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# 数学代写|扭结理论代写Knot Theory代考|MATH332 Computing arc lengths and areas

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## 数学代写|扭结理论代写Knot Theory代考|Computing arc lengths and areas

2.1.2. Computing arc lengths and areas. Now we will use the formulas obtained above to do calculations, in order to better understand the hyperbolic space $\mathbb{H}^2$.

Example 2.1. Fix a height $h>0$, and consider first a horizontal line segment between points $(0, h)=i h$ and $(1, h)=1+i h$ in $\mathbb{H}^2$. We may parameterize the line segment by $\gamma(t)=(t, h)$, for $t \in[0,1]$. Using equation (2.1), we find that the arc length of $\gamma$ is $|\gamma|=1 / h$. Note that because $h$ is fixed, the arc length of $\gamma$ is just its usual Euclidean length rescaled by $1 / h$. Thus when $h=1$, the length of $\gamma$ is 1 . When $h$ becomes large, the arc length becomes very small. In other words, points with the same height become very close together as their heights increase. On the other hand, as $h$ approaches 0 , the length of $\gamma$ approaches infinity. In fact, points near the real line $\mathbb{R}=\left{(x, 0) \in \mathbb{R}^2\right}$ can be very far apart.

Example 2.2. Now consider a vertical line between points $(x, a)$ and $(x, b)$, for $x, a, b$ fixed in $\mathbb{R}, 0<a<b$. Such a line can be parameterized by $\zeta(t)=(x, t)$ for $t \in[a, b]$. So $\zeta^{\prime}(t)=(0,1)$. Thus its arc length is given by
$$|\zeta|=\int_a^b \sqrt{0+1} \frac{1}{s} d s=\log \left(\frac{b}{a}\right) .$$
If we set $b=1$ and let $a$ approach 0 , note that the arc length of $\zeta$ gets arbitrarily large, approaching infinity. Similarly, setting $a=1$ and letting $b$ approach infinity give arbitrarily long lengths.

The real line $\mathbb{R}=\left{(x, 0) \in \mathbb{R}^2\right}$ along with the point at infinity $\infty$ play an important role in the geometry of $\mathbb{H}^2$, although these points are not contained in $\mathbb{H}^2$.

## 数学代写|扭结理论代写Knot Theory代考|Geodesics and isometries

Geodesics and isometries. Recall that a geodesic between points $p$ and $q$ is a length minimizing curve between those points. An infinite geodesic is a curve $\gamma$ from $\mathbb{R}$ to a Riemannian manifold such that for any $s<t \in \mathbb{R}$, the curve $\gamma([s, t])$ minimizes the distance between $\gamma(s)$ and $\gamma(t)$.
Theorem 2.5. The infinite geodesics in $\mathbb{H}^2$ consist of vertical straight lines and semi-circles with center on the real line.

Note these are exactly the circles and lines in the upper half-plane that meet $S_{\infty}^1$ at right angles. See Figure 2.2. Observe that between any two points in the upper half-plane, there is a unique vertical line or semi-circle between them. Thus a geodesic between points $p$ and $q$ in $\mathbb{H}^2$ is a segment of a semi-circle or a vertical straight line. An infinite geodesic can also be viewed as the unique semi-circle or vertical straight line between two points on the boundary at infinity of $\mathbb{H}^2$. We will typically drop the word “infinite” to describe geodesics between points on the boundary at infinity. Thus we use the same word “geodesic” to describe both infinite or bounded arcs, depending on context.

The proof of Theorem $2.5$ is left as an exercise in Riemannian geometry. The simplest way to prove the theorem uses coordinates and a bit more Riemannian geometry than we have reviewed so far. The interested reader can work through the details. The fact that these are the geodesics of $\mathbb{H}^2$ is all we will need going forward.

## 数学代写|扭结理论代写Knot Theory代考|Computing arc lengths and areas

2.1.2. 计算弧长和面积。现在我们将使用上面得到的公式进行计算，以便更好地理解双曲空间 $\mathbb{H}^2$. $\gamma(t)=(t, h)$ ，为了 $t \in[0,1]$. 使用等式 (2.1)，我们发现弧伥 $\gamma$ 是 $|\gamma|=1 / h$. 请注意，因为 $h$ 是固定的，弧长为 $\gamma$ 只是它通常的 欧几里得长度重新缩放 $1 / h$. 因此当 $h=1$ ，的长度 $\gamma$ 是 1 。什么时候 $h$ 变大，弧长变得非常小。换句话脱，具有相同高度的点随着 高度的垾加变得非常接近。另一方面，如 $h$ 接近 0 ，长度为 $\gamma$ 接近无穷大。事实上，靠近实线的点 lleft 的分隔符缺失或无法识别 可以相距很远。

$$|\zeta|=\int_a^b \sqrt{0+1} \frac{1}{s} d s=\log \left(\frac{b}{a}\right) .$$

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