Parametric Representations of Curves

Suppose . What is ?

Notice:

note that is also a single variable function with respect to , since a change in only will change both and .

To better understand this equation, we need to better understand ,
This is a parametric representation of a path, meaning it has a start and a finish.

Example

Example

This is an equation of a circle:

If you set , you go around 3 times. You can't write that as a cartesian function.

We can go backwards by using :

(see function symmetry)

Example

What is the parametric representation of the blue path

solution
We can decompose the path as:

which is a ball in pure rolling.

Rotation motion
We first try and find .

is close, but the light is at (6 o'clock) when , so we should add a phase shift.

This doesn't work because we have when . So we just need to offset the y-component by

(see Trig Identities#Symmetry Identities).

Translational Motion
The centre of the circle moves at the velocity:

What is ?

After , (why? because it makes a full rotation according to ), so

Together,

Derivatives of Parametric Curves

Remark

Like normal vectors, the rate of change represents the next "measurement":
If the path denotes position in time,

Example

The graphs for these two functions are the same:

and

since the ratios between and are the same.

But their velocities are not:

and