$γ(t) = (x(t),y(t))$,What is parameter t mean here?

Question

Website is here Curvature from Wikipedia

In the 1.3 Local expressions,first line.

Can somebody explain parameter t for me?Also in the second line and fourth line,I don't know the different between two formulas which one has absolute value and one has not.

Can somebody help me ?Thank you very much!

Answer 1

The parameter $t$ is used here since the plane curve is given parametrically. You can think of the equations $x(t)$ and $y(t)$ as defining a single point in the Cartesian coordinate system at a "time" $t$. Hence, $t$ acts as a parameter for defining a point $(x,y)$.

As for the differences between the two formulas, one has an absolute value since it represents only the curvature, while the one without represents the signed curvature (note the difference in symbols $\kappa$ and $k$).

An explanation of the difference between the two is provided from the Wiki page:

The sign of the signed curvature k indicates the direction in which the unit tangent vector rotates as a function of the parameter along the curve. If the unit tangent rotates counterclockwise, then k > 0. If it rotates clockwise, then k < 0. So, for example, the sign of the curvature of a graph is the same as the sign of the second derivative.

Hopes this helps.