Everything is simple, until you go underwater – This is what the University of Waterloo Submarine Racing team, or in short ‘WatSub’ coined as their motto. Never mind learning to scuba dive, and dealing with such things as rust, this newly formed team would have to compete against university teams with a decade or more of experience.

But that did not deter the team, and they started work on Ontario’s first submarine racing project. The team approached Maplesoft to be a sponsor and we are proud to have supported this ingenious venture. The team has used Maplesoft technology in the design and testing of the submarine.

“Maple has been our go-to calculations and analysis tool throughout the development of Amy (2015-2016 season), and we will continue using it throughout the development of Bolt (2016-2017 season),” said Gonzalo Espinoza Graham, President of the WatSub Team. “Its familiar interface and computing environment allowed us to set design benchmark targets from early on the design process and follow through with them on the later stage.”

What started as an engineering project in December 2014, becoming officially the first submarine racing team in Ontario. The team soon grew to over 130 general members and a tight core-team, who were eager to tackle new challenges.  The team resides inside the Sedra Student Design Centre, University of Waterloo’s state of the art facility that houses over 25 student teams, the largest of its kind in North America.  

WatSub made its first appearance on the European International Submarine Races (eISR) back in July 2016, with its 1st submarine ‘Amy’, where a single scuba diver piloted the submarine and propelled it through an unforgiving winding course marked by obstacles and turns 10 meters underwater. The team has since then participated in other competitions and is constantly improving the design and performance of the submarine, learning from each competition they participate in.  Next year Amy will participate in the 14th edition of the eISR international competition. “I think the greatest thing we learned is never to give up,” said Ana Krstanovic, a third-year political science student who manages communications for the team. “We’re more motivated now than ever.”

Ojaswi Tagore, Gonzalo Espinoza Graham, and Janna Henzl represented WatSub at the European International Submarine Race in Gosport, UK.

Another example of an innovative project that Maplesoft supported in 2016 is Waterloop: The Canadian SpaceX Hyperloop Competition Team, Canada's only SpaceX Hyperloop Pod Competition team. This project, which could change the way we travel in the future, is driven by a group of dedicated University of Waterloo students who have taken on the challenge to design and build a functional prototype Hyperloop pod. They will test it on a one-mile test track in Hawthorne, California in January 2017, pitting it against 22 of the 1200+ teams who originally entered the competition.

The Hyperloop is a conceptual next generation high-speed transit system that will take commuters between cities at speeds over 1,000 km/h. The technology will differ from previous rail transit by having pods ride on a cushion of air in a reduced pressure tube in order to reach greater speeds with a smoother ride, and is powered entirely by renewable energy.

 The Hyperloop Pod Competition was launched by Elon Musk, the billionaire engineer and founder of SpaceX and Tesla Motors.  The competition is separated into 3 rounds. The first one was held in late December, where selected teams sent in their initial designs to be reviewed. From there, 180 teams were chosen to compete at Texas A&M University. Each team set up a booth and a panel of judges critiqued them and chose 31 teams to move onto the final, build and test stage.

Waterloop Goose I

Waterloop Goose X

The GOOSE I is Waterloop’s half-scale, functional prototype vehicle pod, which will be the one in the competition.  The GOOSE X pod is a conceptual full size Hyperloop vehicle inspired by the prototype they are building. The full size pod will have a capacity of 26 passengers per pod.

"Our prototype has been designed to be as simple and economical as possible, while still performing all necessary functions for the full size Hyperloop. If it is successful, it has the potential to revolutionize the transit industry in the same manner the train and airplane has before it," said Montgomery de Luna, architectural design lead for Waterloop. “We would like to thank Maplesoft for their generous support.  Without sponsors like Maplesoft supporting our vision and encouraging innovative student projects, we wouldn’t be able to achieve our goal.”

Revolutionizing the transportation industry isn’t easy and is at times frustrating and time consuming for these teams, but having the best tools and resources will ensure that the teams have a good chance at excelling in competitions and creating innovative models that could change our future.

