Home  18.013A  Chapter 15 


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 $x=2t,y=3t,z=t$ describe a line that has the direction of the vector $(2,3,1)$ and of the unit vector $\frac{(2,3,1)}{\sqrt{14}}$ .
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 $\left(\frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt}\right)$ , which we will call $\stackrel{\u27f6}{v}(t)$ .
We define $\widehat{T}(t)$ to be a unit vector in the direction of $\stackrel{\u27f6}{v}(t)$
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 $\widehat{T}(t)$ 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 $\widehat{T}(t)$ or $\widehat{T}(t(s))$ .
