### Geometry: Parametric curves

### Vectors and parametric equations

It is useful to think of vectors as arrows in the plane. One can also think of a vector #\blue{\vec x = \cv{ x_0 \\ y_0 }}# as the point #\ivcc{x_0}{ y_0}#, i.e. forgetting the arrow but not the end point. This allows us to relate vectors to parametric equations and by extension to real valued functions.

More precisely, a parametric curve given by equations #\ivcc{\blue{x(t)}}{ \green{y(t)}}# can be viewed as a "vector valued" function that maps #t# to the vector #\cv{ \blue{x(t)} \\ \green{y(t)} }#.

One example of this is the way one can describe a line in the #x,y#-plane by means of a "support vector" and a "direction vector."

**Support vector and direction vector determine a line.**

Let #\orange{ l }\colon { y } = { 3 x +2}# be a line in the #x, y#-plane. The points on this line are given by #\ivcc{t} {3t+2}# for #t# any real number. Define a vector valued function #F(t) = \cv{ t \\ 3t + 2 }#. We now rewrite this function using the rules of vector addition and scalar multiplication. We get

\[\begin{array}{rcl} F(t) & = & \cv{ t \\ {3t + 2 } }\\ && \phantom{xxx} \blue { \text{ copied the vector valued function rule } } \\ & = & \cv{ t \\ 3t }+ \green{\cv { 0 \\ 2 }}\\ && \phantom{xxx} \blue { \text{ used the definition of vector addition } } \\ & = & t \cdot \blue{\cv{ 1 \\ 3 }} +\green{ \cv { 0 \\ 2 } }\\ && \phantom{xxx} \blue { \text{ used the definition of scalar multiplication } }\end{array}\]

If one now thinks of vectors as points in space instead of arrows in space we recover the line #l# as the range of #F#.

We call the vector #\blue{\cv{1 \\ 3 }}# the **direction vector** and the vector #\green{\cv{ 0 \\ 2 }}# the **support vector**. The support vector and direction vector are not unique.

You can see a picture of this vector valued function in the block "Picture".

The previous example can be generalized. It will now be possible to use all notions on vectors to study parametric curves.

Let #\orange{C}# be a parametric curve given by by the parametric equations\[\blue{x=x(t)}\\\green{y=y(t)}\] Then the vectorvalued function \[F(t)=\cv{\blue{x(t)}\\\green{y(t)}}\] describes the curve.

Here is a picture where #\blue{x(t) = \cos(2t)}# and #\green{y(t) =\sin(t) + \cos(t)}#. The point #\orange{P_t}# is the curve at time #t#.

**Picture**

The support vector is # \matrix{ -5 \\ 5 }#.

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