The code for the animation:

L:=[[-0.12,2],[-0.14,0],[0.14,0],[0.12,2]]:
L1:=[[0.05,2],[4,1],[2,4],[3.5,3.5],[1,7],[2,6.5],[0,10]]:
A:=plot(L, color=brown, thickness=10):
B:=plot([op(L1),op(map(t->[-t[1],t[2]],ListTools:-Reverse(L1)))], color="Green", thickness=10):
C:=plottools:-polygon([op(L1),op(map(t->[-t[1],t[2]],ListTools:-Reverse(L1)))], color=green):
Tree:=plots:-display([A, B, C], scaling=constrained, axes=none):
T:=[[-3.2,-2, Happy, color=blue, font=[times,bold,30]], [0,-2,New, color=blue, font=[times,bold,30]], [2.5,-2,Year, color=blue, font=[times,bold,30]], [-5,-3.5, "&", color=yellow, font=[times,bold,30]],[-2.5,-3.5, Merry, color=red, font=[times,bold,30]], [2.3,-3.5, Christmas!, color=red, font=[times,bold,30]], [0,-5, "2017", color=cyan, font=[times,bold,36]]$5]:
F:=k->plottools:-homothety(Tree, k, [0,5]):
A:=plots:-animate(plots:-display, ['F'(k)], k=0..1, frames=60, paraminfo=false):
B:=plots:-animate(plots:-textplot,[T[1..round(i)]], i=0..nops(T), frames=60, paraminfo=false):
plots:-display(A, B, size=[500,550], scaling=constrained);

Christmas_Tree.mw

 Edit.

Parametric equation of second-order curve in 3d. Draghilev method.
PLAN_CURVE_3d_1.mw
Examples:
x1^2+x1*x3+13*x2^2+x3-1=0;
x1+x2+x3=0;

 x1^2+0.1*x2^2+x3^2-9=0;
 x1+3*x3+1=0;

 x1^2-0.1*x2^2+x3^2-9=0;
 x1+3*x3+1=0;

Parametric equation of a circle in 3d by three points. Draghilev method.

CIRCLE_3_POINTS_geom3d_2.mw

In this post I want to present an easy method to obtain a discrete parametrization of a surface S defined implicitly (f(x,y,z)=0).
This problem was discussed here several times, the most recent post is
http://www.mapleprimes.com/posts/207661-Isolation-Of-Sides-Of-The-Surface-On-The-Graph

S is supposed to be the boundary of a convex body having (x0,y0,z0) an interior point and contained in a ball of radius R centered at (x0,y0,z0).
Actually, the procedure also works if the body is only star-shaped with respect to the interior point, and it is also possible to plot only a part of the surface
inside a solid angle centered at (x0,y0,z0).

Usage:
Par3d(f, x=x0, y=y0, z=z0, R, m, n,  theta1 .. theta2,  phi1 .. phi2)

f           is an expression depending on the variables x, y, z
x0, y0, z0  are the coordinates of the interior point
R           is the radius of the ball which contains the surface,
m, n        are the numbers of the grid lines which will be generated
The last two parameters are optional and are used when only a part of S will be parametrized.

The procedure Par3d returns a MESH structure M, which can be plotted with PLOT3D(M).

Par3d :=proc(f,x::`=`,y::`=`,z::`=`,R,m,n,th:=0..2*Pi,ph:=0..Pi)
    local A,i,j, rij,fij,Cth,Sth,Cph,Sph, theta,phi, r;
    A:=Array(1..m+1,1..n+1,1..3,datatype=float[8]);
    for i from 0 to m do for j from 0 to n do
      theta:=op(1,th)+i/m*(op(2,th)-op(1,th));
      phi:=op(1,ph)+j/n*(op(2,ph)-op(1,ph));
      Cth:=evalf(cos(theta)); Sth:=evalf(sin(theta));
      Cph:=evalf(cos(phi));   Sph:=evalf(sin(phi));
      fij:= eval(f, [lhs(x)=rhs(x)+r*Sph*Cth, lhs(y)=rhs(y)+r*Sph*Sth, lhs(z)=rhs(z)+r*Cph]);
      rij:=fsolve(fij,r=0..R);  if [rij]::list(numeric) then rij:=min(rij) fi; 
      if [rij]=[] or not(type(rij,numeric)) then print(['i'=i,'j'=j], fij); rij:=undefined fi; 
      A[i+1,j+1,1]:=evalf(rhs(x)+rij*Sph*Cth);
      A[i+1,j+1,2]:=evalf(rhs(y)+rij*Sph*Sth);
      A[i+1,j+1,3]:=evalf(rhs(z)+rij*Cph);
    od;od:
    MESH(A);
end:

The procedure is not optimized, e.g.
- Cth, etc could be Vectors computed outside the loops
- Some small changes to use evalhf.

