# Finding the arg equation of arc with given endpoints

#### director

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!

#### Prove It

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?

#### director

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.