Create complex definition

Replies to This Discussion

Permalink Reply by Ethan Gross on December 28, 2016 at 11:54pm

Is this helpful?

Attachments:

• Complex intro.gh, 12 KB

Permalink Reply by Wong Wai Hang on December 29, 2016 at 12:44am

Hi Ethan,

Thanks.

Your sample helps me to understand the basic how it works.

However, it seems to me using math & Vector components can get the same results.

So I think the critical thing is what is " imaginary" definition.

Besides, when will we use (or apply) these components?

Permalink Reply by Ethan Gross on December 29, 2016 at 1:08am

You're right, in many cases it is just a matter of choice. However, the example below takes a grid of points in the first quadrant and treats them as complex numbers. Squaring them maps them into the first and second quadrants (making a nice pattern). You could do this without using complex numbers but it would require several additional steps.

Attachments:

• w = z^2.gh, 5 KB

Permalink Reply by Wong Wai Hang on December 29, 2016 at 8:40pm

Hi Ethan,

The new sample is very interesting. It took me a night to figure it. And I am still in the dark side.

I think the main problem is I don't understand squaring the X, Y after created complex will get the negative number. These sets of number do explain how they jump from first quadrant to second quadrant.

Appreciate it if you can spend a little time to direct me.

Cheers

Raymond

Attachments:

• question.PNG, 238 KB

Permalink Reply by Ethan Gross on December 29, 2016 at 10:33pm

Hi Raymond, I'm not certain how familiar you are with complex numbers but if what I say below is confusing, as it very well may be, I suggest you also consult https://www.youtube.com/watch?v=5mPN86W5hbk. He seems to do a pretty good job of explaining it.

One thing that makes complex numbers useful is that they simplify rotations in the plane. If a complex number z = x +i*y, then it can also be expressed as r*exp(i*theta), where r = the modulus sqrt(x^2 + y^2) and theta is the argument or angle with the x axis = arctan(y/x). That is what the grasshopper components above do; they extract this information. If you square the number, then you get a new number (r^2)*exp(2*i*theta), so the angle has been doubled.

In my example, the top right corner of the square was at point r=sqrt(2) and angle 45 degrees= pi/4. Squaring it brings it to the point r = (sqrt(2))^2 = 2 and angle 90 degrees = pi/4 *2 = pi/2. This is equivalent to the point x=0, y= 2 as you can see. Any point in the square whose angle is greater than 45 degrees, once doubled will be rotated into the second quadrant. The entire square sat in the first quadrant going from theta = 0 to 90 degrees. Squaring it takes it from 0 to 180 degrees, covering the first and second quadrants.

One other interesting thing about these kind of maps, they are "conformal", meaning that angles are preserved, so all of the 'squares' in the mapped image still have corners that meet at 90 degrees even though the squares themselves are distorted. This can be very useful.

Hope I haven't confused you further...

Permalink Reply by Wong Wai Hang on January 1, 2017 at 8:42pm

Hi Ethan,

I start to realize why X will move to second quadrant. 

Thanks for your introduction.

:)

Raymond 

Permalink Reply by mükerrem özafşar on March 1, 2017 at 4:50pm

Does anybody know how to use   .i   in evaluate component for calculating  i* (x+iy)

Permalink Reply by Ethan Gross on March 2, 2017 at 1:26am

Attachments:

• multiply by i.gh, 11 KB

Permalink Reply by Wong Wai Hang on March 2, 2017 at 8:21pm

Hi Mukerrem,

I am not familiar with complex number coz I didn't take any apply math in college. However, Ethan has given a website about that explains some basic knowledge. From there, I started to realize how the i works. But I only have a basic idea why the i moves from quadrant to quadrant only. My knowledge is far away from using them to create my design.

you can open Ethan's .gh file in grasshopper.

cheers

Raymond

Permalink Reply by mükerrem özafşar on April 6, 2017 at 12:27pm

thank yo very much