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4. Binormal Vectors

In order for our frame to span three-space, we will need to find a third vector linearly independent from the other two. To do this, at each point along a curve where the acceleration vector is not parallel to the velocity vector, we use the orientation of space to define a unit binormal vector B(t) orthogonal to the osculating plane. There are two possible unit vectors orthogonal to a given plane at a given point, and we choose B(t) such that the three mutually orthogonal unit vectors T(t) , P(t) , and B(t) form a right-handed frame .

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This frame consisting of T , P and B is called the Frenet Frame of the curve.

Demonstration 6: T, P and B Demo
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You can use the "scale" slider to scale from X' , X'' and X' X'' (where the slider equals 0) to T , P and B (where the slider equals 1).

This demo again builds on the last one. Now, the binormal vectors at various points are also shown.

    Look at the twisted cubic, with the slider constant, c as the coefficient the quadratic term, i.e. t->[t, c t t, t t t]. Watch what happens to the binormal vectors as you make the curve approach a plane curve by gradually decreasing c to zero.

    A helix is defined as a curve whose tangent vectors make a constant angle with a fixed direction. Do you notice anything special about the binormal vectors of such a curve? (Input a circular helix: t->[cos(t),sin(t), c t].)

Demonstration 7: Flying Along a Curve in Your Car
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This demo allows the user to examine the Frenet frame of any curve. The window called the Flight shows the curve projected into the plane of the normal and binormal vectors. This is the view one would get if one were sitting on the blue car, looking out the front window. Notice that at the origin (where the car would be in TheFlight ) there is always a cusp. This is because the curve is being projected into a plane perpendicular to the tangent vector.

    In this last demo, look especially at the SpaceCardioid . For what values of the TapeDeck do you get cusps in TheFlight at places other than the origin. Why?

    Do you notice anything special when you fly along a helix?


Next: The Acceleration in the Frenet Frame and the Curvature of a Space Curve