Parametric Curves
Vector-valued functions
A function is called a vector-valued function if the output of the function is a vector (instead of a real number). In particular, a vector-valued function of the following form
where are real-valued functions and , can trace out a curve in as varies over such that each point on the curve has as its position vector for some . This curve is usually called a parametric curve. In other words, the curve is parametrized by .
The parametric equation of a line can be regarded as the following vector-valued function:
,
such that its parametric curve is the line passing through in the direction of .
(Note: If we want to parametrize a line segment instead, we need to choose a finite range of . For example, how can we parametrize a line segment from to ?)
In the applet below, the parametric curve is shown for a given . It is a helix. You can change the functions and the range of to obtain a different curve.
Examples:
An ellipse on the plane : , .
A spiral on the surface of a cone: , .
A "roller coaster" curve: ,
Remarks:
- A curve can be parametrized by different vector-valued functions. For example, with parametrizes the same helix in the above applet.
- When a curve is parametrized by , a positive orientation is given to the curve - the direction in which the curve is generated as increases from to .
- If a vector-valued function gives 2D vectors as output, the function is in the following form: .
Limit and continuity
The limit of a vector-valued function is defined in terms of the limit of its components as follows:
That is to say, suppose , where is a vector in 3D space. Then we have
.
Computing the limit of a vector-valued function boils down to computing the limits of its components. Moreover, exists if and only if exist.
Another equivalent definition of the limit:
Example:
Let . Then we have
Definition: is said to be continuous at if i.e. if are continuous at .
Remarks:
- The definition of limit of a vector-valued function with 2D vectors as outputs can be similarly definied.
- If is continuous function of , then the parametric curve it describes has no breaks or gaps, a property necessitated by the trajectory of an object.