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3. Unit Tangent and Principal Normal Vectors

We have seen in the lab on plane curves how useful the concept of a moving frame can be (cf. lab1: Acceleration and Normal Vectors). In space, too, we want to find ourselves a frame which to resolve the acceleration vector.

At each point along a curve where the velocity vector is not the zero vector, it is possible to divide it by its length to define a unit tangent vector T(t) . So,

    X'( t)=s'(t)T(t)

A vector perpendicular to the velocity vector at a point is said to be normal to the curve at that point. At a point X(t) at which the velocity vector is non-zero, there is an entire circle of unit normal vectors. In the plane there were two choices of unit normal vector, and we used the orientation of the plane to choose one of them to use for the moving coordinate frame. In three-space, we can choose a particular normal vector by using the curve's acceleration vector X''( t) . At a point where the acceleration vector X''( t) is not a multiple of the velocity vector, these two vectors will determine a plane, called the osculating plane. This is the plane the curve is closest to being on at that point. We define the principal normal P(t) to be the unit vector in the osculating plane by the conditions that is perpendicular to T(t) and such that the dot product of X''( t) and P(t) is positive. Another way of defining the principal normal is to say that it is the second vector obtained by applying the Gram-Schmidt orthonormalization process to the linearly independent pair X'( t) and X''( t) . Specifically we obtain the principal normal by subtracting the tangential component of X''( t) from X''( t) . Since the resulting vector is non-zero, by hypothesis, we may divide it by its length to obtain P(t) . The principal normal is not defined when the acceleration is parallel to the velocity.

Demonstration 5: Unit Tangents and Principal Normals
Java not enabled.

You can use the "scale" slider to move from X' and X'' (where the slider equals 0) to T and P (where the slider equals 1).

This demo builds on the previous one by adding normal vectors.


Next: Binormal Vectors