Newtonian mapping equation

Image formation on a lens. The sizes and are marked in red. ${\displaystyle z}$ ${\displaystyle z'}$

The Newtonian mapping equation is a formula of ray optics named after the English physicist Isaac Newton .

It is and is often used instead of the lens equation . Here, z or z 'stands for the difference between the object distance or image distance and focal length . ${\displaystyle f^{2}=z\cdot z'}$ ${\displaystyle {\frac {1}{f}}={\frac {1}{g}}+{\frac {1}{b}}}$

Derivation with the theorem of rays

If one considers the lowest ray emanating from object G in the figure, and the uppermost ray incident to the image (i.e. the rays through the two focal points ), then follows from the ray law

${\displaystyle {\frac {G}{B}}={\frac {z}{f}}={\frac {f}{z'}}}$

Here and are the height of the object or picture. The Newtonian mapping equation results directly from the right equal sign by expanding with . ${\displaystyle G}$ ${\displaystyle B}$ ${\displaystyle fz'}$

Derivation from the lens equation

The Newtonian mapping equation is equivalent to the lens equation:

${\displaystyle {\frac {1}{f}}={\frac {1}{g}}+{\frac {1}{b}}}$

After simple arithmetic transformations it results:

${\displaystyle {\begin{aligned}f&={\frac {g\cdot b}{g+b}}\\f(g+b)&=g\cdot b\\f\cdot g+f\cdot b&=g\cdot b\end{aligned}}}$

Adding both sides gives ${\displaystyle f^{2}-f\cdot g-f\cdot b}$

${\displaystyle {\begin{aligned}f^{2}&=g\cdot b-g\cdot f-b\cdot f+f^{2}\\&=(g-f)(b-f)\end{aligned}}}$

what about

${\displaystyle \ z=g-f}$ and ${\displaystyle \ z'=b-f}$

leads to the desired result.

swell

• Lexicon of physics: mapping equations. Retrieved June 17, 2014 .