Visual complex analysis problem

In summary, the locus of points which subtend a fixed angle from two given points is an arc of a circle joining those two points.
  • #1
raphael3d
45
0

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.
 
Physics news on Phys.org
  • #2
hi raphael3d! :wink:
raphael3d said:
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? :smile:
 
  • #3
an ellipse, is my guess?
 
  • #4
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? :wink:
 
  • #5
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...
 
  • #6
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 :wink:
 
  • #7
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...?
 
  • #8
you mean...meet at a given angle c?

i am stuck, to be honest^^
 
Last edited:
  • #9
hi raphael3d! :smile:

(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 :smile:

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

clearly this is background knowledge which your course assumes you already have​
 
  • #10
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
 
  • #11
hi metta! :smile:
raphael3d said:
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" :wink:
 

1. What is visual complex analysis?

Visual complex analysis is a branch of mathematics that focuses on the geometric and visual interpretation of complex numbers and functions. It is used to study the behavior of complex functions and their properties, such as continuity, differentiability, and analyticity.

2. What are some common problems in visual complex analysis?

Some common problems in visual complex analysis include understanding the behavior of complex functions, finding the roots of complex polynomials, and determining the properties of complex curves and surfaces.

3. How is visual complex analysis used in real life?

Visual complex analysis has many applications in physics, engineering, and other fields. It is used to model and analyze systems with complex variables, such as electrical circuits, fluid dynamics, and quantum mechanics.

4. What are some tools and techniques used in visual complex analysis?

Some common tools and techniques used in visual complex analysis include contour integration, conformal mapping, and the use of complex plots and diagrams. Computer software and graphing calculators can also be helpful in visualizing complex functions.

5. Can visual complex analysis be applied to non-mathematical problems?

Yes, visual complex analysis can be applied to non-mathematical problems, such as analyzing the behavior of systems or processes with complex variables. It can also be used to study patterns and relationships in data sets and to visualize complex data in a more intuitive way.

Similar threads

  • Linear and Abstract Algebra
Replies
14
Views
497
  • Calculus and Beyond Homework Help
Replies
5
Views
969
  • Calculus and Beyond Homework Help
Replies
1
Views
878
  • Calculus and Beyond Homework Help
Replies
13
Views
2K
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
820
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
Back
Top