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Aug
28
comment Calculate the area on a sphere of the intersection of two spherical caps
Yes - but by "plane representation" do you mean the projection onto the $xy$-plane? Because then the two circles would look like ellipses (and that's why I asked the question that you answered for me before).
Aug
28
comment Calculate the area on a sphere of the intersection of two spherical caps
I'm having trouble understanding your diagram - Is the shaded region the area that you're trying to find? What is its relation to the intersection of the two spherical caps?
Aug
28
revised Calculate the area on a sphere of the intersection of two spherical caps
New solution
Aug
28
comment Proving XY is perpendicular on :CD
Is this homework?
Aug
27
accepted How can I describe the area between two ellipses?
Aug
27
comment How can I describe the area between two ellipses?
Okay, @Lubin. That makes sense now. And an intersection between a circle and ellipse is guaranteed to be a simple type II region. Thanks, Christian.
Aug
27
revised How can I describe the area between two ellipses?
Title change for clarity
Aug
27
comment How can I describe the area between two ellipses?
Yes, I suppose I should have been more hesitant to use the word "define" in Math.SE when I meant "describe, delineate" rather than the mathematical interpretation of "define." But to be fair I did say "describe $\Sigma$" when I was presenting the problem in the text under the title.
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