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3. Acceleration and Normal Vectors

In the same way that our position vector traces out a curve as t changes, so does our tangent vector. This suggests that we could discover much from differentiating the tangent vector to obtain the acceleration vector, X''(t).

Demonstration 6: Acceleration Vectors
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The control panels work in much the same way as demonstration on the effects of parametrization on the tangent vector to a curve. The tapedeck allows the user to watch the curve leave a trail of normal vectors. The normals to the curve are displayed in green.

This demonstration allows you to look at the acceleration vectors of a curve using various parametrizations. If the curve is X(u(t)), the first derivative is X'(u(t))u'(t) and so by the chain rule and the product rule, the acceleration vector is X''(u(t))(u'(t))2+ X'(u(t))(u''(t)).

    Look at the acceleration vectors for the curve [t, sin(t)] under parametrizations u(t) = t and u(t) = t2.

The previous demo suggests that a reparametrization can change the length of a velocity vector, but not its direction. The acceleration vector on the other hand can change both length and direction under reparametrization. Nonetheless there are some properties that are independent of reparametrization: if the acceleration vector is in the left-hand half-plane determined by the velocity vector in one parametrization, then it remains so in any (orientation preserving) reparametrization. It turns out that there is another very geometric quantity associated with the acceleration that is independent of reparametrization, namely, the curvature, a function that measures deviation from straightness at each point of the curve. In order to isolate this curvature function, we want to relate the acceleration to a coordinate system given by a specially constructed pair of mutually orthogonal unit vectors that move along the curve, a "moving frame" particularly well adapted to the curve.

To achieve this we do the following: at each point along a curve where the velocity vector is not the zero vector, we divide the velocity vector by its length to define the unit tangent vector,

    T(t) =X'( t)/|| X'(t)||=X '(t)/s'(t)
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A vector perpendicular to the velocity vector at a point is said to be normal to the curve at that point. At each point of a smooth curve, there are two unit normal vectors, pointing in opposite directions. We choose one by defining the unit normal vector U(t) to be the unit tangent vector rotated by 90 degrees in the counter-clockwise direction, so that if T(t) =(a(t),b( t)) , then U(t) =(-b(t),a( t)).

Demonstration 7: Tangent and Normal Vectors of Plane Curves
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You can specify the curve by choosing values for the constant c . The velocity vectors can be scaled to unit tangent vectors with the Scale between X'(t) and T(t) variable, which interpolates the vectors tangent to the curve between X' (variable equals 0) and T (variable equals 1).This demo is similar to the Position and Velocity Vectors on Plane Curves demo . The default function is an ellipse with the length of the major axis controlled by c .

Running the tapedeck in the control panel makes the parameter t run through its domain.

This demo displays a plane curve with its velocity vectors and unit normal vectors emanating from the points on the curve with which they are associated.

    Change the slider c to 1 on the default curve, to make the curve a unit circle. What is the relationship between normal vectors and position vectors? What is the relationship between velocity vectors and unit tangent vectors?

    Choose a curve with an inflection point, like X(t) =(t,t3) . What happens to the normal vectors and to the tangent vectors that emanate from the neighborhood of the inflection point?

    What happens if the curve has a cusp ; for example, X(t) =(t2,t3) ? What happens at the point t=0 for the curve [cos(5*t),t*t*t] ?

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Next: Smooth Curves and Curves with Singular Points