Sampling distances

The z-sampling distance for objects $\Delta Z_{obj}$ is given by the difference:

$$\Delta Z_{obj}\left( i,k\right) =Z_{obj}\left( i+1,k\right) -Z_{obj} \left( i,k\right)$$ (10)

For simplicity let us use now a coordinate system for which the observer's z-position is zero: $Z_{obs}=0$. With respect to the fan projection the viewing center of the observer might be:

$$\underline{V}_{c}=\left( X_{obs},Y_{obs},0\right)$$ (11)

The sampling positions of the screen are given by

$$\underline{S}\left( i_{x},i_{y}\right) =\left( i_{x}\cdot \D... ...creen},i_{y}\cdot \Delta Y_{screen},Z_{screen}\right) \mbox{,}$$ (12)

with normally $\Delta X_{screen}=\Delta Y_{screen}=LS_{pitch}$. The sampling z-planes for objects are derived from $\left( \ref{Eq_Zobs2} \right)$ by

$$Z_{obj}\left( i_{z}\right) =\frac{Z_{screen}\cdot D_{eye}} {D_{eye}-i_{z}\cdot \Delta X_{screen}} \mbox{,}$$ (13)

with

$$-i_{zmax}<i_{z}<\frac{D_{eye}}{\Delta X_{screen}} \mbox{.}$$ (14)

Figure 3: Coordinate system, with viewing planes and representation planes

The sampling coordinates for the objects to be projected on the screen pixels are given by

$$\frac{X_{obj}-X_{obs}}{Z_{obj}\left( i_{z}\right) } =\frac{i_{x}\cdot \Delta X_{screen}-X_{obs}}{Z_{screen}}$$ (15)

or

$$X_{obj}\left( i_{z},i_{x}\right) =X_{obs}+\frac{Z_{obj}\left... ...eft( i_{x}\cdot \Delta X_{screen}-X_{obs}\right) }{Z_{screen}}$$ (16)

The equivalent equation holds for $Y_{obj}$:

$$Y_{obj}\left( i_{z},i_{y}\right) =Y_{obs}+\frac{Z_{obj}\left... ...ft( i_{xy}\cdot \Delta Y_{screen}-Y_{obs}\right) }{Z_{screen}}$$ (17)

The sampling distances $\Delta X_{obj}$, $\Delta Y_{obj}$, $\Delta Z_{obj}$ are given as follows:

$$\Delta X_{obj}\left( i_{z}\right) =\frac{\Delta X_{screen}\cdot Z_{obj} \left( i_{z}\right) }{Z_{screen}}$$ (18)

resp.

$$\Delta Y_{obj}\left( i_{z}\right) =\frac{\Delta Y_{screen}\cdot Z_{obj} \left( i_{z}\right) }{Z_{screen}}$$ (19)

$$\Delta Z_{obj}\left( i_{z}\right) = Z_{obj}\left( i_{z}+1\right) -Z_{obj} \left( i_{z}+1\right)$$

$$= Z_{screen}\cdot \frac{\Delta D}{\left( 1-\left( i_{z}+1\right) \cdot \Delta D\right) \cdot \left( 1+i_{z}\cdot \Delta D\right) }\mbox{,}$$ (20)

with

$$\Delta D=\frac{\Delta X_{screen}}{D_{eye}} \mbox{.}$$ (21)