Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

My interpretation

My first question is what this type of graph (of $x-y-i$) is called since I was unable to find any information about any such graph.

Now for the real question, I used the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and drew the graph on the $xy$ plane and then added an imaginary axis and found out the value $y = bi$. so I plotted it and since it was symmetric I decided to do it for $-b$; then I saw that the values fell sharply so now I think this graph will be an Ellipse. Am I correct?

Is there any research about the xy plane with imaginary axis?

share|improve this question

1 Answer 1

First of all, I would not call it the "imaginary axis" - the name needs to be clearer as to what the value on that axis represents. If I understand your question correctly, it is the imaginary $y$-axis; in other words, you're allowing $y$ to be a complex number $y=y_0+iy_1$, while you are still requiring $x$ to be real.

If I understand your question correctly, you are wondering if your plot of the solutions to $$\frac{x^2}{a^2}-\frac{(y_0+iy_1)^2}{b^2}=1$$ is correct. Well, I'd advise thinking about it like this: $$\frac{x^2}{a^2}-\frac{(y_0+iy_1)^2}{b^2}=\bigg(\frac{x^2}{a^2}-\frac{y_0^2}{b^2}+\frac{y_1^2}{b^2}\bigg)-i\bigg(\frac{2y_0y_1}{b^2}\bigg)=1$$ The only way this is possible is if $y_0=0$ or $y_1=0$; otherwise, the imaginary part of the left side is non-zero, while the imaginary part of the right size is zero.

Thus, you're looking for the solutions to $$\frac{x^2}{a^2}-\frac{y_0^2}{b^2}+\frac{y_1^2}{b^2}=1$$ where either $y_0=0$ or $y_1=0$. The above equation defines a hyperboloid of one sheet, and so you're looking for the intersection of that hyperboloid with the $xy_1$-plane (where $y_0=0$) and $xy_0$-plane (where $y_1=0$).

enter image description here

enter image description here


In conclusion: Your plot seems to be of the right form, though it is not centered correctly (the center of the whole thing should be at the origin).

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.