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Equations for the Central Image

To describe our central image in technical language, we have to define a central curve X(t) over a parameter t. The curve we chose is an arithmetic spiral lifted onto a cone, with equation

X(t) = (tcos(t), tsin(t), pt).

The radius function r(t) that describes the expanding cone is given by 0.8e0.6t.

Thus the image grows not only in size, but but also in dimension.

If we continue to the next section of the curve, from 2p to 5p/2, the slices will be two-dimensional spheres, given by

X(t) = r(t)(cos(v)cos(u)P(t) + cos(v)sin(u)B(t) + sin(v)W(t))

where W(t) is a unit vector in 4-space perpendicular to T(t), P(t), and B(t). We can project this slice into our 3-space by pushing W(t) down to T(t), and this is the view we show on the previous page.

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