]> 15.2 Intrinsic Properties of a Curve

## 15.2 Intrinsic Properties of a Curve

First, what are they? What are the intrinsic local properties of the curve?

A straight line has a direction which we can describe by a unit vector in that direction:

Thus the equations $x = 2 t , y = 3 t , z = t$ describe a line that has the direction of the vector $( 2 , 3 , 1 )$ and of the unit vector $( 2 , 3 , 1 ) 14$ .

A general differentiable curve is one that looks like a straight line when looked at over a sufficiently short interval. Thus at any point it has a slope and that slope will in general be in the direction of the vector $( d x d t , d y d t , d z d t )$ , which we will call $v ⟶ ( t )$ .

We define $T ^ ( t )$ to be a unit vector in the direction of $v ⟶ ( t )$

$T ^ ( t ) = v ⟶ ( t ) | v ⟶ ( t ) |$

We define one more parameter $s ( t )$ which represents the distance along the curve between where you are at $t = 0$ and where you are on it at argument $t$ .

The intrinsic information about the curve is contained in the relation between $T ^ ( t )$ and $s ( t )$ , between the tangent vector and the distance parameter along the curve.

To a first approximation, the curve at any point is characterized by its slope there, which is the direction of $T ^ ( t )$ or $T ^ ( t ( s ) )$ .