###### EXAMPLES ######

f1 := x^2+3*y^2+4*z^2 - x*y - 2*y*z - 10:
plots:-implicitplot3d(f1, x=-5..5, y=-5..5, z=-2..2);

M:=Par3d(f1, x=0,y=0,z=0,5,40,40):
PLOT3D(M);

f2 := x^4+y^4+z^4-1:
M:=Par3d(f2, x=0,y=0,z=0,5,40,40):
PLOT3D(M);

M:=Par3d(f2, x=0,y=0,z=0, 5,40,40, 0..Pi, 0 .. Pi/3): #Plot half of the top only
plots:-display(PLOT3D(M), scaling=constrained);

M:=Par3d(f2,      x=0,y=0,z=0, 5,30,30, 0..Pi, 0 .. Pi):
N:=Par3d(f2+0.01, x=0,y=0,z=0, 5,30,30, 0..Pi, 0 .. Pi):
plots:-display(PLOT3D(M), color=red):
plots:-display(PLOT3D(N), color=green):
plots:-display(%,%%, orientation=[-40,65,10]);

f3 := (x^2+y^2-1)^2+(z+sin(x*y+z))^4-120:
plots:-implicitplot3d(f3, x=-4..4,y=-4..4,z=-5..5, numpoints=10000);

Par3d(f3, x=0,y=0,z=0,5, 30,30):
PLOT3D(%);

Note.
The procedure could be used to plot locally around a point (x0,y0,z0)
One may use the spherical coordinates (theta0,phi0) and then call the procedure taking theta0-a .. theta0+a,  phi0-b, .. phi0+b  for the trailing parameters
The spherical coordonates can be computed using:

ThetaPhi :=proc(x,y,z, X,Y,Z)
    local r:=sqrt((X-x)^2+(Y-y)^2+(Z-z)^2);
    ['theta'=arctan(Y-y,X-x), 'phi'=arccos((Z-z)/r)]
end:

ThetaPhi(10,20,30, 11,21,28);evalf(%);

One way is coloring a surface on both sides. We build equidistant surface with very small radius and stain the equidistant surface in color different from the color of the original surface.
Examples coloring of surfaces on both sides.  Radius equal to abs (0.0001).
x3-0.5*exp(sin(x1+2.5*x2+x3))=0;
(x1^2+x2^2-0.4)^2+(x3+sin(x1*x2+x3))^4-0.1=0;
2_COLORS.mw

Let us consider

restart; Digits := 20; evalf(Int(abs(cos(1/t)), t = 0 .. 0.1e-1), 3);
   -0.639e-2

Pay your attention to the minus sign. Simply no words. Mma produces 0.006377.

evalf@Int.mw

A population  governed by the logistic equation with a constant rate of harvesting satisfies the initial value problem . This model is typically analyzed by setting the derivative equal to zero and finding the two equilibrium solutions . A sketch of solutions  for different values of a suggests that the larger equilibrium is stable; the smaller, unstable.

When a is less that the unstable equilibrium,  becomes zero at a time , and the population becomes extinct. If  is not interpreted as pertaining to a population, its graph exists beyond , and actually has a vertical asymptote between the two branches of its graph.

In the worksheet "Logistic Model with Harvesting", two questions are investigated, namely,

1. How does the location of this vertical asymptote depend on on a and h?
2. How does the extinction time , the time at which , depend on a and h?

To answer the second question, an explicit solution , readily provided by Maple, is set equal to zero and solved for . It turns out to be difficult both to graph the surface  and to obtain a contour map of the level sets of this function. Instead, we solve for  and obtain a graph of  with  as a slider-controlled parameter.

