Recall from Section 6.2, that the human visual system has neural structures dedicated to detecting the motion of visual features in the field of view; see Figure 8.13. It is actually the images of these features that move across the retina. It is therefore useful to have a mathematical concept that describes the velocities of moving points over a surface. We therefore define a *vector field*, which assigns a velocity vector at every point along a surface. If the surface is the plane, then a velocity vector

(8.32) |

is assigned at every point . For example,

(8.33) |

is a

(8.34) |

is non-constant, and assigns at each point ; see Figure 8.14(b). For this vector field, the velocity direction is always diagonal, but the length of the vector (speed) depends on .

To most accurately describe the motion of features along the retina, the vector field should be defined over a spherical surface that corresponds to the locations of the photoreceptors. Instead, we will describe vector fields over a square region, with the understanding that it should be transformed onto a sphere for greater accuracy.

Steven M LaValle 2020-11-11