Curves & Parametric Equations: Parametric Forms, Arc Length, Curvature

TAKEAWAYS:

• Parametric equations express curves as $x=f(t),y=g(t)$ where $t$ is a parameter (e.g., time, angle), enabling modeling of complex paths like projectile motion or cycloid curves.
• Arc length for parametric curves is $L={\int }_{t_1}^{t_2}\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt$, derived from the Pythagorean theorem for infinitesimal segments.
• Curvature $\kappa =\frac{|x^{\prime }{y}^{\prime \prime }-y^{\prime }{x}^{\prime \prime }|}{(x^{\prime 2}+y^{\prime 2}{)}^{3/2}}$ measures how sharply a curve bends at a point, critical for designing roads, springs, and computer-generated shapes.
• Cycloids, cycloids, and involutes are classic parametric curves with real-world applications in gear teeth design, roller coasters, and robot arm trajectories.
• Converting between Cartesian and parametric forms is essential for solving optimization problems (e.g., finding extrema of $y=f(x)$ via $x=g(t),y=h(t)$).
• Exam questions often test visualization (sketching curves), calculation (arc length/curvature), and applications (e.g., path optimization for drones or traffic flow).

1. Parametric Equations: Definition and Basics

Parametric equations define a curve by expressing coordinates as functions of a third variable $t$ (parameter). For example:

• Projectile motion: $x(t)=v_0\cos (\theta )t$, $y(t)=v_0\sin (\theta )t-\frac{1}{2}g{t}^2$.
• Circle: $x(t)=r\cos (t)$, $y(t)=r\sin (t)$.

Key Concepts

• Domain of $t$: Determines the portion of the curve traced (e.g., $t\in [0,2\pi ]$ for a full circle).
• Eliminating the parameter: Convert to Cartesian form $y=f(x)$ if possible (e.g., for a circle, $x^2+y^2=r^2$).
• Direction of traversal: Increasing $t$ may trace the curve clockwise or counterclockwise.

Worked Example 1: Sketch the curve and find Cartesian form Given $x(t)=t^2$, $y(t)=t^3-2t$, $t\in \mathbb{R}$.

1. Eliminate $t$: From $x=t^2$, $t=\sqrt{x}$ (for $t\ge 0$). Substitute into $y$: $y=(\sqrt{x}{)}^3-2\sqrt{x}=x^{3/2}-2x^{1/2}$.
2. Sketch:
   • At $t=0$: $(0,0)$
   • At $t=1$: $(1,-1)$
   • At $t=2$: $(4,4)$
   • Symmetry: For $t<0$, $x$ is same but $y$ changes sign (e.g., $t=-1$ → $(1,1)$).

2. Arc Length of Parametric Curves

The arc length $L$ of a curve from $t_1$ to $t_2$ is: $$L={\int }_{t_1}^{t_2}\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt$$ Derivation: For a small change $\Delta t$, the displacement is $\Delta x=x^{\prime }(t)\Delta t$, $\Delta y=y^{\prime }(t)\Delta t$. The infinitesimal arc length $ds$ satisfies: $$ds=\sqrt{(\Delta x{)}^2+(\Delta y{)}^2}=\sqrt{x^{\prime }(t{)}^2+y^{\prime }(t{)}^2}dt$$

Worked Example 2: Arc Length of a Cycloid

A cycloid is generated by a circle of radius $a$ rolling along the x-axis: $$x(t)=a(t-\sin t),y(t)=a(1-\cos t)$$ Find the arc length for one full rotation ($t\in [0,2\pi ]$).

1. Compute derivatives: $$\frac{dx}{dt}=a(1-\cos t),\frac{dy}{dt}=a\sin t$$
2. Plug into the arc length formula: $$L={\int }_0^{2\pi }\sqrt{a^2(1-\cos t{)}^2+a^2{\sin }^{2}t}dt=a{\int }_0^{2\pi }\sqrt{2-2\cos t}dt$$
3. Simplify using $1-\cos t=2{\sin }^2(t/2)$: $$L=a{\int }_0^{2\pi }2\left|\sin \left(\frac{t}{2}\right)\right|dt=4a{\int }_0^{\pi }\sin \left(\frac{t}{2}\right)dt=8a$$ (The integral evaluates to $8a$ for one full rotation.)

