3.4: The Binormal Vector and the Frenet Frame
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:| B(t) = T(t)×P(t) |  |
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| P(t) = B(t)×T(t) | ||
| T(t) = P(t)×B(t) |
This frame consisting of T(t), P(t) and B(t) is called the Frenet Frame of the curve.
This demonstration allows you to input a parameterized curve X(t) and travel along it by using the tapedeck for t0 in the control panel. The curve window shows the Frenet Frame at the point X(t0). The default curve is a closed curve thats looks like the boundary of a potato chip. A second tapedeck in the control panel allows you to look at the curve for different values of c.
1. Look at the twisted cubic, with the slider constant, with c as the coefficient the quadratic term, i.e. X(t) = (t, c*t^2, t^3). Watch what happens to the binormal vectors as you make the curve approach a plane curve by gradually decreasing c to zero.
2. 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: X(t) = (cos(t),sin(t),c*t).)
2. 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: X(t) = (cos(t),sin(t),c*t).)
Driving Along a Curve
One way to think about the Frenet frame is to imagine being in a car driving along the curve X(t). In this case it is not ideal to use the regular x,y,z frame since you are moving relative to its origin. By switching to a Frenet frame, you and your car become the origin. Analogously, the tangent vector T(t) runs along the drive shaft; the principal normal vector P(t) is parallel to the axle; and the binormal vector B(t) points straight up like your radio antenna.
This demo allows the user to examine the Frenet frame of any curve. The window titled "View From Car" shows the curve projected into the plane of the principal normal and binormal vectors (i.e. the P-B plane). This is the view one would get if one were sitting in the yellow car, looking out the front window. Notice that at the origin, the driver always sees a cusp at his location. This is because the curve is being projected into a plane perpendicular to the tangent vector.
1. In this last demo, look especially at the Space Cardioid:
X(t) = (c + cos(t))*(cos(t),sin(t),f(t)).
For what values of t0 do you get cusps in The Flight at places other
than the origin. Why?
2. Do you notice anything special when you fly along a circular helix(X(t) = (cos(t),sin(t),c*t))?
2. Do you notice anything special when you fly along a circular helix