Decomposition of Acceleration
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) Text Suppose some object is traveling along a curve. Geodesic curvature describes quantitatively the change in the direction of the object's travel at some point on the curve.
Consider a parameterized curve X(t) = (x(t),y(t)) with velocity vector X'(t) = (x'(t),y'(t)). The unit tangent vector is defined as T(t) = (x'(t),y'(t))/s'(t) and the unit normal is defined as U(t) = (y'(t),x'(t))/s'(t). As long as T(t) and U(t) exist and are nondegenerate, they form a basis for R^{2}. Therefore, we can write the accleration vector as a linear combination of T(t) and U(t):
X''(t) = a(t)T(t) + b(t)U(t)
By differentiating the expression for the velocity vector, X'(t)=s'(t)T(t), we obtain
X''(t) = s''(t)T(t) + s'(t)T'(t)
Observe that T(t) is a unit vector, so that T(t)·T(t) = 1 and taking the derivative of both sides gives 2T(t)·T'(t) = 0. So T'(t) is perpendicular and T(t) and it follows that T'(t) must be a multiple of U(t). We define T'(t) = κg(t)s'(t)U(t) and write the acceleration vector as
X''(t) = s''(t)T(t) + κg(t)s'(t)2U(t)
The function κg(t) is the geodesic curvature of the curve at the point X(t).
Demos
Decomposition of Acceleration
 
This demo allows you to examine the curvature of a curve X(t) for various values of t0 which you can scroll through using the tapedeck in the control panel. In the Curve window, you can see the acceleration vector of the curve at X(t0) in magenta as well as its components in the tangent and normal directions. In a second window, we graph the geodesic curvature function κg(t). Observe the relationship between the acceleration vector and the unit normal vector at the current position on the curve. The red vector, which is the projection of the acceleration vector onto the unit normal vector, is related to the curvature of the curve. Watch how the change in this projection varies while the point (t,κg(t)).

Exercises 1. Examine the acceleration vectors and curvature of the ellipse X(t) = (c*cos(t), sin(t)).
2. Examine the acceleration vectors and curvature of the cardioid X(t) = (1 + cos(t))*(cos(t), sin(t)).
3. Examine the acceleration vectors and curvature of the exponential spiral: X(t) = e^t*(cos(t),sin(t)).
