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 .
f
2
=
z
⋅
z
′
{\ displaystyle f ^ {2} = z \ cdot z '}
1
f
=
1
G
+
1
b
{\ 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
G
B.
=
z
f
=
f
z
′
{\ 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 .
G
{\ displaystyle G}
B.
{\ displaystyle B}
f
z
′
{\ displaystyle fz '}
Derivation from the lens equation
The Newtonian mapping equation is equivalent to the lens equation:
1
f
=
1
G
+
1
b
{\ displaystyle {\ frac {1} {f}} = {\ frac {1} {g}} + {\ frac {1} {b}}}
After simple arithmetic transformations it results:
f
=
G
⋅
b
G
+
b
f
(
G
+
b
)
=
G
⋅
b
f
⋅
G
+
f
⋅
b
=
G
⋅
b
{\ 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
f
2
-
f
⋅
G
-
f
⋅
b
{\ displaystyle f ^ {2} -f \ cdot gf \ cdot b}
f
2
=
G
⋅
b
-
G
⋅
f
-
b
⋅
f
+
f
2
=
(
G
-
f
)
(
b
-
f
)
{\ displaystyle {\ begin {aligned} f ^ {2} & = g \ cdot bg \ cdot fb \ cdot f + f ^ {2} \\ & = (gf) (bf) \ end {aligned}}}
what about
z
=
G
-
f
{\ displaystyle \ z = gf}
and
z
′
=
b
-
f
{\ displaystyle \ z '= bf}
leads to the desired result.
swell
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