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A parametric surface
Bézier surface example

Parametric forms for lines and vectors

In many situations, it is useful to have an alternative way of describing a curve besides having an equation for it in the plane. A parametric form for a line occurs when we consider a particle moving along it in a way that depends on a parameter , which might be thought of as time. Thus both and become functions of . The simplest parameterisation are linear ones.

In this step we

  • look at parameterisations of the simplest curves: lines

  • show how to go from parametric equations to Cartesian equations

  • introduce vectors.

Parameterisations of a line

A simple example of a parameterisation is a linear parameterisation:

As varies, the point moves along a line. When , , when , , and when , . Clearly also fractional values of are allowed.

An animation showing the trajectory p(t)=[t+1, 2t-3]

How do we go from a parametric form for a line to one of the more usual Cartesian forms involving and ? We eliminate the parameter . For example for the parametric equation we have

so that or . Please check that this is the same line – with a different equation.

Graph of y=2x-5

Q1 (E): Find a Cartesian equation for the parametric line .

How to go from a Cartesian equation to a parametric form?

On the other hand, how can we go from a Cartesian equation of a line, say to a parametric form? It is important to realise that although the Cartesian form is more or less unique, this is not at all the case for a parametric form.

One way is to introduce a parameter in a simple way, say by setting . Then the original equation gives . Putting these together gives the parameterisation .

Q2 (E): Find another parameterisation of by setting .

Using vectors to parameterise lines

The rest of this step is about vectors, for those who are familiar with them. A vector is a directed line segment, or geometrically a difference between two points in the plane. We will denote vectors by round brackets, such as . This represents a relative displacement of to the right in the -direction and up in the -direction. This plays a similar role to the notion of slope, which is the ratio of the relative displacement of the and -directions or “rise over run”. Using vectors allows us to be more precise about the notion of direction of a line.

We agree that vectors are not fixed in place, as they are only determined in a relative sense.

Suppose that we consider the line through the point which goes in the direction . This line can be expressed as

which is now in parametric form with parameter .

What is the Cartesian equation of this line? We may eliminate from the equations , to get so the equation is .

Graph of y=4x-10 showing also the point [3,2] and the vector (1,4)

Now imagine that we would like to traverse the same line, only faster. To move through the same line at double the speed, we replace with :

Notice that the direction vector has doubled, but the Cartesian equation of this line remains unchanged. We can also move backwards along the line by relpacing with :

Now the direction vector has been negated, but again the Cartesian equation remains unchanged.

So we see that the parameterisation of a curve describes more than the Cartesian equation, a parameterisation describes how we traverse the curve: fast or slow, forwards or backwards.

Answers

A1. This is also .

A2. Another parameterisation is .

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Maths for Humans: Linear and Quadratic Relations

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