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## A geometric construction for the Summer Holiday

by: Maple

A geometric construction for the Summer Holiday

Does every plane simple closed curve contain all four vertices of some square?

This is an old classical conjecture. See:
https://en.wikipedia.org/wiki/Inscribed_square_problem

Maybe someone finds a counterexample (for non-analytic curves) using the next procedure and becomes famous!

 > SQ:=proc(X::procedure, Y::procedure, rng::range(realcons), r:=0.49) local t1:=lhs(rng), t2:=rhs(rng), a,b,c,d,s; s:=fsolve({ X(a)+X(c) = X(b)+X(d),             Y(a)+Y(c) = Y(b)+Y(d),             (X(a)-X(c))^2+(Y(a)-Y(c))^2 = (X(b)-X(d))^2+(Y(b)-Y(d))^2,             (X(a)-X(c))*(X(b)-X(d)) + (Y(a)-Y(c))*(Y(b)-Y(d)) = 0},           {a=t1..t1+r*(t2-t1),b=rng,c=rng,d=t2-r*(t2-t1)..t2});  #lprint(s); if type(s,set) then s:=rhs~(s)[];[s,s[1]] else WARNING("No solution found"); {} fi; end:

Example

 > X := t->(10-sin(7*t)*exp(-t))*cos(t); Y := t->(10+sin(6*t))*sin(t); rng := 0..2*Pi;
 (1)
 > s:=SQ(X, Y, rng): plots:-display(    plot([X,Y,rng], scaling=constrained),    plot([seq( eval([X(t),Y(t)],t=u),u=s)], color=blue, thickness=2));

## how to find back the input if slope is this?...

sph := <R*cos(u)*cos(v)|R*sin(u)*cos(v)|R*sin(v)>;
GK(sph); #Gauss Curvature
MK(sph); #Mean Curvature

how to find sph if slope is tan(u) ?

## how to guess the variables?...

A. how to find xx1,xx2,xx3,yy1,yy2,yy3 that
Determinant(Matrix([[xx1,yy1,1],[xx2,yy2,1],[xx3,yy3,1]])) =(1/2)*aa*d*s*u+(1/2)*aa*d*s*a*t+(1/2)*d*v*u*t+(1/4)*d*v*a*t^2;

B. how to find x1,x2,x3,x4,y1,y2,y3,y4 that expand(
(x2 - x1)*(y4 - y3) - (y2 - y1)*(x4 - x3)) = (1/2)*d*s*aa*v+(1/2)*d*aa*v*u*t+(1/4)*d*aa*v*a*t^2+(1/2)*aa*d*s*u+(1/2)*aa*d*s*a*t+(1/2)*d*u^2*t+(3/4)*d*u*a*t^2+(1/4)*d*a^2*t^3;

## Geometry of the Canada 150 logo maple leaf

by: Maple

When we first started trying to use Maple to create a maple leaf like the one in the Canada 150 logo, we couldn’t find any references online to the exact geometry, so we went back to basics. With our trusty ruler and protractor, we mapped out the geometry of the maple leaf logo by hand.

Our first observation was that the maple leaf could be viewed as being comprised of 9 kites. You can read more about the meaning of these shapes on the Canada 150 site (where they refer to the shapes as diamonds).

We also observed that the individual kites had slightly different scales from one another. The largest kites were numbers 3, 5 and 7; we represented their length as 1 unit of length. Also, each of the kites seemed centred at the origin, but was rotated about the y-axis at a certain angle.

As such, we found the kites to have the following scales and rotations from the vertical axis:

Kites:

1, 9: 0.81 at +/-

2, 8: 0.77 at +/-

3, 5, 7: 1 at +/-, 0

4, 6: 0.93 at +/-

This can be visualized as follows:

To draw this in Maple we put together a simple procedure to draw each of the kites:

# Make a kite shape centred at the origin.
opts := thickness=4, color="#DC2828":
MakeKite := proc({scale := 1, rotation := 0})
local t, p, pts, x;

t := 0.267*scale;
pts := [[0, 0], [t, t], [0, scale], [-t, t], [0, 0]]:
p := plot(pts, opts);
if rotation<>0.0 then
p := plottools:-rotate(p, rotation);
end if;
return p;
end proc:

The main idea of this procedure is that we draw a kite using a standard list of points, which are scaled and rotated. Then to generate the sequence of plots:

shapes := MakeKite(rotation=-Pi/4),
MakeKite(scale=0.77, rotation=-2*Pi/5),

MakeKite(scale=0.81, rotation=-Pi/2),
MakeKite(scale=0.93, rotation=-Pi/8),
MakeKite(),
MakeKite(scale=0.93, rotation=Pi/8),
MakeKite(scale=0.81, rotation=Pi/2),
MakeKite(scale=0.77, rotation=2*Pi/5),
MakeKite(rotation=Pi/4),
plot([[0,-0.5], [0,0]], opts): #Add in a section for the maple leaf stem
plots:-display(shapes, scaling=constrained, view=[-1..1, -0.75..1.25], axes=box, size=[800,800]);

