Now that we have familiarize ourselves with the terms velocity and acceleration in 3 dimensional vectors, we will introduce a new term called curvature, and later another closely related one called radius of curvature.
While it may still be premature to describe these new terms geometrically, we will slowly introduce such geometric properties which will be built upon in the later sections.
VECTOR DIFFERENTIAL CALCULUS >
CURVATURE.1
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CURVATURE.2
VELOCITY AND ACCELERATION
ACCELERATION COMPONENTS
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CONCEPT
We recap from the previous section that when
we have
I have previously shown that
provided that We will now define the new term. The curvature
The Greek symbol
assuming that
Next, we shall use the curvature and radius of curvature to define another two terms. The unit normal vector to the curve at the point P is given by
We can verify that this is a unit vector, by simply taking its magnitude like so
Morever, N is orthogonal to T. To prove this, we recall a basic property of the dot product which is the a vector dot by itself is equal to it’s magnitude squared. Thinking along this lines and differentiating the equation, we get
But since Next, we shall see how we can write the acceleration vector in terms of these two components, All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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, 
and 
is in the direction of the tangent to the trajectory at the point 
defined by parameter t. I have then later shown that if we parameterize the curve in terms of its arc length s along the curve from some initial point, then 
has unit length and thus is the unit tangent vector. In cases where it is inconvenient to introduce the variable s as finding the equation 
may prove difficult, we obtain the unit tangent vector by dividing 
by its length. And so we can then defined the unit tangent vector 
as,
. If 
, we do not define a tangent vector to the curve at 
. If 
, and this equation gives us 
, which again is consistent with our previous definitions.
of a curve in three-space is the magnitude of the rate of change of the tangent vector with respect to s, 
is pronounced as kappa. By looking at the equation, we suspect that the value of 
gives us the value of what we call the “amount of bending” or curvature. For example, the greater the magnitude of 
, the more the curve bends for a given change in arc length. A straight line has a constant tangent vector and so it’s curvature is 0. By using calculus, we can show that in terms of the position vector 
,
. We will rarely use the above equation.
, the reciprocal of the curvature, is called the radius of curvature of a curve, provided that 
. The Greek symbol 
is called rho. It’s value gives us the radius of a circle that best approximates the curve C at the point P on its concave side. Such a circle is called the osculating circle to the curve at P.
is orthogonal to 
, and 
is a positive scalar multiple of 
and so in the same direction as 
, we conclude that 
is orthogonal or perpendicular to 
.
and 
.
