Parametric Equations

Introduction to Parametric Equations

In mathematics, equations are often written to show how one variable depends directly on another. For example, the Cartesian equation y = 2x + 1 describes a straight line by expressing y in terms of x.

However, some relationships are easier to describe when both x and y depend on a third variable. This approach uses parametric equations.

A parametric equation describes the coordinates of a point using a separate variable called a parameter, usually represented by t.

A pair of parametric equations has the general form:

x = f(t)

y = g(t)

where:

• x represents the horizontal coordinate.

• y represents the vertical coordinate.

• t is the parameter.

• f(t) and g(t) describe how the coordinates change as t changes.

In many applications, t represents time. This makes parametric equations especially useful for describing moving objects, trajectories, circular motion, and changing positions.

Understanding Parametric Equations

Consider the equations:

x = 2t

y = t + 1

Both coordinates depend on t.

By choosing different values of t, we can determine different points on the graph.

Plotting these points produces a straight line.

As t increases, the point moves from the lower-left region of the graph toward the upper-right region.

This introduces an important feature of parametric equations: they describe not only the shape of a curve, but also the direction in which it is traced.

The Role of the Parameter

The parameter t may represent:

• Time in seconds.

• An angle in radians.

• Distance traveled along a path.

• A position within a sequence.

• Another quantity that controls the coordinates.

The parameter does not always represent time, although time is especially common in physics and engineering.

Parameter Restrictions

Sometimes the parameter is restricted to a particular interval.

For example:

x = 2t

y = t + 1

0 ≤ t ≤ 3

This describes only a line segment, not the entire line.

The starting point is:

t = 0 → (0, 1)

The ending point is:

t = 3 → (6, 4)

The restriction on t determines which part of the curve is traced.

Converting Parametric Equations to Cartesian Form

Parametric equations can often be converted into Cartesian equations by eliminating the parameter.

The goal is to find a relationship between x and y without using t.

Worked Example: Converting a Linear Relationship

Given:

x = 3t + 2

y = 2t − 1

Step 1: Rearrange one equation to isolate t.

From x = 3t + 2:

x − 2 = 3t

t = (x − 2)/3

Step 2: Substitute into the second equation.

y = 2((x − 2)/3) − 1

Step 3: Simplify.

y = (2x − 4)/3 − 1

y = (2x − 7)/3

Final answer:

y = (2/3)x − 7/3

This is a straight line with slope 2/3 and y-intercept −7/3.

If t can take any real value, the parametric equations describe the entire line.

Worked Example: Converting a Parabolic Relationship

Given:

x = t + 1

y = t² − 4

Step 1: Isolate t.

t = x − 1

Step 2: Substitute into y.

y = (x − 1)² − 4

Final answer:

y = (x − 1)² − 4

This is a parabola with vertex (1, −4).

Converting Cartesian Equations to Parametric Form

A Cartesian equation can be represented parametrically in many different ways.

For example:

y = x²

One simple parameterization is:

x = t

y = t²

Another valid parameterization is:

x = 2t

y = 4t²

Both describe the same parabola when t ranges over all real numbers.

However, they trace the curve at different rates as the parameter changes.

Worked Example: Parameterizing a Straight Line

Convert:

y = 3x − 2

into parametric form.

Choose:

x = t

Then:

y = 3t − 2

Final answer:

x = t

y = 3t − 2

where t ∈ ℝ.

Parameterizing a Circle

A circle centred at the origin with radius r has Cartesian equation:

x² + y² = r²

Using trigonometric functions, we can write:

x = r cos t

y = r sin t

For example, a circle with radius 5 can be represented by:

x = 5cos t

y = 5sin t

For 0 ≤ t ≤ 2π, the point traces one complete circle counterclockwise, beginning at (5, 0).

Sketching Parametric Curves

Sketching a parametric curve involves finding points and determining the order in which they are traced.

