Curvature of a curve

Krümmung ist ein Begriff aus der Mathematik, der in seiner einfachsten Bedeutung die lokale Abweichung einer Kurve von einer Geraden bezeichnet. Der gleiche Begriff steht auch für das Krümmungsmaß, welches für jeden Punkt der Kurve quantitativ angibt, wie stark diese lokale Abweichung ist. In this section we want to briefly discuss the curvature of a smooth curve (recall that for a smooth curve we require is continuous and ). In general, there are two important types of curvature : extrinsic curvature and intrinsic curvature. The formal definition of .

Let a plane curve C be defined parametrically by the radius vector r(t). While a point M moves along the curve C, the direction of the tangent changes (Figure 1). This ratio ΔφΔs is called the . Consider a plane curve defined by the equation y=f(x). Suppose that the tangent line is drawn to the curve at a point M(x,y). At the displacement Δs along the arc of the curve , the point M moves to the point M1.

Equally true, your resulting trajectory will be bent in a tighter curve. Thus the rate of change of direction-facing measures curvature of your .

For a parametrically defined curve we had the definition of arc length. Since vector valued functions are parametrically defined curves in disguise, we have the same definition. We are now going to look at a very important property of a curve defined by a space curve known as the curvature at a point on the curve.

Curvature at a Point on a Curve. In essence, the curvature at a point on is defined to be the rate at which the unit tangent vector changes direction. Proof: Let be a plane polar curve.

To prove Theorem we will compute the necessary components and plug them into the formula for curvature. We first compute as: (1). We now compute the cross product as follows: (3). We want to find a measure of how 'curved' a curve is. Therefore we have that is: (4).

Further, the measure of curvature should agree with our intuition in simple special cases. Straight lines themselves have . Since the tangent line or the velocity vector shows the direction of the curve, this means that the curvature is, roughly, the rate at which the tangent line or velocity vector is turning. Let $ f$ be a function and $ P$ be a point on the graph of $ f$.

If C is traversed exactly once as t increases from a to b, then its length is.

Example: Find the length of the curve defined by R( t) .