coordinate transformation and allocentric space

People see objects in space as external and stationary (allocentric) [Rizzolatti et al., 1997] [Velmans, 1993]. Cerebellum and forebrain anticipate, coordinate, and compensate for movements.

Frontal-lobe topographic maps can represent egocentric space [Olson et al., 1999], with vertical, right-left, and front-back directions. Coordinate-origin egocenter is in head center, on a line passing through nosebridge. Space points have directions and distances from egocenter. All points make vector space.

As body, head, or eyes move, egocentric space moves, spatial axes move, and point coordinates and geometric figures transform linearly to new coordinate values [Shepard and Metzler, 1971]. Transformations are translation, rotation, reflection, inversion, and scaling (zooming). Motor processing uses tensor transform functions to describe changes from former to current output-vector field [Pellionisz and Llinás, 1982]. To maintain stationary allocentric space, so point coordinates do not change when body moves, visual processing must cancel egocentric spatial-axis coordinate transformations {coordinate transformation and allocentric space}. Visual processing inverts motor-system tensors to transform egocentric coordinate systems in opposite directions from body movements [Pouget and Sejnowski, 1997]. Topographic maps can describe tensors that transform from egocentric to allocentric space. Topographic maps can represent allocentric space.

example

Translating and rotating make spatial axes change direction. After movement, new axes relate to old axes by coordinate transformations. For example, two-dimensional vector (0,1) can translate on y-axis to make vector (0,0), rotate both axes to make vector (1,0), or reflect y-axis to make vector (0,-1). Coordinate transformations do not change dimension number.

stationary space

Perception typically maintains an absolute spatial reference frame. Stationary space allows optimum feature tracking during object and/or body motions. Moving reference frames make all motions three-dimensional, but stationary space makes many movements one-dimensional or two-dimensional.

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