A useful method is:

1. Identify the permitted values of t.

2. Calculate x and y for selected parameter values.

3. Plot the resulting coordinates.

4. Connect the points with a smooth curve where appropriate.

5. Add arrows showing the direction as t increases.

6. Identify starting points, endpoints, and any repeated sections.

Worked Example: Sketching a Parabola

Given:

x = t

y = t² − 1

For −2 ≤ t ≤ 2.

Construct a table:

The Cartesian equation is:

y = x² − 1

The graph is an upward-opening parabola.

The point begins at (−2, 3), moves downward toward (0, −1), then moves upward toward (2, 3).

Why Direction Matters

A Cartesian equation describes the set of points on a curve.

A parametric representation can additionally describe:

• The starting point.

• The ending point.

• The direction of movement.

• Whether a point is visited more than once.

• How position changes as the parameter changes.

For example:

x = cos t

y = sin t

traces a circle counterclockwise as t increases.

However:

x = cos t

y = −sin t

traces the same circle clockwise when t increases from 0 to 2π.

The curve is the same, but the direction is different.

Parametric Equations and Circular Motion

Circular motion is one of the most important applications of parametric equations.

For an object moving around a circle centred at the origin:

x = r cos(ωt)

y = r sin(ωt)

where:

• r is the radius.

• ω is the angular speed in radians per second.

• t is time in seconds.

The angle traveled after time t is:

θ = ωt

For constant angular speed, the object completes one revolution in:

T = 2π/|ω|

where T is the period and ω ≠ 0.

Worked Example: A Rotating Wheel

A point on a wheel moves in a circle of radius 4 m.

The wheel rotates counterclockwise at 2 rad/s.

Its position is modeled by:

x = 4cos(2t)

y = 4sin(2t)

Find the position at t = π/4 seconds.

Step 1: Calculate the angle.

θ = 2(π/4)

θ = π/2

Step 2: Calculate x.

x = 4cos(π/2)

x = 0

Step 3: Calculate y.

y = 4sin(π/2)

y = 4

Final answer:

The point is at (0, 4).

It has rotated 90° counterclockwise from its initial position.

Parametric Equations and Motion

In physics, parametric equations are used to describe the position of a moving object over time.

For two-dimensional motion:

x = x(t)

y = y(t)

The position at any time can be determined by substituting the time value into both equations.

Velocity Components

The horizontal and vertical velocity components are the rates of change of position.

vₓ = dx/dt

vᵧ = dy/dt

The speed is:

v = √(vₓ² + vᵧ²)

These relationships are particularly useful when studying motion in two dimensions.

Acceleration Components

Acceleration describes the rate of change of velocity.

aₓ = dvₓ/dt = d²x/dt²

aᵧ = dvᵧ/dt = d²y/dt²

These equations allow parametric motion to be analyzed using calculus.

Projectile Motion

Projectile motion is a classic example of a parametric model.

Ignoring air resistance, the horizontal and vertical positions of a projectile are:

x = x₀ + v₀ cos θ · t

y = y₀ + v₀ sin θ · t − ½gt²

where:

• x₀ and y₀ are the initial coordinates.

• v₀ is the initial speed.

• θ is the launch angle.

• g is gravitational acceleration, approximately 9.8 m/s².

• t is time.

The horizontal position changes linearly with time, while the vertical position changes quadratically.

Worked Example: Projectile Motion

A ball is launched from ground level at 20 m/s at an angle of 30° above the horizontal.

Find its position after 1 second.

Step 1: Write the parametric equations.

x = 20cos 30° · t

y = 20sin 30° · t − 4.9t²

Step 2: Substitute t = 1.

x = 20cos 30°

x ≈ 17.32 m

y = 20sin 30° − 4.9

y = 10 − 4.9

y = 5.1 m

Final answer:

After 1 second, the ball is approximately 17.32 m horizontally from its starting point and 5.1 m above the ground.

