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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 ) ) .

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