In the Real World:

• Pathao/Daraz Delivery Routes: Parametric equations model optimal delivery paths for riders, minimizing distance (arc length) while avoiding traffic. For example, a rider’s path from Kathmandu to Lalitpur can be parameterized by time $t$, with $x(t)$ and $y(t)$ representing longitude/latitude. The arc length integral ensures the shortest route is calculated dynamically.
• NTC Power Line Design: Overhead power lines follow parametric curves to minimize material cost (arc length) while maintaining clearance. Engineers use calculus to optimize the shape of sagging cables between poles.
• YouTube Video Thumbnails: Computer graphics use parametric curves (e.g., Bézier curves) to design smooth thumbnails. The arc length ensures the curve’s "length" matches the designer’s specifications.

3. Curvature of Parametric Curves

Curvature $\kappa$ measures how sharply a curve bends at a point. For parametric curves: $$\kappa =\frac{|x^{\prime }{y}^{\prime \prime }-y^{\prime }{x}^{\prime \prime }|}{(x^{\prime 2}+y^{\prime 2}{)}^{3/2}}$$ Intuition:

• High $\kappa$ → sharp turn (e.g., a circle has $\kappa =1/r$).
• Low $\kappa$ → gentle curve (e.g., a straight line has $\kappa =0$).

Worked Example 3: Curvature of a Helix

A helix is parameterized by: $$x(t)=\cos t,y(t)=\sin t,z(t)=t$$ Find $\kappa$ at any point.

1. Compute first and second derivatives: $$x^{\prime }=-\sin t,y^{\prime }=\cos t,z^{\prime }=1$$ $$x^{\prime \prime }=-\cos t,y^{\prime \prime }=-\sin t,z^{\prime \prime }=0$$
2. For 3D curves, curvature is: $$\kappa =\frac{\sqrt{(x^{\prime }{y}^{\prime \prime }-y^{\prime }{x}^{\prime \prime }{)}^2+(y^{\prime }{z}^{\prime \prime }-z^{\prime }{y}^{\prime \prime }{)}^2+(z^{\prime }{x}^{\prime \prime }-x^{\prime }{z}^{\prime \prime }{)}^2}}{(x^{\prime 2}+y^{\prime 2}+z^{\prime 2}{)}^{3/2}}$$ Plugging in: $$\kappa =\frac{\sqrt{(-\sin t)(-\cos t)-(\cos t)(-\sin t)+0+0+0}}{(1{)}^{3/2}}=\frac{\sqrt{2\sin t\cos t}}{1}=\sqrt{2}|\sin t|$$ (Simplified using $\sin t\cos t=\frac{1}{2}\sin 2t$.)

In the Real World:

• Bank Loan Interest Calculation: While not directly curvature, parametric equations model loan repayment schedules. For example, the "amortization curve" of a loan can be parameterized by time $t$, with $x(t)$ = principal remaining and $y(t)$ = interest paid. The curvature of this curve helps banks assess risk (sharp bends indicate high variability in payments).
• Kathmandu Traffic Flow: Roads can be modeled as parametric curves where $t$ = time, $x(t)$ = position along the road, and $y(t)$ = traffic density. Curvature analysis helps identify congestion hotspots (high $\kappa$ = sudden changes in traffic flow).
• WhatsApp Message Encryption: Parametric curves are used in cryptographic algorithms to generate pseudorandom paths for data packets. The curvature of these paths ensures security by making it hard to predict the "bending" of the data route.

4. Special Parametric Curves

classDiagram
    class Cycloid {
        +x(t) = a(t - sin t)
        +y(t) = a(1 - cos t)
        +Arc length: 8a per rotation
    }
    class Involute {
        +x(t) = a(cos t + t sin t)
        +y(t) = a(sin t - t cos t)
        +Used in gear design
    }
    class Spiral {
        +x(t) = a t cos t
        +y(t) = a t sin t
        +r = a t
    }
    Cycloid --> "Generates" GearTeeth
    Involute --> "Used in" ClockGears
    Spiral --> "Models" DNA

Worked Example 4: Astroid Curve For the astroid $x={\cos }^{3}t$, $y={\sin }^{3}t$:

1. Find Cartesian form: $x^{2/3}+y^{2/3}={\cos }^{2}t+{\sin }^{2}t=1$.
2. Arc length for one quadrant ($t\in [0,\pi /2]$): $$\frac{dx}{dt}=-3{\cos }^{2}t\sin t,\frac{dy}{dt}=3{\sin }^{2}t\cos t$$ $$L={\int }_0^{\pi /2}\sqrt{9{\cos }^{4}t{\sin }^{2}t+9{\sin }^{4}t{\cos }^{2}t}dt=3{\int }_0^{\pi /2}\sin t\cos tdt=\frac{3}{2}$$ (Total arc length for full astroid: $4\times \frac{3}{2}=6a$.)

