Finding the arg equation of arc with given endpoints

May 2010
63
0
Hi guys,

I'm kind of struggling to understand the ARG representation of a (complex) arc maybe if I can see a properly solved example, it'll be become clearer.
Problem: find an ARG equation for the arc with end points \(\displaystyle -1\) and \(\displaystyle 1\) passing through \(\displaystyle i\).

The equation is in the form arg[(z - A) / (z - B)] = Theta

Taking \(\displaystyle z = i\), \(\displaystyle A = -1\), B = 1

I get agr[(i + 1) / (i - 1)] = -PI/2
I suppose this makes sense that the angle is 90 deg. Is this right?

Then how do I present the final equation of this arc?

arg[(z - A) / (z - B)] = -PI/2 ...............?

How do I manipulate the equation to find some other point on this arc besides the three already given?
Thanks!
 
Aug 2008
12,883
4,999
You do realise that "arg" is used to represent the angle that is swept out from the positive x axis in the anticlockwise direction, right?

So you'd be starting at -1 + 0i and going to 1 + 0i, passing through 0 + i. It should be obvious that it's a semicircle centred at the origin of radius 1. How big is the angle in a semicircle?
 
May 2010
63
0
Thanks, Prove It.

I didn't think about it this way... my brain was fixated on the angle between the line segments from 1 to i and -1 to i.