This looked pretty similar to the original logo, however the kites 2, 4, 6, and 8 all needed to be moved behind the other kites. This proved somewhat tricky, so we just simply turned on the point probe in Maple and drew in the connected lines to form these points.

shapes := MakeKite(rotation=-Pi/4),
plot([[-.55,.095],[-.733,.236],[-.49,.245]],opts),

MakeKite(scale=0.81, rotation=-Pi/2),
plot([[-.342,.536],[-.355,.859],[-.138,.622]],opts),
MakeKite(),
plot([[.342,.536],[.355,.859],[.138,.622]],opts),
MakeKite(scale=0.81, rotation=Pi/2),
plot([[.55,.095],[.733,.236],[.49,.245]],opts),
MakeKite(rotation=Pi/4),
plot([[0,-0.5], [0,0]], opts):
plots:-display(shapes, scaling=constrained, view=[-1..1, -0.75..1.25], axes=box, size=[800,800]);

## Classical differential geometry...

Unfortunately, the Differential Geometry package is too difficult for non-mathematicians. Is there a package in the Maple for classical differential geometry?

## computation geometry...

Hi

Is there any idea using Maple  to compute the leg lengths  in Stewart-Gough platform ( see the following figure)

I would like to make a code using cross-product and the unit Normal N

·       P is of length 13 and displaced in the Y direction by 10 degrees from the vertical (Z axis)

·       N is displaced in the X direction by 18 degrees from the vertical (Z axis)

·       LB is position [7 5] from the bottom plate centre in the XY plane

·       LT is in position [3.5 4.2] from the top-plate centre in the AB plane

Many thanks for any help

## Computation of geodesic connecting two points on a...

Respected members!

I downloaded this file because I've to study the geodesics over the cone, the sphere and the cylinder (and in general on a Riemannian manifold). But when I modify the equation, putting one of that I'm interested to, the file doesn't work (for example, it doesn't recognize the "assign" command). Could you help me, please?

Best,

Marzio

## How do I calculate the wedge product of differenti...

Dear all,

I have the following question, this code:

restart:
with(DifferentialGeometry):
DGsetup([w1,w2],N):
eq1 := ExteriorDerivative(w1);
eq2 := ExteriorDerivative(w1) &wedge ExteriorDerivative(w2);
eq1 &wedge eq2;

Gives the error:
Error, (in DifferentialGeometry:-Tools:-DGzero)  given degree, 3, exceeds that of frame dimension, 2

Unfortunately, I am not so familiar with differential geometry but as far as I know dw1  (dw1  dw2) = 0 should be correct.

best
baustamm1

## How to symbolically solve the parametric equations...

I want to calculate the ratio of the length of day and night for every latitude on earth ?
but i confused on using Maple in a wise way for finding the formula !
this is my demonstration :

Equations

the grat circle that divides the earth's surface into two dark and bright sides

 >
 (1.1)

circle of revolving of a point on earth in 24 hours

 >
 (1.2)

Visualization of dark and bright side the of earth

 >
 >

## Simple operation on geometry (area, equation)...

Hello :)

When points of triangle, A, B, C (R^2) are given, how to find area of triangle; equation and radius of circle which passes traingle(A,B,C) and height(AH).

or where can i find information about how to do this?

thank you so much:)

## Graph geometry cut off by a plane...

Hi.

This might not be a maple question. But I know maple has very smart and many plotting tools as well as this comunity is full of very skillfull people. So I concluded that I might get an answer here.

Lets say I got some sphere that is cut off by any abirutary plane, how would I go about plotting this?

For the purpose of this example the sphere is centred at the origin with radius of 4 and are cut of by the plane z=2-y.

## Possible to plot/draw triangles knowing only the s...

Hi, I was wondering, as stated in the title, if it is possible to plot/draw a triangle knowing only the sides and angles, and not the coordinates for each point making up its corners. If not, what would be the easiest way, to calculate the coordinates using the sides and angles (assuming I know the value for each side and corner) and then plot/draw it?

I'm rather green when it comes to using Maple, so if you could explain it in a simple way that would be appreciated.

## prove of circumference of a circle ...

how to Prove that the circumference of a circle of radius r is 2πr
on maple ?????

## how to find the intersection points of calabi yau ...

n := 5:
z1 := exp(2*3.14*I*k1/n)*cosh(z)^(2/n);
z2 := exp(2*3.14*I*k2/n)*sinh(z)^(2/n);
xx := Re(z1);
yy := Re(z2);
uu := cos(alpha)*Im(z1) + sin(alpha)*Im(z2);

i find that the 3d graph has many intersection points to itself

how to find these intersection points of calabi yau ?

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