Parametric Equations

What are Parametric Equations?

Parametric equations: Express coordinates as functions of a parameter (usually t)

Format:

x = f(t)
y = g(t)

Parameter t: Independent variable (often represents time)

Why use: Some curves easier to describe parametrically

Example: Simple Line

Parametric:

x = t
y = 2t + 1

As t varies, traces out line y = 2x + 1

Graphing Parametric Curves

Steps:

1. Make table of t-values
2. Calculate corresponding (x, y) points
3. Plot points
4. Connect in order of increasing t
5. Show direction with arrows

Example 1: Create Table

x = t, y = t² for -2 ≤ t ≤ 2

Curve is parabola y = x², traced from left to right

Example 2: Circle

x = 3cos(t), y = 3sin(t) for 0 ≤ t ≤ 2π

Circle of radius 3, traced counterclockwise

Converting to Rectangular Form

Eliminate parameter t to get equation in x and y

Methods:

• Solve one equation for t, substitute into other
• Use trigonometric identities
• Algebraic manipulation

Example 1: Solve for t

x = t + 2, y = 3t - 1

From first equation: t = x - 2

Substitute into second:

y = 3(x - 2) - 1
y = 3x - 6 - 1
y = 3x - 7

Rectangular form: y = 3x - 7

Example 2: Using Trig Identity

x = 2cos(t), y = 2sin(t)

From equations:

cos(t) = x/2
sin(t) = y/2

Use sin²(t) + cos²(t) = 1:

(y/2)² + (x/2)² = 1
y²/4 + x²/4 = 1
x² + y² = 4

Rectangular form: Circle with radius 2

Example 3: Parabola

x = t, y = t² - 4t + 3

Since x = t:

y = x² - 4x + 3

Rectangular form: Parabola

Converting from Rectangular to Parametric

Infinitely many parametrizations possible!

Common approach: Let x = t, then express y in terms of t

Example 1: Line

Rectangular: y = 2x + 3

Parametric:

x = t
y = 2t + 3

Or alternatively:

x = 2t
y = 4t + 3

Example 2: Circle

Rectangular: x² + y² = 9

Parametric (using trig):

x = 3cos(t)
y = 3sin(t)

Or (using other parameter):

x = 3cos(2t)
y = 3sin(2t)

(Traces circle twice as fast)

Orientation

Direction curve is traced as t increases

Show with arrows on graph

Same curve can have different orientations

Example: Opposite Orientations

Clockwise circle:

x = 2cos(t)
y = -2sin(t)

Counterclockwise circle:

x = 2cos(t)
y = 2sin(t)

Same circle, opposite directions

Parametric Curves Not Functions

Advantage: Can represent curves that fail vertical line test

Example: Vertical Line

Rectangular: x = 3 (not a function of x!)

Parametric:

x = 3
y = t

Easily represented parametrically

Example: Circle

Rectangular: x² + y² = 25 (not a function)

Parametric:

x = 5cos(t)
y = 5sin(t)

Clean parametric form

Ellipses

General ellipse:

x = a·cos(t)
y = b·sin(t)

Traces ellipse x²/a² + y²/b² = 1

Example: Ellipse

x = 4cos(t), y = 3sin(t)

Rectangular form:

cos(t) = x/4
sin(t) = y/3

(x/4)² + (y/3)² = 1
x²/16 + y²/9 = 1

Ellipse with a = 4, b = 3

Cycloids

Path traced by point on rolling circle

Parametric equations:

x = r(t - sin(t))
y = r(1 - cos(t))

Where r = radius of circle

Famous curve in physics and mathematics!

Example: Unit Cycloid

r = 1:

x = t - sin(t)
y = 1 - cos(t)

Creates arch-like pattern

Lissajous Curves

Combination of perpendicular oscillations

Form:

x = A·sin(at + δ)
y = B·sin(bt)

Creates beautiful symmetric patterns

Example: Simple Lissajous

x = sin(t), y = sin(2t)

Figure-eight pattern

Real-World Applications

Physics: Projectile motion

x = v₀t·cos(θ)
y = v₀t·sin(θ) - (1/2)gt²

Engineering: Gear tooth profiles, cam design

Computer graphics: Animation paths, curves

Astronomy: Planetary orbits

Robotics: Path planning

Example: Projectile Motion

Ball thrown at 45° with initial velocity 20 m/s

Parametric equations:

x = 20t·cos(45°) = 14.14t
y = 20t·sin(45°) - 4.9t² = 14.14t - 4.9t²

Horizontal position and vertical position as functions of time

Restrictions on Parameter

Domain of t affects curve traced

Example: Partial Circle

x = cos(t), y = sin(t) for 0 ≤ t ≤ π

Only traces semicircle (top half)

Full circle needs 0 ≤ t ≤ 2π

Speed and Arc Length

Parametric form useful for calculating arc length

Speed:

v(t) = √[(dx/dt)² + (dy/dt)²]

Arc length: Integrate speed

Example: Speed on Circle

x = 3cos(t), y = 3sin(t)

Derivatives:

dx/dt = -3sin(t)
dy/dt = 3cos(t)

Speed:

v = √[9sin²(t) + 9cos²(t)]
  = √9
  = 3

Constant speed of 3 (radius × angular velocity)

Practice