Visual complex analysis problem

[raphael3d]

Homework Statement

Explain geometrically why the locus of z such that

arg [ (z-a)/(z-b) ] = constant

is an arc of a certain circle passing through the fixed points a and b.


i tried to visualize the equation in a cartesian co-system but in doing so, i was not very successful.

 

[tiny-tim]

hi raphael3d!

i tried to visualize the equation in a cartesian co-system but in doing so, i was not very successful.

no, visualise these problems as a diagram in Euclidean geometry, not as an equation …

what do you get? ​

 

[raphael3d]

an ellipse, is my guess?

 

[tiny-tim]

no, i mean describe what "arg [ (z-a)/(z-b) ] = constant" means as a piece of geometry …

what lines is it telling you to draw? ​

 

[raphael3d]

two lines from two distinctive points a,b to one point z. whereas those lines form angles with the horizontal and the difference between those angles is constant. all the points lie on a circle...
i have drawn the lines and points and angles, but i don't know how to proceed from here... what kind of circle and so forth...

 

[tiny-tim]

hmm …

a better way of putting it is that from two points a and b, we draw a pair of lines that meet at a given angle …

you should be able to prove that all such points (for a fixed angle) form an arc of a circle

 

[raphael3d]

if a is 1 and b is i on the unit circle, then z lies in the first quadrant? i would guess the angle where a and b meet z doesn't change as long as z lies between them...?

 

[raphael3d]

you mean...meet at a given angle c?

i am stuck, to be honest^^

 

[tiny-tim]

hi raphael3d!

(just got up :zzz: …)

there's a well-known theorem that the locus of points which subtend a fixed angle from two given points is an arc of a circle joining those two points

you need to find a book of geometry (sorry, i don't know any online ones ) which gives you all the theorems for a circle, and their proofs …

clearly this is background knowledge which your course assumes you already have​

 

[raphael3d]

clearly we live in shifted time zones =)

well thank you, i will look into that...surely there will be a wiki or something similar.

this is a problem of "visual complex analysis" by tristan needham. a wonderful book :)

here is it:
http://www.mathsisfun.com/geometry/circle-theorems.html

now i would love to show it with some complex algebra ;)

thanks for the help
keep up the good work, with that many qualitative posts you could easily have written a book.

metta

 

[tiny-tim]

hi metta!

this is a problem of "visual complex analysis" by tristan needham. a wonderful book :)

here is it:
http://www.mathsisfun.com/geometry/circle-theorems.html

yes, that looks good

the theorem you need is the third diagram on that page, marked "Angles Subtended by Same Arc Theorem"