Curve

Curve

Definition 1. We call the vector function $\textbf{r}: (a, b) \rightarrow \mathbb{R}^3$ a curve. We can parametrize curves by $\textbf{r}(t) = \langle f(t), g(t), h(t) \rangle$ where $t \in (a, b)$. 

Velocity, Speed, Unit Tangent Vector

Definition 2. Let $\textbf{r}$ be a curve. Then
(1) $\textbf{v}(t) = \frac{d \textbf{r}}{dt}$ is the velocity vector or the tangent vector of $\textbf{r}$,
(2) $|\textbf{v}|$ is the speed of $\textbf{r}$,
(3) $\textbf{T} = \frac{\textbf{v}}{|\textbf{v}|}$ is the unit tangent vector.
(4) the tangent line is the line passing through $\textbf{r}(t)$ in the direction $\textbf{r}'(t)$. 

Simple Curve

Definition 3. The curve $\mathbf{r}$ is simple if it has no further self-intersections; that is, if $t_1, t_2 \in (a, b), t_1 \neq t_2$, then $\mathbf{r}(t_1) \neq \mathbf{r}(t_2).$ ^[각주:1]

Limit and Continuity of Curves

Definition 4. Let $\textbf{r}(t) = \langle f(t), g(t), h(t) \rangle$ be a curve with domain $D$, and let $\mathbf{L} \in \mathbb{R}^3$. We say that $\mathbf{r}$ has limit $\mathbf{L}$ as $t$ approaches $t_0$ and write $$\lim_{t \to t_0} \mathbf{r}(t) = \mathbf{L}$$ if, for every number $\epsilon > 0$, there exists a corresponding number $\delta > 0$ such that for all $t \in D$ $$|\mathbf{r}(t) - \mathbf{L}| < \epsilon \text{  whenever  } 0 < |t-t_0| < \delta.$$

Definition 5. A curve $\textbf{r}(t)$ is continuous at a point $t = t_0$ in its domain if $$\lim_{t \to t_0} \textbf{r}(t) = \textbf{r}(t_0).$$ The curve is continuous if it is continuous at every point in its domain.

Theorem 1

Theorem 1. Let $\textbf{r}(t) = \langle f_1(t), f_2(t), f_3(t) \rangle$ be a curve with differentiable functions $f_1, f_2, f_3$. Then $$\lim_{t \to t_0} \textbf{r}(t) \text{ exists } \iff \forall j \in \{1, 2, 3\}, \lim_{t \to t_0} f_i(t) \text{ exists. }$$ Moreover, $$\lim_{t \to t_0} \textbf{r}(t) = \left< \lim_{t \to t_0} f_1(t), \lim_{t \to t_0} f_2(t), \lim_{t \to t_0} f_3(t) \right>.$$

Proof. $(\Longrightarrow)$ Let $\lim_{t \to t_0} \textbf{r}(t) = \textbf{L} = \langle L_1, L_2, L_3 \rangle$. Then $\forall \epsilon, \exists \delta > 0$ such that $|\textbf{r}(t) - \textbf{L}| < \epsilon$ whenever $0 < |t - t_0| < \delta$. This means that $|f_i(t) - L_i| < \epsilon, \forall i$ whenever $0 < |t - t_0| < \delta$. Thus limits of each component exist and $\lim_{t \to t_0} f_i(t) = L_i, \forall i$.

$(\Longleftarrow)$ Suppose that $\lim_{t \to t_0} f_i(t) = L_i, \forall i$. Then $\forall \epsilon > 0, \exists \delta_i$ such that $|f_i(t) - L_i| < \frac{\epsilon}{\sqrt{3}}$ whenever $0 < |t - t_0| < \delta_i$. Take $\delta = \min\{\delta_1, \delta_2, \delta_3 \}$. If $\textbf{L} = \langle L_1, L_2, L_3 \rangle$, then we have $$|\textbf{r}(t) - \textbf{L}| \leq \sqrt{\sum_{i=1}^3 (f_i(t) - L_i)^2} < \sqrt{\frac{\epsilon^3}{3} \cdot 3} = \epsilon \text{ whenever } 0 < |t - t_0| < \delta. \blacksquare$$

Differentiability of Curves

Definition 6. The curve $\text{r}(t) = \langle f(t), g(t), h(t) \rangle$ has a derivative (is differentiable) at $t$ if $f, g$ and $h$ have derivatives at $t$. The derivative is the curve $$\textbf{r}'(t) = \frac{d \textbf{r}}{dt} = \lim_{\Delta t \to 0} \frac{\textbf{r}(t + \Delta t) - \textbf{r}(t)}{\Delta t} = \langle f'(t), g'(t), h'(t) \rangle.$$ The curve is differentiable if it is differentiable at every point of its domain.

Theorem 1의 존재로 인해 우리는 곡선의 극한, 연속, 미분가능성은 component-wise하게 정의할 수 있다. 때문에 각 component function인 scalar function들이 극한을 가지는지, 연속인지, 미분가능한지 체크해서 취합해주면 된다.

Smooth

Definition 7. The curve $\textbf{r}$ is smooth if the velocity vector, i.e., the velocity of $\mathbb{r}$, that is, $\mathbb{v} = \frac{d \textbf{r}}{dt}$ is continuous and never $\textbf{0}$. 

속도가 영벡터가 된다면 곡선을 따라 움직이는 입자가 멈췄다가 다시 움직인다고 물리적으로 해석할 수 있다. 그러면 말그대로 부드럽지 않고 방향을 전환하여 곡선이 꺾이는 등의 불상사가 발생할 수도 있다. 때문에 이러한 상황이 발생하지 않는 곡선을 smooth하다고 말한다.

Length

Definition 8. The length of a smooth curve $\textbf{r}(t) = \langle f(t), g(t), h(t) \rangle, a \leq t \leq b$, that is traced exactly once as $t$ increases from $t=a$ to $t=b$, is $$L = \int_a^b |\textbf{v}| dt = \int_a^b \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 + \left( \frac{dz}{dt} \right)^2} dt.$$

곡선의 '길이'로 생각할 수도 있고, 물리적으로는 곡선을 따라 움직이는 입자의 '이동거리'라고 생각할 수 있다. 이 정의에 근거해서 arc length parameter $s$를 정의하자.

Arc Length Parameter

Definition 9. If we choose a base point $P(t_0)$ on a smooth curve $C$ parametrized by $t$, each value of $t$ determines a point $P(t) = (f(t), g(t), h(t))$ on $C$ and a directed distance $$s(t) = \int_{t_0}^t |\mathbb{v}(\tau)| d \tau,$$ measured along $C$ from the base point. We call $s$ an arc length parameter for the curve. 

Integrals

Definition 10. The definite integral of a continuous curve $\textbf{r}(t) = \langle f(t), g(t), h(t) \rangle$ is defined by $$\int_a^b \textbf{r}(t) dt = \left< \int_a^b f(t) dt, \int_a^b g(t) dt, \int_a^b h(t) dt \right>.$$ $\textbf{R}(t)$ is antiderivative of $\textbf{r}(t)$ if $\textbf{R}'(t) = \textbf{r}(t)$. We use the notation $$\int \textbf{r}(t) dt = \textbf{R}(t) + \textbf{C}$$ for some constant vector $\textbf{C}$. 

1. https://product.kyobobook.co.kr/detail/S000002271869 [본문으로]