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TRMM satellite image

I have the above projection from this paper Projection coordinates

O'($x_0$,$y_0$) is the projection of the nadir directed vector onto the x-y plane (nadir is a technical word for direction directly below)

P(x',y') is the intersection of the off nadir incidence vector at an angle of $\theta$ with the x-y plane

R(x,y) is the projection of the off nadir incidence vector at a height h onto the x-y plane.

o(x,y,z) refers to a Cartesian coordinate system whose origin is at the center of the earth and the location of the satellite above is defined in terms of three coordinate ($\lambda$,$\phi$,R) assuming the earth to be an ellipsoid.

It is not clear to me me why $\Delta x $ should be along the y axis and $\Delta y$ should be along the x axis and also how the formula for $\Delta x $ and $\Delta y $ are arrived at.

$$ x = x' + \Delta x $$

$$ y = y' + \Delta y $$

$$\Delta x = h \tan \theta \, \, \frac {x' - x_0}{\sqrt{(x'- x_0)^2 +(y' - y_0)^2 }} $$

and

$$\Delta y = h \tan \theta \, \, \frac {y' - y_0}{\sqrt{(x'- x_0)^2 +(y' - y_0)^2 }} $$

1 Answers1

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I’d say that $\Delta x$ and $\Delta y$ are mislabeled in the illustration, since the two formulas reference distances measured along the $x$- and $y$-axes, respectively.

As far as the formulas themselves are concerned, by inspection, $|RP| = h\tan\theta$ and $|O'P| = \sqrt{(x'-x_0)^2+(y'-y_0)^2}$, and the rest is a straightforward application of similar triangles: $${\Delta x \over |RP|} = {x'-x_0 \over |O'P|}$$ and similarly for $\Delta y$.

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