk? (I like to call this the “Canada theorem” since this is like Canada, which has most of its population quite close to its boundary.)

hbghlyj

hbghlyj 发表于 2022-11-14 15:17
Think of a geometric way of finding the circumference of a disk by taking a derivative.

由6，圆的周长$C(\theta)=2\pi\sinh(\theta)$
圆面积可以视为一圈一圈的小圆环面积 $\rmd A=C(r)\rmd r$ 沿半径的累积，即
\[A(\theta)=\int_0^{\theta } 2 \pi  \sinh (r)\rmd r=2 \pi  (\cosh (\theta )-1)\]

hbghlyj

后文“most of its population quite close to its boundary”提示了大部分面积在半徑$θ$较大的区域
如果$D(θ)$是半径为$θ$的圆盘面积，按第7题$D(θ+1)-D(θ)=2π(\cosh(θ+1)-\cosh(θ))\to\infty$

hbghlyj

hbghlyj 发表于 2024-4-3 23:49
后文“most of its population quite close to its boundary”提示了大部分面积在半徑$θ$较大的区域

如果您尝试将一张纸包裹在马鞍面上，您会发现必须将其撕开才能使其展平在表面上。
这是因为，在具有负高斯曲率的表面上，圆的周长大于$π$乘以其直径，因此，要使平板沿着这样的表面放置，您必须将其撕裂以增加周长。

在自然界中，植物增加叶子边缘的生长速率，就可以产生弯曲或皱纹的叶子，从而控制叶子曲率，以调节水分流失，如这张观赏羽衣甘蓝的图片所示：

Is there any easy way to understand the definition of Gaussian Curvature?