Alexey Ivanov

## 1160 Reputation

12 years, 43 days

## Angle of a triangle in geom3d[FindAngle...

Maple 17

Why in geom3d[FindAngle] we cannot get the value of the angle of a triangle greater than Pi / 2?
For example, I build a chord of a circle of unit radius along the sides of the triangle and calculate the center angle that corresponds to the given angle of the triangle. But it's not very convenient.
TR_ANGLE.mw

## Is there a tetrahedron or not...

Recently found out about Steiner ellipsoids. This is an ellipsoid inscribed in a tetrahedron, and an ellipsoid described near a tetrahedron. One ellipsoid touches the faces of the tetrahedron in centroids, the other ellipsoid touches the planes in vertices of the tetrahedron, and these planes are parallel to the faces.
steiner_in_ellipsoid.mw
steiner_out_ellipsoid.mw

There is a desire to find out if 4 vertices determine the tetrahedron or not. And it seems Maple answer this question:

```restart: with(geom3d):
point(A, 0, 0, 0), point(B, 1, 0, 0), point(C, 0, 0, 1), point(F, 0, 1, 0):
gtetrahedron(T1, [A, B, C, F]):
i := nops(op(0, detail(T1)));
if i = 1 then print("Ok gtetrahedron") else print("No gtetrahedron") end if:
```

I want to stop further execution of the program if the tetrahedron is not determine, for example, when point
B = [0,0,0], but I don’t know how to do it.

## Smooth cylinder surface...

Maple

Source of interest in this topic.  There is a problem with drawing a cylinder. I draw a cylinder using the arrow function as follows:

cylinder := seq(arrow(`<,>`((L[1][i]+L[4][i])*(1/2), (L[2][i]+L[5][i])*(1/2), (L[3][i]+L[6][i])*(1/2)), `<,>`(-(L[1][i]+L[4][i])*(1/2)+(L[7][i]+L[10][i])*(1/2), -(L[2][i]+L[5][i])*(1/2)+(L[8][i]+L[11][i])*(1/2), -(L[3][i]+L[6][i])*(1/2)+(L[9][i]+L[12][i])*(1/2)), width = 1, color = green, head_length = 0, head_width = 0, length = VectorNorm(`<,>`((L[1][i]+L[4][i])*(1/2)-(L[7][i]+L[10][i])*(1/2), (L[2][i]+L[5][i])*(1/2)-(L[8][i]+L[11][i])*(1/2), (L[3][i]+L[6][i])*(1/2)-(L[9][i]+L[12][i])*(1/2)), 2), transparency = 0.), i = 1 .. N+1):

(L [j], j = 1..12 - the coordinates of the red points on the cylinder).

But, as you can see, because of the faces, everything does not turn out very smoothly. Is there a way to display the cylinder smoothly, but so that without too much computing resources, ie no more resource than a "arrow"?

## Solution of the boundary value problem f...

Maple

Example of Duffing equation with boundary conditions.
y'' + 0.2y' + y^3 - 0.3cos(s) = 0;
y(0) = y (2Pi);
y'(0) = y'(2Pi);
For convenience, we replace the original equation with a system of two first order equations:
--------------------------------------------------------------------------
x1'(t) = 2*Pi*x2(t);
x2'(t) = - 0.4*Pi*x2(t) - 2*Pi*x1(t)^3 +0.6*Pi*cos(2*Pi*t);
x1(0) = x1(1);
x2(0) = x2(1);
--------------------------------------------------------------------------
I have long wanted to apply an optimization package to solve a boundary value problem for ODE. The decision helped procedure for solving ODE, written by forum member vv.
It seems to me that two solutions have been found and that the solutions are weakly sensitive to the initial approximations. These are two closed trajectories. For example, these are points that belong to these solutions:
(0.5966963,  1.0482816) , ( - 0.3132584, 0.0664941).
I am wondering: are the solutions right, and how justified is the use of optimization methods for such tasks?
At the end of the program, the solution is checked on the original Duffing equation using standard Maple functions.   Duffing_equation_BC.mw

(In the figures, the trajectory bypass occurs three times.)

## To paint a triangle on a sphere...

Maple 15

My profile picture was formerly animation and looked like this:

It would be interesting to paint a triangle on a sphere. How can I do that?

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