To answer the first question, the explicit solution, which has the form , exhibits its vertical asymptote when . Solving this equation for  gives the time at which the vertical asymptote is located, a function that is as difficult to graph as . Again the remedy is to solve for, and graph, , with  as a slider-controlled parameter.

Download the worksheet: Logistic_with_Harvesting.mw

Let us consider 

Student[Precalculus]:-LimitTutor(sqrt(x), x = 2);

One expects a nice illustration of the result sqrt(2). But instead of that one reads "f(x) approaches 1.41 as x approaches 2". This is simply clueless and forms a wrong understanding of limits. It should also be noticed that all the entries (left, 2-sided, and right) produce the same animation. The same issue with other limits I tried, e.g.

Student[Precalculus]:-LimitTutor(sqrt(x), x = 1);

. I think this command should be completely rewritten or excluded from Maple. 

A number of MaplePrimers have asked how one might use the section and subsections of a Maple worksheet to structure the source code of an extended Maple package.  The usual answer is that it cannot be done; a module-based Maple package must be assigned in a single input region in a worksheet.  A recommended alternative is to write the source in text files and use either command line tools or the Maple read command from a worksheet to assign the package.  Because the read command handles Maple preprocessor macros, specifically the $include macro, the source can be conveniently split into smaller files.

I prefer this file-based method for development because text files are generally more robust than Maple worksheets, can be edited with the user's preferred editor, can be put under version control, and can be searched and modified by standard Unix-based tools.  However, not everyone is familiar with this method of development.  With that in mind, I wrote a small Maple package, CodeBuilder, that permits splitting the source of a Maple package (or any Maple code) into separate code edit regions in a standard Maple worksheet, using $include macros to include the source of other regions.  To build the package, the code edit regions are written to external files, using the names of the regions as the local file name relative to a temporary directory.

The package includes a method to run mint on the source code.  The result can be either printed in the worksheet or displayed in a pop-up maplet that allows selecting the infolevel and the region to check.

CodeBuilder includes help pages and a simple example (referenced from the top-level help page) demonstrating the usage.  To install the package, unzip the attached zip file and follow the directions in the README file.

CodeBuilder-1-0-3.zip

Errata Just noticed that a last minute change broke some of the code.  Do not bother with the 1-0-1 version; I'll upload a new version shortly.  The latest version (1-0-3) is now available.

Let us look in RealDomain and then in the RealDomain:-solve command. One is addressed to the usual solve command. The commands of the RealDomain package are not still documented since Maple 7 when the package was introduced. There is a general description only 

• By default, Maple performs computations under the assumption that the underlying number system is the complex field. The RealDomain package provides an environment in which computations are performed under the assumption that the basic underlying number system is the field of real numbers.
• Results returned by procedures are postprocessed by discarding values containing any detectable non-real answers or replacing them with undefined where appropriate.

The above is not enough. Here is an example which confuses me: 

RealDomain:-solve(exp(I*x) = -1, AllSolutions);
NULL

though 

solve(exp(I*x) = -1, AllSolutions);
                         Pi (2 _Z1 + 1)

and 

RealDomain:-solve(exp(I*x) = -1);
                               Pi

I lie awake thinking about that. Maplesoft staff help me!

Let us consider 

Statistics:-Mode(Binomial(n, p));
                        floor((1 + n) p)

Up to Wiki, the output is not correct. Simply no words.

There seems to be a bug in determining the folowing integral analytically:

Maple gives as a result

However, numerically integrating it

gives

In fact, integrating it from a to b,

gives

suggesting that Maple thinks the integrand is just 3/2. If one plots it, then it becomes obvious that this is not the case.

There were recently submitted a dozen Maple bugs by me and others. Maplesoft have brought no responses. They keep strategic silence. True merit is not afraid of criticism.

Download Bug_in_Statistics_PDF.mw

Let us consider 

restart; J := int(cos(a*x)^2/(x^2-1), x = -infinity .. infinity, CPV);
-(1/4)*Pi*sin(2*a)*csgn(I*a)-(1/4)*Pi*sin(2*a)*csgn(I/a)

This result is not true for a=I:

eval(J, a = I);
                               0

In this case the integral under consideration diverges because of 

cos(I*x)^2;
                                
                            cosh(x) ^2