Markiyan Hirnyk

Markiyan Hirnyk
8 years, 346 days


These are questions asked by Markiyan Hirnyk

Let a planar polygon P without selfintersections be given through the plottools:-polygon command.
How to find its triangulation as a set of triangles (The indication of common sides is desired too.) in an optimal way with Maple? This is used in the finite element method.

Triangulation

Let a finite set of closed intervals in the plane be given.
How to find all the intersections of these, outputing the intersection points together with the intersecting intervals?
This is a problem of computational geometry
(see http://en.wikipedia.org/wiki/Line_segment_intersection).
In other words, how to realize the sweep line algorithm in Maple?

PS. I'd like to note that computational geometry has serious applications, in particular, in robotics.

How to calculate the integral of (z-z0)*z/sqrt((x-x0)^2+(y-y0)^2+(z-z0)^2)
over the unit sphere {(x,y,z):x^2+y^2+z^2<=1}
under the assumtion x0^2+y0^2+z0^2<=1 (x0^2+y0^2+z0^2>1)?
Its physical interpretation suggests the integral can be expressed through  elementary functions of the parameters.

My tries are
VectorCalculus:-int((z-z0)*z/sqrt((x-x0)^2+(y-y0)^2+(z-z0)^2),[x,y,z]=Sphere(<0,0,0>,1)) assuming x0^2+y0^2+z0^2<=1;

and

VectorCalculus:-int(eval((z-z0)*z/sqrt((x-x0)^2+(y-y0)^2+(z-z0)^2),
[x=r*sin(psi)*cos(theta),y=r*cos(psi)*sin(theta),z=r*cos(psi)])*r^2*sin(psi),
[r,psi,theta]=Parallelepiped(0..1,0..Pi,0..2*Pi)) assuming x0^2+y0^2+z0^2<=1;

The both are spinning on my comp. Also

VectorCalculus:-int((z-1/4)*z/sqrt((x-1/2)^2+(y-1/3)^2+(z-1/4)^2),[x,y,z]=Sphere(<0,0,0>,1),numeric);

is spinning.
Edt. The omitted part of the code assuming x0^2+y0^2+z0^2<=1 is added.

How to calculate floor(10^(10^(10^(10^(10^(-10^10))))))? My simpleminded try is

Error, numeric exception: underflow
PS. I think that is about 10^(10^10).

Let us consider the expression

restart;
f := log[x](1+(x^a-1)*(x^b-1)/(x-1));
Does it define a convex function on the interval 0..1 and on the interval 1..infinity if the parameters a>0, a< 1, b >0, b <1?

My try is

and

.

At the same time I have got problems in the general case. For example,

Error, (in @) too many levels of recursion

and

That's all right if

convex.mw

 

 

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