The crossed ladder problem and optics mirror/lens equation

[Vriska]

the 1/a + 1/b = 1/c keeps cropping up all over the
place like the lens/mirror formula, parallel resistors but why? All these somewhat seem related to me as some kind of harmonic average, like they're the constraints of some kind of crossed ladder problem:
https://en.m.wikipedia.org/wiki/Crossed_ladders_problem

My apologies if this post isn't quite coherent, I'm just looking for more insight into the 1/a + 1/b = 1/c form that keeps cropping up

 

[sophiecentaur]

the 1/a + 1/b = 1/c keeps cropping up all over the
place like the lens/mirror formula, parallel resistors but why? All these somewhat seem related to me as some kind of harmonic average, like they're the constraints of some kind of crossed ladder problem:
https://en.m.wikipedia.org/wiki/Crossed_ladders_problem

My apologies if this post isn't quite coherent, I'm just looking for more insight into the 1/a + 1/b = 1/c form that keeps cropping up

If you are happy when you meet
p=q+r
and the units for pqr are defined in one way (say speeds) then, if you happen to have measured or defined those three quantities in reciprocal terms such as 'time taken to go 1m' then p=1/a q=1/b r=1/c then you could re-write the first equation as
1/a = 1/b + 1/c
I know I have only stated the (mathematical) obvious but you have already accepted the idea of a mathematical representation of a situation (total speed = speed1 +speed2). So you really could accept all the consequences of where the maths takes you, which is that 'harmonic equation'.

In electrical calculations, it is particularly useful to be 'ambidextrous' and to be able to talk in terms of Impedance or Admittance (or just Resistance and Conductance) to suit the particular problem you are dealing with.
In the case of your crossed ladder problem, I wonder whether it could be worth while approaching it by giving the ladders and the wall 'reciprocal lengths'. I am not geeky enough (too dumb) to do that but I am sure someone could manage to solve the problem in just a couple of lines with an 'inside out' solution. It did strike me that there could be a related problem in Xray crystallography where the wave number is used in working out reflections in a lattice. (??)

 

[pixel]

I'm not sure if there's anything fundamental about 1/a + 1/b = 1/c. As sophiecentaur points out, the form of the equation can change by defining alternative variables. For example, the thin lens equation can be written in the Newtonian Form as x₀x_i=f² where the object and image distances are measured from the front and rear focal points.