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Thread: The meaning of the curvature

  1. #1
    Dec 2008

    The meaning of the curvature

    Basically, the curvature of a curve at a given point measures how quickly the curve changes direction at that point. But what exactly does that mean? What "direction" changes.... is it an angle? Suppose the curvature is 2. How would I interpret that number as the rate of how a curve changes? There must be some physical meaning behind the number.

    Another question - am I correct to assume that a vector function, when differentiated twice, can't denote direction in 3D as opposed as in 2D. In 2D, the acceleration denotes the rate of how the slope changes, negative or positive. But in 3D.... the acceleration function is basically denotes the rate of how the vector rate changes with no data based on 3D direction?

    You getting my drift?

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  2. #2
    MHF Contributor Calculus26's Avatar
    Mar 2009
    Actually both your questions are related.

    given r(t) r " (t) is the acceleration.

    There are 2 factors in acceleration change in direction and change in speed.

    The change in speed is simply d(|dr/dt|)/dt this is called the tangential component of acceleration denoted aT. If s is the distance traveled along the curve then ds/dt is the speed and d^2 s/dt^2 is the tangential component

    The change in direction is related to the curvature k = |dtheta/ds|
    where theta is the angle the tangent vector to the curve makes wrt
    the horizontal. So curvature is the rate at which this angle changes with respect to the distance s traveled along the curve. If K is large the direction changes rapidly with respect to a small distance traveled along the curve.

    The normal component of acc denoted aN is k(ds/dt)

    You should soon study these in detail when you consider the arclength parameterization of a curve and the details will be filled in.
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