## Center and radiius of a circle in 3d...

restart;
_local(D, O);
with(Student:-MultivariateCalculus);
A := [0, 0, 0];
B := [a, 0, 0];
C := [a, b, 0];
D := [0, b, 0];
S := [0, 0, h];
O := [x, y, z];
lineSC := Line(S, C);
lineSD := Line(S, D);
H := Projection(A, lineSC);
K := Projection(A, lineSD);
OH := H - O;
OK := K - O;
OC := C - O;
M := Matrix([OH, OK, OC]);
O := eval(O, %);
R := simplify(Distance(O, H));
O

Student:-MultivariateCalculus:-LineApplyFunction([0,0,h],

[Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(a,

rowalign = "", columnalign = "", groupalign = "",

rowspan = "1", columnspan = "1"), rowalign = "",

columnalign = "", groupalign = ""), Typesetting:-mtr(

Typesetting:-mtd(b, rowalign = "", columnalign = "",

groupalign = "", rowspan = "1", columnspan = "1"),

rowalign = "", columnalign = "", groupalign = ""),

Typesetting:-mtr(Typesetting:-mtd(uminus0h, rowalign = "",

columnalign = "", groupalign = "", rowspan = "1",

columnspan = "1"), rowalign = "", columnalign = "",

groupalign = ""), foreground = "[0,0,0]", readonly = "false",

align = "axis", rowalign = "baseline", columnalign = "center",

groupalign = "{left}", alignmentscope = "true",

columnwidth = "auto", width = "auto", rowspacing = "1.0ex",

columnspacing = "0.8em", rowlines = "none",

columnlines = "none", frame = "none",

framespacing = "0.4em 0.5ex", equalrows = "false",

equalcolumns = "false", displaystyle = "false", side = "right",

minlabelspacing = "0.8em")],variables=[x,y,z],parameter=t,id=1)

Student:-MultivariateCalculus:-LineApplyFunction([0,0,h],

[Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(0,

rowalign = "", columnalign = "", groupalign = "",

rowspan = "1", columnspan = "1"), rowalign = "",

columnalign = "", groupalign = ""), Typesetting:-mtr(

Typesetting:-mtd(b, rowalign = "", columnalign = "",

groupalign = "", rowspan = "1", columnspan = "1"),

rowalign = "", columnalign = "", groupalign = ""),

Typesetting:-mtr(Typesetting:-mtd(uminus0h, rowalign = "",

columnalign = "", groupalign = "", rowspan = "1",

columnspan = "1"), rowalign = "", columnalign = "",

groupalign = ""), foreground = "[0,0,0]", readonly = "false",

align = "axis", rowalign = "baseline", columnalign = "center",

groupalign = "{left}", alignmentscope = "true",

columnwidth = "auto", width = "auto", rowspacing = "1.0ex",

columnspacing = "0.8em", rowlines = "none",

columnlines = "none", frame = "none",

framespacing = "0.4em 0.5ex", equalrows = "false",

equalcolumns = "false", displaystyle = "false", side = "right",

minlabelspacing = "0.8em")],variables=[x,y,z],parameter=t,id=2)

/     2            2         / 2    2\ /
|    h  a         h  b     h \a  + b / |
HAssign|------------,------------,------------|
| 2    2    2  2    2    2  2    2    2|
\a  + b  + h  a  + b  + h  a  + b  + h \

/    2         2  /
|   h  b    h b   |
KAssign|0,-------,-------|
|   2    2  2    2|
\  b  + h  b  + h \

/                2                       2
|               h  a                    h  b
OHAssign|uminus0x + ------------,uminus0y + ------------,uminus0z
|            2    2    2             2    2    2
\           a  + b  + h             a  + b  + h

/ 2    2\ /
h \a  + b / |
+ ------------|
2    2    2|
a  + b  + h \

/                      2                    2  /
|                     h  b               h b   |
OKAssign|uminus0x,uminus0y + -------,uminus0z + -------|
|                     2    2             2    2|
\                    b  + h             b  + h \

