The Frenet Frame
Before we continue on to describing how the shape operator can be used in brain modelling, let's take a moment to describe one last concept that may be able to simplify the way that we've looked at surface geometry.
We have already looked at the normal and tangent vectors and planes of a curve as functions depending on the derivative of the curve at a point. We can combine these two vectors with a third orthonormal vector called the binormal vector to form an orthonormal basis at a given point on the curve called the Frenet frame.
A series of formulas developed by Frenet and fellow French mathematician Serret showed that these orthonormal vectors could be described purely in terms of a matrix multiplication involving the curvature and torsion of the curve at the point. This breakthrough is important because it turns these vectors into intrinsic properties of the curve.
In the scene below, we have graphed the curve x(t) = (t, sin(t), cos(3t)) from 0 to 2pi. The frenet frame contains three orthonormal vectors: the tangent vector, the normal vector, and the binormal vector.
Use the slider to move the frenet frame along the (parameterized) curve. In addition, the following controls are available to manipulate the image:
- Mouse: rotate camera
- Shift + mouse: zoom in/out
- Ctrl + mouse: pan x/y
- Shift + ctrl + mouse: adjust camera's focal length