highmath said:
(1) Is there a pi in ellipse entity?Why not or yes?
(2) Is there a pi in polygons entities (e.g square)? not or yes?
(3) If there is pi in some geometries and other not - What is the reason to that?
(4) How cloud I know that are no hidden formula of pi in a square figure that the expression in formula, his value is: 0 in addition and 1 in multiplicatoin and etc
We can roughly divide geometry in angles and coordinates.
We see $\pi$ in the domain of the angles and not in the domain of the coordinates.
In a simple right angle on the unit circle we have the simplest possible coordinates with $0$ and $1$.
Its angle is $\smash{\frac \pi 2}$.
Its hypotenuse is $\sqrt 2$, which is an
algebraic number and not
transcendental like $\pi$.
So $\pi$ comes along whenever an angle or turn is involved.
To find the area of a circle, we integrate over the angle so that we see $\pi$ back in the result.
An ellipse is a circle scaled in one direction by a factor, which scales its area by the same factor.
So yes, there is $\pi$ in the area of an ellipse.
The circumference of an ellipse is so complicated that an elliptic integral function was invented to describe it.
Still, there is probably a $\pi$ hidden in there somewhere.
The coordinates of a polygon and its sides are typically algebraic numbers.
Its angles contain $\pi$, but $\pi$ only shows up if we map the polygon into the domain of a circle, since that is how angles have been defined. If we would identify angles by their coordinate ratios (slopes) there would be no $\pi$.
Through projective geometry a circle is transformed, or rather is equivalent to an ellipse, a parabola, and a hyperbola. These are the so called conic sections.
We already saw $\pi$ in the area of an ellipse.
A parabola and a hyperbola don't have an area though as they are unbounded. If we bound a parabola with a line at an algebraic coordinate, we don't see $\pi$, but we see an algebraic number. A hyperbola bounded by a line gives an area that contains $e$ instead of $\pi$.
The arc length of a parabola also contains $e$, while the arc length of a hyperbola is too complicated to tell.
We see $\pi$ a lot in physics formulas as well, and usually as $2\pi$ or $4\pi^2$. What these formulas have in common is that they deal with a full turn. As we know, the arc length of a full turn is $2\pi r$. All those physics formulas would be simplified if we used $\tau$ for a full turn instead of $2\pi$.
