Smooth Parametric Curves and Their Slopes

Smooth Parametric Curves and Their Slopes

Definition and Properties of Smoothness

A smooth parametric curve is one where the tangent line exists at every point and turns smoothly as the point moves along the curve. In contrast, a curve is not smooth at points where the tangent line does not exist, such as at cusps or corners.

• Smooth Curve: Tangent line exists everywhere; the curve transitions smoothly.

• Not Smooth Curve: At certain points (e.g., point P), the tangent line does not exist, indicating a cusp or corner.

Example: Cusp in Parametric Curves

Consider the parametric curve:

• $x = t^2$

• $y = t^3$

• $t \in \mathbb{R}$

For $t > 0$: $y = x^{3/2}$ For $t < 0$: $y = -x^{3/2}$ At $t = 0$, both derivatives vanish:

• $\frac{dx}{dt} = 2t \rightarrow 0$ as $t \rightarrow 0$

• $\frac{dy}{dt} = 3t^2 \rightarrow 0$ as $t \rightarrow 0$

Although $\lim_{t \to 0^+} \frac{dy}{dx} = 0$ and $\lim_{t \to 0^-} \frac{dy}{dx} = 0$, there is no tangent at $t = 0$ due to the cusp at $(0,0)$.

Theorem: Smoothness and Tangents of Parametric Curves

Conditions for Smoothness

Let $\ell$ be a parametric curve:

• $x = f(t)$

• $y = g(t)$

• $t \in I$

where $f'$ and $g'$ are continuous on $I$.

• If $f'(t) \neq 0$ on $I$: The curve is smooth and has a tangent line at $t$ with slope $\frac{dy}{dx} = \frac{g'(t)}{f'(t)}$.

• If $g'(t) \neq 0$ on $I$: The curve is smooth and has a normal line at $t$ with slope $-\frac{dx}{dy} = -\frac{f'(t)}{g'(t)}$ (the reciprocal of the tangent slope).

• If $f'(t) = 0$ but $g'(t) \neq 0$: The curve has a vertical tangent.

Example: Horizontal and Vertical Tangents

For the curve:

• $x = t^3 - 3t$

• $y = t^3 - 12t$

• $t \in \mathbb{R}$

Horizontal Tangent: $\frac{dy}{dt} = 0$ and $\frac{dx}{dt} \neq 0$ $\frac{dy}{dt} = 3t^2 - 12 = 3(t-2)(t+2) \Rightarrow t = \pm2$ $\frac{dx}{dt}|_{t=\pm2} = 9 \neq 0$

Vertical Tangent: $\frac{dx}{dt} = 0$ and $\frac{dy}{dt} \neq 0$ $\frac{dx}{dt} = 3t^2 - 3 = 3(t-1)(t+1) \Rightarrow t = \pm1$ $\frac{dy}{dt}|_{t=\pm1} = -9 \neq 0$

Example: Slope at a Specific Point

For the curve:

• $x = t^4 - t^2$

• $y = t^3 - 2t$

• $t \in \mathbb{R}$

At $t = -1$:

• $\frac{dy}{dx}|_{t=-1} = \frac{g'(t)}{f'(t)} = \frac{3t^2 - 2}{4t^3 - 2t} = \frac{1}{-2} = -\frac{1}{2}$

Parametric Equations of Tangent and Normal Lines

General Formulas

Let $\ell$ be a smooth parametric curve at $t = t_0$:

• $x = f(t)$

• $y = g(t)$

• $f'(t_0)^2 + g'(t_0)^2 \neq 0$

Tangent Line:

• $x = f(t_0) + f'(t_0) \cdot (t - t_0)$

• $y = g(t_0) + g'(t_0) \cdot (t - t_0)$

Normal Line:

• $x = f(t_0) + g'(t_0) \cdot (t - t_0)$

• $y = g(t_0) - f'(t_0) \cdot (t - t_0)$

Example: Tangent and Normal Lines at $t = \frac{\pi}{4}$

For the curve:

• $x = t - \cos t$

• $y = 1 - \sin t$

• $t \in \mathbb{R}$

At $t = \frac{\pi}{4}$:

• Point: $\left(\frac{\pi}{4} - \frac{1}{\sqrt{2}}, 1 - \frac{1}{\sqrt{2}}\right)$

• Derivatives: $x' = 1 + \sin t$, $y' = -\cos t$

• At $t = \frac{\pi}{4}$: $x' = 1 + \frac{1}{\sqrt{2}}$, $y' = -\frac{1}{\sqrt{2}}$

Tangent Line:

• $x(t) = \frac{\pi}{4} - \frac{1}{\sqrt{2}} + \left(1 + \frac{1}{\sqrt{2}}\right)(t - \frac{\pi}{4})$

• $y(t) = 1 - \frac{1}{\sqrt{2}} + \left(-\frac{1}{\sqrt{2}}\right)(t - \frac{\pi}{4})$

Normal Line:

• $x(t) = \frac{\pi}{4} - \frac{1}{\sqrt{2}} + \left(-\frac{1}{\sqrt{2}}\right)(t - \frac{\pi}{4})$

• $y(t) = 1 - \frac{1}{\sqrt{2}} - \left(1 + \frac{1}{\sqrt{2}}\right)(t - \frac{\pi}{4})$

Concavity of Parametric Curves

Determining Concavity

The concavity of a parametric curve is determined by the sign of $\frac{d^2y}{dx^2}$:

• $\frac{d^2y}{dx^2} > 0$: Curve is concave up

• $\frac{d^2y}{dx^2} < 0$: Curve is concave down

• $\frac{d^2y}{dx^2} = 0$: Curve has an inflection point

The formula for the second derivative is:

• $\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d}{dt}\left(\frac{g'}{f'}\right) \cdot \frac{dt}{dx}$

• $= \frac{g''f' - f''g'}{(f')^2} \cdot \frac{1}{f'} = \frac{g''f' - f''g'}{(f')^3}$

Example: Concavity and Tangents of a Parametric Curve

For the curve:

• $x = t^3 - 3t$

• $y = t^2$

• $t \in [-2, 2]$

Key Points:

• $x = 0$: $t(t^2 - 3) = 0 \Rightarrow t = 0, \pm\sqrt{3}$

• $y = 0$: $t = 0$

• Horizontal tangent: $g' = 0 \Rightarrow t = 0$

• Vertical tangent: $f' = 0 \Rightarrow t = \pm1$

• No cusps

• Concavity: $\frac{d^2y}{dx^2} = \frac{-2(1 + t^2)}{9(t^2 - 1)^3}$

• Concave up: $t \in (-1, 1)$

• Concave down: $t \in [-2, -1) \cup (1, 2]$

• No inflection points

• As $t \to \pm\infty$: $y \approx (x^2)^{1/3}$

• Symmetry: $(x, y) \leftrightarrow (-x, y)$