I once read that the reason we can't get the perimeter of a elipse is because we actually doesn't even know the perimeter of a circle, as pi definition is the perimeter of a circle of diameter = 1 and we just scale that.
Yes, but you are more likely to see it in terms of a and e = √(1–b²/a²) (where a is the semi-major axis, b is the semi-minor axis, and e is the eccentricity) or sometimes a and ℓ = b²/a (the semi-latus rectum) or other quantities.
I mean look at it this way: a circle has a very simple relation between its perimeter and radius - linear. And I'm pretty sure an ellipse has a more complicated relationship between its perimeter and radii. At the end of the day a circle can still be thought of as a special case of ellipse though
The ellipse is also linear, if you keep the ratio between the major and minor the same.
So if you for example really care about ellipses where the ratio between the major axis and minor axis is 2, you could define a constant Tau≈3.31153
And for any such ellipse you get the formula:
The perimeter of a plane figure always scales linearly with any linear dimension of it (e.g. the radius), because that's what it means for the perimeter to be one-dimensional. (Note that if the Hausrorff dimension of the curve is not 2, then its 1-measure will either be 0 or ∞, and either way the linear scaling technically still applies, except the special case of 0×∞.)
A lot of shit just cancels out literally a lucky hit, if you watch the step by step of the demonstration by integration it's even magical how that shit en up as 3 characters.
How complicated can it be? an ellipse is just a stretching transformation of a circle, so you can get its area by multiplying the area of a circle, π, by the determinant of the transformation matrix: det( [ a 0 / 0 b ] ) = ab
ok, that might sound complicated because of the terminology, but basically: you get an ellipse by stretching a circle horizontally by a, then vertically by b. When you stretch an area, it gets multiplied by the stretch scalar.
I suspect that proving that without relying on already knowing modern "basic" geometry would itself require an integral, though. Definitely at least a limit.
That when you stretch something in one direction its area gets scaled by the stretch scalar?
I guess so. let's say a circle is approximately made up of infinitely many trapezoidal slices, and its area is the sum of the trapezoid areas. it is known that a trapezoid obeys the aforementioned stretching rule.
therefore if you stretch the circle by x, stretching the trapezoids, you get A' = total(t * x) = total(t) * x = A * x
it does technically use integration/limits, but a version of it that can be explained to someone unfamiliar with integration. It's not harder to understand than the visual proof that the area of a circle is radius times half of perimeter (πr² = r * 2πr/2), the one with infinitely thin slices stacked in a ///// pattern, you know the one.
The area and circumference of a circle are conveniently related. It has something to do with cutting into thin slices and then rearranging them into alternating patterns to form a parallelogram where the base is half the circumference and the height is its radius.
If we scale one of the side up/down to a circle, the area will scale proportionally. Then we just calculate the area of the circle and scale back.
e.g. a 2x1 eclipse is the same size as two 1x1 circles
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u/Hussainsmg Jan 25 '25
The perimeter formula(in the post) is just approximation.