Motion in Space

If a particle is moving according to the path r(t) (it's also named as position vector), then its velocity, v(t) is found by differentiating r(t) and its acceleration is by differentiating v(t):

The unit vector of the velocity has a special name: unit tangent, T. As we know, tangent line is the line that gives us an idea about the slope of a curve, similar to tangent vector. 

In addition to this, there is a principal unit normal, N, that points to the direction to where the unit tangent T is turning.


In order to calculate the rate at which 𝐓 turns per unit of length along the curve, we look for curvature, 𝜅 (“kappa”).

The graph below gives an example for the calculation mentioned above. Slide the value of t in the upper box to move point A along the curve. Observe the changes on the unit tangent and the pricipal unit normal. In this case, the curvature remain constant throughout all values of t. You can always click here if you want to rotate the graph.

For every curve, there exists a circle of curvature with radius . This circle of radius satisfies the following critiria:

1. It is tangential to the respective curve.
2. It has the same curvature as the respective.
3. It lies towards the concave or the inner side of the curve.

The green curve in the graph below is the graph of . The blue line is the tangent line of the curve, and the red circle is the circle of curvature. Try to slide the value of t to see the changes of the radius. You can get the same effect by moving point A.

Betty, created with GeoGebra and Thomas' Calculus, Pearson.