Tuesday, July 14, 2015

Circle through two points on the Complex Plane

This formula makes a clockwise circle on the complex plane starting at point (x,y) and theta = 0,  and going through (a,b) at theta = pi, then back around to (a,b) at  theta=2*pi...

So you can use it to solve certain problems like what if you go 1/5th of the way from point (3,5) to point (-4,2) along a circular arc? The circular arc connecting the two points is:

But that is all the way with theta going to pi, one fifth of that would be pi/5 so:

One fifth of the way between those tow points along the circle is roughly at 4.38, 2.27...

**Solving for angles between unit vectors**
Another thing to do is say we have the unit vector ((2^1/2) / 2, (2^1/2) / 2), and we want to know the angle between that and ((3^1/2)/2), (1/2)) we can first make a circle with that start point and negative times each coordinate as the final point like so:

Then solve for when the formula above equals the chosen point:

It's pi/12 between those two vectors... If the angle had been the same but counterclockwise it would have read as -pi/12!

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