5. Converting Between Parametric and Cartesian Forms

Strategy:

1. Solve one equation for $t$ (if possible) and substitute into the other.
2. Use trigonometric identities (e.g., ${\sin }^{2}t+{\cos }^{2}t=1$).
3. For implicit curves, eliminate $t$ algebraically.

Worked Example 5: Convert to Cartesian Given $x=e^t+e^{-t}$, $y=e^t-e^{-t}$:

1. Let $u=e^t$, then $x=u+\frac{1}{u}$, $y=u-\frac{1}{u}$.
2. Solve for $u$: $$x^2-y^2={\left(u+\frac{1}{u}\right)}^2-{\left(u-\frac{1}{u}\right)}^2=4⟹x^2-y^2=4$$ This is a hyperbola.

6. Applications in Engineering

1. Robot Arm Trajectories: Parametric equations define the path of a robotic arm’s end-effector. For example: $$x(t)=L_1\cos t+L_2\cos (t+\theta ),y(t)=L_1\sin t+L_2\sin (t+\theta )$$ where $L_1,L_2$ are link lengths and $\theta$ is a joint angle.

   • Arc length ensures the arm moves smoothly without jerks.
   • Curvature helps avoid sharp turns that could damage the arm.

2. Computer-Aided Design (CAD): Parametric curves (e.g., Bézier curves) are used to design car bodies, airplane wings, and 3D-printed objects. The curvature controls the "smoothness" of the design.

3. Physics: Projectile Motion: As shown earlier, parametric equations model the trajectory of a thrown object. The arc length can represent the total distance traveled, while curvature indicates how much the path deviates from a straight line.

flowchart TD
    A["Parametric Equations"] --> B["Robot Arm Path"]
    A --> C["CAD Design"]
    A --> D["Projectile Motion"]
    B --> E["Optimize Arc Length"]
    C --> F["Control Curvature"]
    D --> G["Calculate Range"]

Exam Tip

1. Sketch the Curve:

   • Always draw the curve based on parametric equations. Label key points (e.g., at $t=0,\pi /2,\pi$).
   • For example, in the cycloid problem, show the generating circle and the path traced.

2. Arc Length Formula:

   • Memorize the formula: $L=\int \sqrt{(dx/dt{)}^2+(dy/dt{)}^2}dt$.
   • Common pitfalls:
     • Forgetting to take the square root.
     • Misapplying limits (e.g., $t\in [0,2\pi ]$ for a full circle).
   • Shortcut: For circles/spirals, use symmetry to simplify the integral.

3. Curvature:

   • For 2D curves, use $\kappa =\frac{|x^{\prime }{y}^{\prime \prime }-y^{\prime }{x}^{\prime \prime }|}{(x^{\prime 2}+y^{\prime 2}{)}^{3/2}}$.
   • For 3D (e.g., helix), extend to include $z$ terms.
   • Check units: Curvature should be in $1/\text{length}$ (e.g., $1/\text{m}$).

4. Parametric to Cartesian:

   • If you can’t eliminate the parameter easily, leave it in parametric form but verify consistency (e.g., check if $x^2+y^2=r^2$ holds for a circle).
   • Tip: For trigonometric parameters, use identities like ${\sin }^{2}t+{\cos }^{2}t=1$.

5. Real-World Context:

   • Questions may ask for "practical applications." Link parametric equations to:
     • Optimization: Minimizing arc length (e.g., shortest path for drones).
     • Design: Curvature in road design or product shapes.
     • Physics: Projectile motion or pendulum paths.

6. Common Exam Questions:

   • Part (a): Sketch the curve and find Cartesian form.
   • Part (b): Calculate arc length for a given interval.
   • Part (c): Find curvature at a specific point (e.g., $t=\pi /2$).
   • Part (d): Apply to a real scenario (e.g., "A cycloid is used in gear design. Find the arc length for one rotation.").

Practice Problems

1. Sketch and Convert: Given $x(t)=t^2-1$, $y(t)=t^3-t$, sketch the curve and find $y$ as a function of $x$.

2. Arc Length: Find the arc length of the curve $x(t)=t^2$, $y(t)=\ln t$ from $t=1$ to $t=2$.

3. Curvature: For the curve $x(t)=e^t$, $y(t)=e^{-t}$, find $\kappa$ at $t=0$.

4. Application: A bead slides along a wire shaped like a cycloid $x=t-\sin t$, $y=1-\cos t$. Find the distance traveled by the bead from $t=0$ to $t=\pi$.

Summary Table

Final Note: Mastering this unit requires visualization (sketching curves) and calculation (arc length/curvature). Practice converting between parametric and Cartesian forms, and always verify your results by checking key points. For exams, allocate time to sketch curves—it often hints at the correct approach!