OCAssign(uminus0x + a,uminus0y + b,uminus0z)

M := Matrix(3, 3, {(1, 1) = Typesetting:-mrow(Typesetting:-mo("&\

uminus0;"), Typesetting:-mi("x"), Typesetting:-mo("&plus;"),

Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-msup(Typese\

tting:-mi("h"), Typesetting:-mn("2")), Typesetting:-mo("&Invis\

ibleTimes;"), Typesetting:-mi("a")), Typesetting:-mrow(Typeset\

ting:-msup(Typesetting:-mi("a"), Typesetting:-mn("2")),

Typesetting:-mo("&plus;"), Typesetting:-msup(Typesetting:-mi(\

"b"), Typesetting:-mn("2")), Typesetting:-mo("&plus;"),

Typesetting:-msup(Typesetting:-mi("h"), Typesetting:-mn("2"))\

))), (1, 2) = Typesetting:-mrow(Typesetting:-mo("&uminus0;"),

Typesetting:-mi("y"), Typesetting:-mo("&plus;"), Typesetting:\

-mfrac(Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi("h"\

), Typesetting:-mn("2")), Typesetting:-mo("&InvisibleTimes;"),\

Typesetting:-mi("b")), Typesetting:-mrow(Typesetting:-msup(Ty\

pesetting:-mi("a"), Typesetting:-mn("2")), Typesetting:-mo("&p\

lus;"), Typesetting:-msup(Typesetting:-mi("b"), Typesetting:-m\

n("2")), Typesetting:-mo("&plus;"), Typesetting:-msup(Typesett\

ing:-mi("h"), Typesetting:-mn("2"))))), (1, 3) = Typesetting:-\

mrow(Typesetting:-mo("&uminus0;"), Typesetting:-mi("z"),

Typesetting:-mo("&plus;"), Typesetting:-mfrac(Typesetting:-mr\

ow(Typesetting:-mi("h"), Typesetting:-mo("&InvisibleTimes;"),

Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-msup(Type\

setting:-mi("a"), Typesetting:-mn("2")), Typesetting:-mo("&plu\

s;"), Typesetting:-msup(Typesetting:-mi("b"), Typesetting:-mn(\

"2"))))), Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi(\

"a"), Typesetting:-mn("2")), Typesetting:-mo("&plus;"),

Typesetting:-msup(Typesetting:-mi("b"), Typesetting:-mn("2"))\

, Typesetting:-mo("&plus;"), Typesetting:-msup(Typesetting:-mi\

("h"), Typesetting:-mn("2"))))), (2, 1) = Typesetting:-mrow(Ty\

pesetting:-mo("&uminus0;"), Typesetting:-mi("x")), (2, 2) =

Typesetting:-mrow(Typesetting:-mo("&uminus0;"), Typesetting:-\

mi("y"), Typesetting:-mo("&plus;"), Typesetting:-mfrac(Typeset\

ting:-mrow(Typesetting:-msup(Typesetting:-mi("h"),

Typesetting:-mn("2")), Typesetting:-mo("&InvisibleTimes;"),

Typesetting:-mi("b")), Typesetting:-mrow(Typesetting:-msup(Ty\

pesetting:-mi("b"), Typesetting:-mn("2")), Typesetting:-mo("&p\

lus;"), Typesetting:-msup(Typesetting:-mi("h"), Typesetting:-m\

n("2"))))), (2, 3) = Typesetting:-mrow(Typesetting:-mo("&uminu\

s0;"), Typesetting:-mi("z"), Typesetting:-mo("&plus;"),

Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mi("h"),

Typesetting:-mo("&InvisibleTimes;"), Typesetting:-msup(Typese\

tting:-mi("b"), Typesetting:-mn("2"))), Typesetting:-mrow(Type\

setting:-msup(Typesetting:-mi("b"), Typesetting:-mn("2")),

Typesetting:-mo("&plus;"), Typesetting:-msup(Typesetting:-mi(\

"h"), Typesetting:-mn("2"))))), (3, 1) = Typesetting:-mrow(Typ\

esetting:-mo("&uminus0;"), Typesetting:-mi("x"), Typesetting:-\

mo("&plus;"), Typesetting:-mi("a")), (3, 2) = Typesetting:-mro\

w(Typesetting:-mo("&uminus0;"), Typesetting:-mi("y"),

Typesetting:-mo("&plus;"), Typesetting:-mi("b")), (3, 3) =

Typesetting:-mrow(Typesetting:-mo("&uminus0;"), Typesetting:-\

mi("z"))})