Determining Maximum Height

For vertical projectile motion:

vᵧ = v₀ sin θ − gt

At maximum height:

vᵧ = 0

Therefore:

t = (v₀ sin θ)/g

For the previous example:

t = 10/9.8

t ≈ 1.02 s

The maximum height is approximately:

h = (10²)/(2 × 9.8)

h ≈ 5.10 m

The parametric model allows both the position and motion of the projectile to be analyzed.

Modeling Real-World Situations Parametrically

Parametric equations are valuable when a situation involves movement along a path.

Application: A Ferris Wheel

A Ferris wheel has a radius of 12 m and its centre is 15 m above the ground.

The wheel completes one revolution every 40 seconds.

A passenger begins at the lowest point.

The angular speed is:

ω = 2π/40

ω = π/20 rad/s

The passenger's position relative to the ground can be modeled by:

x = 12sin(πt/20)

y = 15 − 12cos(πt/20)

At t = 0:

x = 0

y = 3

The passenger begins 3 m above the ground.

At t = 20 seconds:

x = 0

y = 27

The passenger reaches the highest point, 27 m above the ground.

At t = 40 seconds, the passenger returns to the starting position.

This model demonstrates how parametric equations can describe both horizontal and vertical position as time changes.

Application: A Moving Robot

A robot travels along a straight path according to:

x = 2t + 1

y = 3t − 2

where t is measured in seconds.

At t = 0:

(x, y) = (1, −2)

At t = 4:

(x, y) = (9, 10)

The robot's horizontal velocity is:

vₓ = 2 units/s

Its vertical velocity is:

vᵧ = 3 units/s

Its speed is:

v = √(2² + 3²)

v = √13

v ≈ 3.61 units/s

This type of model can be used in robotics, computer animation, and automated navigation.

Parametric Curves That Cross Themselves

Some parametric curves form loops or intersect themselves.

For example:

x = sin(2t)

y = sin t

For 0 ≤ t ≤ 2π.

This produces a figure-eight-shaped curve.

The point passes through the origin more than once.

Such curves demonstrate why the parameter is important: the same position may be reached at different times or parameter values.

Interpreting Repeated Positions

When a curve crosses itself, the object may pass through the same point at different parameter values.

However, its direction of travel may be different each time.

This is important in motion analysis because position alone does not fully describe the motion.

Velocity and direction must also be considered.

Comparing Cartesian and Parametric Representations

Neither representation is always better.

Cartesian equations are often convenient for studying functions and algebraic relationships.

Parametric equations are particularly useful when position, direction, and time are important.

Common Mistakes

Treating the parameter as an ordinary coordinate

The parameter t is not necessarily an x- or y-coordinate. It controls both coordinates.

Forgetting to eliminate the parameter completely

When converting to Cartesian form, the final relationship should involve x and y without t.

Ignoring parameter restrictions

A restricted parameter interval may produce only part of a line, parabola, or circle.

Sketching the curve without direction arrows

A complete parametric sketch should indicate the direction as t increases.

Confusing speed with velocity

Velocity includes direction, while speed is the magnitude of velocity.

Using degrees instead of radians

Calculus-based parametric motion formulas involving trigonometric functions require angles measured in radians.

Assuming every curve can be written as a single function y = f(x)

Circles and curves with loops may require implicit Cartesian equations or multiple functions.

Forgetting initial conditions

In motion problems, the initial position and direction determine the correct parametric model.

Failing to check physical restrictions

Time is often restricted to t ≥ 0, and projectile motion may end when the object reaches the ground.

Key Terms

• Parametric equation: An equation that expresses a coordinate or quantity in terms of a parameter.

• Parameter: An independent variable used to control the values of other variables.

• Parametric curve: A curve formed by plotting coordinates determined by parametric equations.

• Cartesian equation: An equation expressing a relationship between x and y coordinates.

