Perspective Ellipse Math for SoK'22

[Srirupa Datta]

Hey Tiar,

Sorry it took me some time to respond back. Thanks for helping out, your
answer was pretty detailed! I went through your explanation, and the
reduction by *a *(*1 * x^2 + a * xy + b * y^2 + c * x + d * y + e = 0*)
seems convincing. I will try this out.

Thanks,
Srirupa


On Mon, Mar 14, 2022 at 12:32 PM Tymon Dąbrowski <tamtamy.tymona at gmail.com>
wrote:

> Hi @Srirupa,
>
> ***Method 1***
> I checked your PDF and in my opinion, the result you got from Wolfram
> Alpha is consistent with your data and your approach. If you use the values
> for a, b, c,..., into the generic ellipse equation, you'll see that it
> results in an equation:
> a*x^2 + a*y^2 - a = 0
> which is the same as:
> a*(x^2 + y^2 - 1) = 0
> And you can see that no matter what value "a" is, it represents the same
> points on the plane (except when a is equal to 0, then it represents the
> whole plane and not just the ellipse).
>
> The problem in your equation is that every part of the equation has a
> parameter/variable next to it. It's probably easiest to see if you reduce
> by *f *(here, with the assumption that *f* != 0):
> f * (a/f * x^2 + b/f * xy + c/f * y^2 + d/f * x + e/f * y + 1) = 0
> as you can see, no matter what value f is, it will represent the exact
> same ellipse. So you could represent it this way:
> a/f * x^2 + b/f * xy + c/f * y^2 + d/f * x + e/f * y + 1 = 0
> or, since *a/f* is just as unknown as *a*, this way:
> a * x^2 + b * xy + c * y^2 + d * x + e * y + 1 = 0
> and this reduces the dimension of the linear equations system to just 5.
>
> NOW, I'm not 100% sure if you can just reduce *f* like that, because I'm
> not sure whether *f* (in the original equation) can or cannot be 0. I
> haven't explored that possibility before writing this mail. If* f* *can*
> be 0, it would have to be considered somewhere separately.
> Thinking about it, while I'm not sure whether we can reduce *f*, we
> *probably* could reduce for example *a*.
> I'm not 100% sure about it either, you'd have to check that, but I'm
> somewhat quite sure you couldn't have an ellipse with equation like that:
> 0*x^2 + ... = 0
> so I think it would be fine if you reduce by *a* and change the original
> equation to be:
> a * (1 * x^2 + b/a * xy + c/a * y^2 + d/a * x + e/a * y + f/a) = 0
> which means you could make an equation like that:
>
> *1 * x^2 + a * xy + b * y^2 + c * x + d * y + e = 0*
>
> which also means just 5 variables/equations, not 6, and less chance of
> infinite solutions.
>
> Think about it this way: it's similar to as if you had an equation for a
> line looking like this: "ax + by + c = 0", which looks correct, but then
> you'd try having three equations to find all three a, b and c, while the
> line only needs two points to be defined, which means just two equations.
> To represent a line, at least one of a, b, or c must not be 0, so there is
> always a way to reduce the number of equations to 2 (though it depends on
> the data in that case because sometimes you'd have "y = 0" (so a = 0, b =
> 1, c = 0) and sometimes you'd have "x = 0" (a = 1, b = 0, c = 0)).
>
> -----
>
> Also
> > I need to solve a system of linear equations for six variables and this
> is *probably* complicated.
> Actually, you'd probably just have to put it all into a matrix and a
> vector and ask Eigen to solve it for you. It's a very basic operation (it's
> not trivial to write yourself, though it wouldn't be very difficult either,
> but it's so common of a need that I would be surprised to learn that some
> library offering matrix operations doesn't provide it out of the box. As an
> example, in languages "octave" (foss) and "matlab" (proprietary) you can
> solve a system of linear equations just by writing a few characters: "x =
> A\b" (note the backslash instead of slash) when A is the matrix of
> coefficients and b is the vector of constant terms. And x is the solution
> vector. The original equation system is Ax = b).
> In Eigen you'll probably have to write a few more characters than that,
> but it should be fairly easy, too :)
> For Eigen, here you have documentation that explains what needs to be
> written: https://eigen.tuxfamily.org/dox/group__TutorialLinearAlgebra.html
> And for usage of Eigen in Krita, you can refer to the file:
> libs/global/kis_algebra_2d.cpp
>
> ***Method 2***
> I think there should be a way, but I don't know yet, but also you have
> Method 1 already figured out, so maybe it's not necessarily needed.
>
> ---
>
> Hope that helps,
> Tiar
>
> PS. If anything is unclear, feel free to ask, either here or on IRC
> (tiar).
>
>
>
> niedz., 13 mar 2022 o 19:04 Srirupa Datta <srirupa.sps at gmail.com>
> napisał(a):
>
>> I have a quadrilateral, whose 4 corners are known, and an inscribed
>> ellipse whose touching points with the quadrilateral and also known. I need
>> to find out the extreme points of the major axis of that ellipse.
>>
>> *Method 1*
>> I decided to first find out the equation of the inscribed ellipse. This
>> <https://drive.google.com/file/d/1mMT9to50lJN1eo2SPQh3jJB_puvb-SBj/view?usp=sharing>
>> is what I did so far. However this approach has a challenge. I need to
>> solve a system of linear equations for six variables and this is
>> *probably* complicated. So I'm not sure if this is at all the right
>> approach.
>>
>> Also, I got stuck since I expected the solution to be a unique one.
>>
>> *Method 2*
>> I have the transformation matrix that maps a unit square to my
>> quadrilateral. Is there any way I can use transformation mathematics to
>> solve this problem easily?
>>
>> Thanks,
>> Srirupa
>>
>>
>>
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