solve(4).sm (110.53 KiB) downloaded 586 time(s).

Hi Valery. It seems that solve doesn't know how to order the interval.

The 6 solutions numerically



And symbolically



solve(4).sm (158.3 KiB) downloaded 591 time(s).

Best regards.
Alvaro.

Thanks, Alvaro

It is one task from the article
https://disk.yandex.ru/i/59sq-AYpKky-3Q

Do you know the best solution in SMath?

Hi Valery. Two ways.



heart area.sm (13.36 KiB) downloaded 524 time(s).

Mathcad 11:



mc11 double int.mcd (4.77 KiB) downloaded 520 time(s).

Best regards.
Alvaro.

And what about this. We can create an animation.

It's possible, although speed isn't one of SMath's main features. And with so few points, there isn't much accuracy.





heart area.sm (26.99 KiB) downloaded 517 time(s).

Best regards.
Alvaro.

I can write a function that returns a set of closed isolines. The area can be calculated from the points on the curve. To construct implicit graphs, a function is used that returns the coordinate sets of the found isolines.

Wrote

I can write a function that returns a set of closed isolines. The area can be calculated from the points on the curve. To construct implicit graphs, a function is used that returns the coordinate sets of the found isolines.

Вячеслав, привет!
Брось все силы на создание инструментов оптимизации с ограничением типа Given... Minimize/Maximize в Mathcad 15. Этого очень не хватает в SMath.
И Андрей на это настроен.

Using Dragilev Method.



Area and Arc Length of an Implicit Planar Closed Curve - Dragilev Method.sm (16.28 KiB) downloaded 519 time(s).

Best regards.
Alvaro.

Yes, something like this with the Gaussian area formula.

Wrote

Yes, something like this with the Gaussian area formula.

Hi Viacheslav. In that case, I'd like to point out that dividing by the norme (Δ) means the curve is parameterized with respect to arc length, and that if you're going to improve the points using the original function, you must also do so with the parameter variable, which in the program's notation is T. This is because Dragilev's method ultimately obtains the parametric expressions x(t) and y(t) from the implicit function f(x,y) when starting from a point (xo,yo) on the curve.

This is one way to correct the arc length given by T if you choose to improve the curve points, although there must be better methods involving splines for curves in the plane, not just single-valued functions. There are also better approximations for the area enclosed by the curve; the one used here is simply the area defined by a closed polygon.



Area and Arc Length of an Implicit Planar Closed Curve - Dragilev Method.sm (17.78 KiB) downloaded 10 time(s).

Best regards.
Alvaro.