Error, invalid input: eval received Matrix(3, 3, {(1, 1) = -x+h^2*a/(a^2+b^2+h^2), (1, 2) = -y+h^2*b/(a^2+b^2+h^2), (1, 3) = -z+h*(a^2+b^2)/(a^2+b^2+h^2), (2, 1) = -x, (2, 2) = -y+h^2*b/(b^2+h^2), (2, 3) = -z+h*b^2/(b^2+h^2), (3, 1) = -x+a, (3, 2) = -y+b, (3, 3) = -z}), which is not valid for its 2nd argument, eqns
/     1       / 2  2      2        2  2    2  2    2  2
R := |------------ \a  h  - 2 a  h z + a  x  + a  y  + a  z
| 2    2    2
\a  + b  + h

2      2  2      2        2  2    2  2    2  2
- 2 a h  x + b  h  - 2 b  h z + b  x  + b  y  + b  z

2      2  2    2  2    2  2\\
- 2 b h  y + h  x  + h  y  + h  z /|^(1/2)
|
/
Why this error ? Thank you.

## loops in MapleFlow...

Hi,

I like to know if it is possible to use a for loops in MapleFlow. I've tried a couple of times but it didn't work. For example,

``````n:=6
a:=Matrix(n,n)

for l from 1 to n do

for c from 1 to n do

a[l,c]:=2^(l+c)

end do

end do

a=``````

doesn't produce any result for a

Best regards,

## ERROR: Flow does not evaluate to a module...

I have Maple 2021 and Mapleflow 2021 installed. The Maple 2021 is working fine. However, MapleFlow gives the error as follows. ERROR: Flow does not evaluate to a module. My system OS is Windows 10 updates till April 2022

## Errors (red boxes) in Maple Flow...

I get this error when typing a sigma symbol with some subscripts.

Is there anyway to get rid of it?

If mapleflow doesn't allow typing in this way, how can I type it in an unexcutable Math format like document mode in Maple?

## Critical fix for Maplesoft products on macOS

Maple 2021 , MapleSim 2021 , Maple Flow 2021

We have just issued a critical fix to Maple, MapleSim, and Maple Flow running on macOS.

We have heard from some users who were experiencing serious problems with doubled characters while using Maplesoft products on macOS, including these reports on MaplePrimes. Further investigation determined that these problems appear specifically on macOS 11 and macOS 12.  I am happy to report that we have now corrected the problem, and a patch is available.

Anyone who uses macOS 11 or macOS 12 should install this update immediately. We also strongly recommend that all macOS users install this update, to avoid problems that may be triggered by future updates to your operating system.

To obtain this update:

For those who have experienced problems, we apologize for the inconvenience and thank you for your patience while we worked to find a solution.

## Maple-flow, cost, Linux version...

Hi.  I have not been able to find the individual cost of Maple Flow.  How much does it cost?  (in CDN)  Is it inlcuded in Maple?

Can Maple Flow be installed on Linux?  If not, will a Linux version be released on the future?

Regards

## Imaginary numbers in mapleflow...

Hi,

I tried a simple expressioin using imaginary numbers in MapleFlow,

a:=1 + 2*i

but got the following error:

¨'mn' is not numeric"

does anyone know what is happening?

thanks very much.

## shown palettes in mapleflow...

hi,

I looked at the file mapleflowrc and saw that it lists several palettes that are not visible in the workshheet dock.

Does anyone know if these palettes are available in mapleflow? And if so, how to make them visible in the worksheet dock?

## How can I get it to stop?...

I've written a Mapleflow document that is supposed to use a microphone array to find the intersection of three equations to locate the source of a gunshot.  But the calculation is "Evaluating," and has been for half an hour.  How do I interrupt the calculation?

I'd up load the file, but

So:

## After fitting a curve to data, how do I plot BOTH ...

When I fit a function to data [e.g., the logistic function to population data], I want to be able to plot both the function and the data. How do I do this in Maple Flow?

## Announcing Maple Flow

by: Maple Flow 2021

I’m excited to announce the launch of a new math tool called Maple Flow. Here, I’ll outline our motivation for developing this product, and talk about its features.

A large fraction of Maple users are professional engineers .