• Parameterization: The process of representing a curve or relationship using a parameter.

• Eliminating the parameter: Removing the parameter to obtain a Cartesian relationship.

• Parameter interval: The permitted range of values for the parameter.

• Direction of travel: The direction in which a point moves along a curve as the parameter increases.

• Starting point: The position corresponding to the initial parameter value.

• Endpoint: The position corresponding to the final parameter value.

• Trajectory: The path followed by a moving object.

• Position function: A function describing an object's position as time changes.

• Velocity: The rate of change of position, including direction.

• Speed: The magnitude of velocity.

• Acceleration: The rate of change of velocity.

• Circular motion: Movement along a circular path.

• Angular speed: The rate at which angular position changes.

• Period: The time required to complete one cycle or revolution.

• Projectile motion: Motion under the influence of gravity, often modeled parametrically.

• Initial conditions: The starting values of position, velocity, or other quantities.

• Self-intersecting curve: A curve that passes through the same point at different parameter values.

• Mathematical model: A mathematical representation of a real-world system or process.

Key Takeaways

• Parametric equations describe coordinates using a third variable called a parameter.

• A parametric curve is commonly represented by x = f(t) and y = g(t).

• The parameter often represents time, making parametric equations useful for motion.

• Parametric equations can be converted into Cartesian form by eliminating the parameter.

• Cartesian equations can often be parameterized in several different ways.

• The same geometric curve may have different parametric representations.

• Parametric equations describe not only the shape of a curve but also its direction and the order in which points are visited.

• Parameter restrictions determine which portion of a curve is traced.

• Tables of parameter values help construct and interpret parametric graphs.

• Circular motion can be modeled using sine and cosine functions.

• Projectile motion can be described using separate horizontal and vertical position equations.

• Velocity and acceleration can be calculated from parametric position functions using differentiation.

• Parametric equations are important in physics, robotics, engineering, navigation, animation, and mathematical modeling.

• Checking parameter intervals, initial conditions, units, and physical restrictions is essential when interpreting results.

Check Your Understanding

Recall and Understanding

1. What is a parameter in a parametric equation?

2. Explain how parametric equations differ from Cartesian equations.

3. Why are parametric equations useful for describing motion?

4. What does the parameter interval determine when graphing a parametric curve?

5. Why is it important to indicate direction when sketching a parametric curve?

Application

6. Convert x = 2t + 3 and y = 4t − 1 into Cartesian form.

7. Convert x = t − 2 and y = t² + 1 into Cartesian form.

8. Write parametric equations for the line y = 5x − 3.

9. Sketch the curve x = t, y = t² for −2 ≤ t ≤ 2, showing the direction of travel.

10. Write parametric equations for a circle centred at the origin with radius 7.

Analysis and Challenge

11. A particle moves according to x = 3t + 2 and y = 4t − 1. Determine its position and speed at t = 5 seconds.

12. A wheel of radius 6 m rotates counterclockwise at 3 rad/s. Write parametric equations for a point on its rim beginning at (6, 0).

13. A projectile is launched at 25 m/s at an angle of 40° above the horizontal. Write parametric equations for its motion and calculate its position after 2 seconds.

14. Explain how two different parametric equations can describe the same curve but produce different directions of travel.

15. Investigate the curve x = sin(2t), y = sin t for 0 ≤ t ≤ 2π. Identify where the curve intersects itself and explain how the parameter reveals the order in which the curve is traced.

Final Summary

Parametric equations provide a flexible way to describe mathematical relationships by expressing coordinates in terms of a shared parameter.

They are especially useful for representing curves, describing motion, and modeling situations where position changes over time. Unlike ordinary Cartesian equations, parametric equations can communicate both the shape of a path and the direction in which it is followed.

By learning to convert between Cartesian and parametric forms, sketch curves, analyze motion, and construct mathematical models, students develop an important foundation for advanced mathematics, calculus, physics, and engineering.