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Aug
27
comment How can I describe the area between two ellipses?
Also, you're assuming that the intersection $S$ is a type I region. I can draw a circle and ellipse intersecting in a way such that they are not a type I region
Aug
27
comment How can I describe the area between two ellipses?
Why can we assume that $E_1$ is a circle, not an ellipse?
Aug
27
comment How can I describe the area between two ellipses?
@HenningMakholm No, I am not looking for the explicit area of the intersection, but a definition with inequalities (which I can use in a surface integral's limits). Perhaps an example will get my point across better. The area of a rectangle $R$ is $ab$, but to define its area with inequalities would be to use $0\le x\le a$ and $0 \le y\le b$. The former I can't use to find $\int\int_R f(\vec r) dS$ for some $f$ and position vector $\vec r$, whereas the latter would be perfect. I didn't want to complicate the question with unnecessary information about surface integration.
Aug
27
revised How can I describe the area between two ellipses?
Clarification of the goal
Aug
27
comment How can I describe the area between two ellipses?
@HenningMakholm I'm not familiar with the concept of the Lebesgue measure beyond a Wikipedia-level examination, but if that way of saying it makes more sense I'll revise.
Aug
27
revised How can I describe the area between two ellipses?
Removed premature "RESOLVED" notes
Aug
27
comment How can I describe the area between two ellipses?
(cont.) have imaginary $y_0$ values. That being said, unless someone can introduce the information about two intersections in an algebraic manner, I don't think I can find a single expression for $x_0$, instead forcing me to rely on specific scenarios. I will take off any "RESOLVED" notes that I put up in my question.
Aug
27
comment How can I describe the area between two ellipses?
Actually - something I just noticed - for the term that I mentioned to become imaginary, only $(x_0-c_2)^2>x_1^2$. Thus, only the value that is closer to $c_2$ can be real. That really summarizes the scenario, though: since I didn't bring in any information about exactly two intersections, the statement holds true for ellipses that intersect once, twice, or even four times (three is not applicable in this scenario). As the distance from $c_2$ is more of a concern, we can't tell which $x$-values to choose until we have a set of numbers to work with - either the larger or smaller x could
Aug
27
awarded  Critic
Aug
27
revised How can I describe the area between two ellipses?
added 161 characters in body
Aug
27
comment How can I describe the area between two ellipses?
No. This is a problem I made up. I understand the approach that you have, but, unfortunately, it is not finding the area of the region $\Sigma$ that I'm interested in, it's more about defining it. I am more interested in finding a definition of $\Sigma$ with $u$ and $v$.
Aug
27
revised How can I describe the area between two ellipses?
Resolved one of my questions, formatting
Aug
27
comment How can I describe the area between two ellipses?
FYI - The image I posted used (1) $\frac{(x-5)^2}{3^2}+\frac{y^2}{6^2}=1$ and (2) $\frac{(x-7)^2}{12}+\frac{y^2}{3^2}$. Should I put that in there?
Aug
27
comment How can I describe the area between two ellipses?
@BenCrowell I agree with Lubin - most likely, the term $(1-\frac{(x_0-c_2)^2}{x_1^2})$ in $y_0=\sqrt{y_2^2(1-\frac{(x_0-c_2)^2}{x_1^2})}$ becomes negative, causing the smaller $x$ values to become imaginary. Therefore, the $x$ that we're looking at has to be the larger one. I guess that answers that. I don't think we need to go through a numerical case if my explanation sounds convincing enough. That still only gets me down to where I ended in my question, though.
Aug
27
asked How can I describe the area between two ellipses?
Aug
26
comment Numerical Analysis
There was no complex conjugate. Just saying.
Aug
26
answered Can you approximate a vector field?
Aug
26
revised Calculate the area on a sphere of the intersection of two spherical caps
added 173 characters in body
Aug
26
revised Calculate the area on a sphere of the intersection of two spherical caps
Minor LaTeX corrections, grammar
Aug
26
revised Calculate the area on a sphere of the intersection of two spherical caps
Minor LaTeX corrections