Is it possible to calculate the area of a rectangle projected

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>>8570814
Reposting because I made a mistake in my last post.

The equivalent of my picture in three dimensions would be mapping a circle with the same diameter as the sphere to the surface of the sphere.

So a surface (circle) of area [math] \pi (\frac{d}{2})^2 [/math] where d is the diameter of the sphere is mapped to a surface area of [math]\frac{ 4 \pi (\frac{d}{2})^2}{2} [/math] which is the surface area of the sphere over two.

Therefore, if we have a rectangle of the same area (examples would be that the width is pi and the length is radius squared) then it is fair to say that the area of that rectangle, projected to the sphere would be equal to the area of the projection made by our circle. In fact it should be defined that way as we would want objects of different dimensions but same area to have the same area regardless, when projected to a sphere.

So now lets say that you have a smaller rectangle.

Lets say that you have a sphere of diameter 1 and a rectangle of area 0.5

First you need to compare this area of 0.5 to the area of the circle with the diameter of the sphere.

This circle has area [math] \pi (\frac{1}{2})^2 [/math] = [math] \frac{ \pi }{4} [/math]

0.5 = [math] \frac{ \pi }{4} * \frac{2}{ \pi} [/math] so the proportion is [math] \frac{2}{ \pi} [/math]

So now take [math]\frac{ 4 \pi (\frac{1}{2})^2}{2} [/math] which is [math]\frac{ 4 \pi}{8} [/math]

and compute [math]\frac{ 4 \pi}{8} * \frac{2}{ \pi } = \frac{8}{8} = 1[/math]

So a rectangle with area 0.5, projected to a sphere has an area of 1. Good to know.

>>8570814
measure the spere and divide by 360 remember this measurment
measure the length from the centre of the shadow to the edge of the shadaow
now take the length and times it by by the first measurement

>>8570922
>measure the spere

By what? Volume? Diameter? Surface area?

>measure the length from the centre of the shadow to the edge of the shadaow

I never heard of this "shadow" you talk about in mathematics? Is this a graduate topic? Never heard of mathematical shadows before.

>>8570927
well seeing as we are trying to work out the area of something i would hazard a wild guess that you would want to measure the spheres area

also the op asked us to find the area of a PROJECTED rectangle on a sphere you can project this any way you want i projected it as a shadow

you dont need algebra and daddys education to work this out when all you really have to do is place silk over the sphere with no creases or gaps project the rectangle on the sphere and draw around it take the silk off the sphere cut around the lines you drew and lay it down flat and measure it

>>8570934
>you dont need algebra and daddys education to work this out when all you really have to do is place silk over the sphere with no creases or gaps project the rectangle on the sphere and draw around it take the silk off the sphere cut around the lines you drew and lay it down flat and measure it

But this is wrong.

If your idea of "projecting" is to put the rectangle over the sphere then what you are doing is not projecting. It is deforming the rectangle. You are just curving it a little and that will not change the area.

Take a look at the pic in >>8570847

See how by projecting it through perpendicular lines, the projection is bigger than the original line?

Well, if you were to take the straight line and then push it against the circle... it would be the same length. You would just be curving the line.

Yes and it's called non-Euclidean geometry anon.

>>8571035
Bullshit.

A rectangle is a parallelogram and in spherical geometry there is no equivalent for a parallelogram because there are no parallel lines.

At best you can enclose an area and arbitrarily say it looks "rectangle-y" but then that would just be wishful thinking.

>>8570814
There's something wrong with that pic.

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>>8571120
yeah something is missing

cant you just parametrize the surface of the sphere and take the surface integral of it over the rectangle in cartesian coordinates?

>>8570814
Jesus, what is wrong with her?
I mean, I would still would but seriously?

>>8572282
Long legs

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>>8571035
Iä! Iä! Cthulhu ftaghn!

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>>8572726

>>8571183
I wonder if she fired him after that picture became a meme.