All use Maple, but very few say that they do math for a living, in much the same way a plumber wouldn’t say they use a wrench for a living.

They say things like:

• I design concrete retaining walls
• I simulate the transients on a transmission line
• I design heat exchangers
• I model the absorbency of diapers
• I design subsea pipelines
• I need to optimize the trajectory of a space shuttle
• I work for a power generation company doing load flow analysis
• I model how a robot arm needs to move

Some of these applications are mathematically simple (but are based on scientific principles, such as the conservation of heat, mass and momentum). The equations consist of basic arithmetic operations, trig and log functions, sprinkled with the occasional numeric integration.

Sometimes, the equations are already formalized in design guides, published by organizations like the IEEE, ASME or ISO. Given the specific physical context, engineers just need to implement the calculations in the right order (this is especially true for Civil and Structural engineering). These applications require you to think at an engineering level.

These are what we call design calculations, done by design engineers.

On the other end of the spectrum, some of these applications are mathematically complex. You might need to derive equations, manipulate PDEs, work with quaternions or transformation matrices, or do some programming. These applications require you to think at a mathematical level.

Let’s call the engineers doing this type of work research engineers. Research engineers are often more closely aligned with mathematicians than design engineers.

So we have design engineers and research engineers (and of course we have engineers with feet in both camps, to a varying degree).

Research engineers and design engineers do different mathematical things, and have different mathematical needs. Both groups use Maple, but one size doesn’t always fit well. Either the toe pinches a little, or the shirt is a mite too baggy.

This is where Maple Flow enters stage right.

Maple Flow is a new tool that we’ve built (and are continuing to expand and improve) with the needs of design engineers in mind.

• The worksheet lets you put math anywhere – just point, click and type
• The evaluation model is forward-in-space (unlike Maple’s forward in time evaluation model). This means the execution order is explicitly given by the position of the math on the canvas.
• The worksheet updates automatically, so results are never stale
• We’ve made several simplifications to massage away some of the complexity of the Maple programming language.
• You can use nearly all of tools in the Maple programming language.

Here’s how we see people using Maple Flow. They

• Enter a few major equations somewhere, followed by some parameters scattered around
• Make the equations “see” the parameters by moving the parameters above the equations
• Insert any parameters or equations you’ve forgotten, and move them into position, shifting the existing content out of the way to make room
• Add text, and perhaps an image or plot
• Finally, align math and format text for a presentable document

I’ve been using Maple Flow for a while now. I like the fact that the nature of Maple Flow means that you don’t have to start with a grand plan, with every computational detail planned out in advance. You’re encouraged to make things up as you go along, and gradually sculpt your calculations into shape.

Basically, Maple Flow doesn’t issue stiff penalties for making mistakes. You fix them, and then move on.

I also like that Maple Flow makes you feel like you’re “touching” your equations, shifting things about easily with either the mouse or the keyboard. There’s a certain tactility and immediacy to Maple Flow that gives me a micro dose of dopamine every time I use it.

Maple Flow’s freeform interface lets you experiment with space, alignment and layout, drawing attention to different groups of equations.

For example, you can design calculation documents that look like this.

You can use nearly all of the Maple programming language in Flow. Here’s a command from the plots package.

Here’s fsolve in action.