Parametrization of Curves

Parametrization of Curves

Definition 1. If $x$ and $y$ are given as functions $$x = f(t), \text{  } y=g(t)$$ over an interval $I$ of $t$-values, then the set of points $(x, y) = (f(t), g(t))$ defined by these equations is a parametric curve. The equations are parametric equations for the curve.
The variable $t$ is a parameter for the curve, and its domain $I$ is the parameter interval. If $I$ is a closed interval, $a \leq t \leq b$, the point $(f(a), g(a))$ is the initial point of the curve and $(f(b), g(b))$ is the terminal point. When we give parametric equations and a parameter interval for a curve, we say that we have parametrized the curve. The equations and interval together constitute a parametrization of the curve.

Differentiable Parametric Curve

Definition 2. A Parametrized curve $x = f(t)$ and $y = g(t)$ is differentiable at $t$ if $f$ and $g$ are differentiable at $t$. 

Derivatives for Parametric Curves

Theorem 1. Let a parametrized curve given by $x = f(t), y = g(t)$ be differentiable at $t$. If y is also a differentiable function of $x$, then the derivatives $dy / dt, dx / dt$, and $dy / dx$ exist and if $dx / dt \neq 0$, then $$\frac{dy}{dx} = \frac{\frac{dy}{dx}}{\frac{dx}{dt}}.$$
If $y$ is defined as a twice-differentiable function of $x$, then at any point where $dx/dt \neq 0$ and $y' = dy / dx$, $$\frac{d^2y}{dx^2} = \frac{\frac{dy'}{dt}}{\frac{dx}{dt}}.$$

Length of a Parametric Curve

Definition 3. If a curve $C$ is defined parametrically by $x = f(t)$ and $y = g(t), a \leq t \leq b$, where $f′$ and $g′$ are continuous and not simultaneously zero on $[a, b]$, and $C$ is traversed exactly once as $t$ increases from $t = a$ to $t = b$, then the length of $C$ is the definite integral $$L = \int_{a}^b \sqrt{[f'(t)]^2 + [g'(t)]^2} dx.$$

일반적인 함수의 arc length를 정의할 때와 동일하다. 우선 parametric curve가 smooth, 즉 $f', g'$이 모두 연속이고 cusps, 즉 첨점이 존재하지 않는다고 가정하자. 동일하게 구간 $[a, b]$를 잘게 쪼개고 각 subinterval을 이은 line segment들의 길이를 구하자. Mean Value Theorem에 의해 각 subinterval $[t_{k-1}, t_k]$ 안에 $t^*_k, t^{**}_k$가 존재하여 $$\Delta x_k = f'(t^*_k) \Delta t_k, \\ \Delta y_k = g'(t^{**}_k) \Delta t_k$$ 를 만족시킨다. 이제 위 사실들을 이용하여 line segment들의 길이를 모두 합해주면 $$\sum_{k=1}^n \sqrt{(\Delta x_k)^2 + (\Delta y_k)^2} \\ = \sum_{k=1}^n \sqrt{[f(t_k) - f(t_{k-1})]^2 + [g(t_k) - g(t_{k-1})]^2} \\ \sum_{k=1}^n \sqrt{[f'(t^*_k)^2 + [g'(t^{**}_k)]^2]} \Delta t_k$$ 이다. 이제 partiton의 norm이 0으로 가는 극한을 취해면 우리가 원하는 definite integral의 형태가 된다. 

Area of Surface of Revolution for Parametrized Curves

Definition 4. If a smooth curve $x = f(t), y = g(t), a \leq t \leq b$, is traversed exactly once as $t$ increases from $a$ to $b$, then the areas of the surfaces generated by revolving the curve about the coordinate axes are as follows: $$S = \int_a^b 2\pi y \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} dx \text{  (Revolution about the } x \text{ axis)}$$