Posts - MaplePrimes

Maple and MapleSim 2024.2 updates

by: TechnicalSupport 750 Products: Maple 2024 , MapleSim 2024

October 31 2024

3 0

We have just released updates to Maple and MapleSim.

Maple 2024.2 includes ability to tear away tabs into new windows, improvements to scrollable matrices, corrections to PDF export, small improvements throughout the math engine, and more. We recommend that all Maple 2024 users install this update.

This update also include a fix to the problem with the simplify extension mechanism, as first reported on MaplePrimes. Thanks, as always, for helping us make Maple better.

This update is available through Tools>Check for Updates in Maple, and is also available from the Maple 2024.2 download page, where you can find more details.

At the same time, we have also released an update to MapleSim, which contains a variety of improvements to MapleSim and its add-ons. You can find more information on the MapleSim 2024.2 download page.

Announcing the MaplePrimes Art Gallery

by: John May 2991 Product: MaplePrimes

October 21 2024

13 1

With the 2024 Maple Conference coming up this week, I imagine one of two of you have noticed something missing. We chose not to have a Conference Art Gallery this year, because we have been working to launch new section of MaplePrimes: The MaplePrimes Art Gallery. This new section of MaplePrimes is designed for showing off your Maple related images, in a gallery format that puts the images up front, like Instagram but for Math.

To get the ball rolling on the gallery, I have populated it with many of the works from past years' Maple Art Galleries, so you can now browse them all in one place, and maybe give "Thumbs Ups" to art pieces that you felt should have won the coveted "People's Choice Award".

Once you are inspired by past entries, we welcome you to submit your new artwork! Just like the conference galleries, we are looking for all sorts of mathematical art ranging from computer graphics and animations, to photos of needlework, geometrical sculptures, or almost anything! Art Gallery posts are very similar to regular MaplePrimes posts except that you are asked to supply an Image File and a Caption that will displayed on the main Gallery Page, the post itself can include a longer description, Maple commands, additional images, and whatever else you like. See for example this Art Gallery post describing Paul DeMarco's sculpture from the 2021 Maple Conference Gallery - it includes an embedded worksheet as well as additional images.

I can't wait to see what new works of art our MaplePrimes community creates!

Vector Fundamentals and Visualization with Maple

by: Paras31 230 Product: Maple 2024

October 16 2024

0 1

I recently prepared a worksheet to teach vector fundamentals in one of my classes, and I wanted to share it with you all. It's nothing special, but I found Maple really helpful in demonstrating the concepts visually. Below is a breakdown of what the worksheet covers, with some Maple code examples included.

Feel free to take a look and use it if you find it useful! Any feedback or suggestions on how to improve it would be appreciated.

v := `<|>`(`<,>`(2, 3)); w := `<|>`(`<,>`(4, 1))

(1)

Basic Vector Operations

•	Addition and Subtraction

We can add and subtract vectors easily if they are of the same dimension.

u_add := v+w; u_sub := v-w

(2)

"The famous triangle law can be used for the addition of vectors and this method is also called the head-to-tail method,As per this law,two vectors can be added together by placing them together in such away that the first vector's head joins the tail of the second vector. Thus,by joining the first vector's tail to the head of the second vector,we can obtain the resultant vector sum."

Parallelogram Law of Addition of Vectors

"An other law that can be used for the addition of vectors is the parallelogram law of the addition of vectors*Let's take two vectors v and u,as shown below*They form the two adjacent sides of aparallelogram in their magnitude and direction*The sum v+u is represented in magnitude and direction by the diagonal of the parallelogram through thei rcommon point."

•	Scalar Multiplication

We can multiply a vector by a scalar. To multiply a vector by a scalar (a constant), multiply each of its components by the constant.

Suppose we let the letter λ represent a real number and u⃗ = (x,y) be the vector

v_scaled := 3*v

(3)

•

Error, (in LinearAlgebra:-Multiply) invalid arguments

Two vectors are opposite if they have the same magnitude but opposite direction.

•	Dot Product

Sometimes the dot product is called the scalar product. The dot product is also an example of an inner product and so on occasion you may hear it called an inner product.

Given the two vectors and
the dot product is,

(4)

(5)

(6)

(7)

•	Vector Norm (Magnitude)

To find the magnitude (or length) of a vector, use Norm.

norm_a := Norm(`#mover(mi("a"),mo("→"))`, 2); norm_b := Norm(`#mover(mi("b"),mo("→"))`, 2)

(8)

•	Calculate the Cosine Between Two Vectors

cos_theta := dot_product/(norm_a*norm_b)

(9)

(10)

(11)

We can determine whether two vectors are parallel by using the scalar multiple method or the determinant (area of the parallelogram formed by the vectors) method.

•	Scalar Multiple Method

a := Vector([5, -8]); b := Vector([10, -16]); k := a[1]/b[1]; is_parallel := a[2]/b[2] = k

(12)

•	Determinant Method

determinant := a[1]*b[2]-a[2]*b[1]; result := determinant = 0

(13)

Two vectors a and b are perpendicular (vertical)

•	if and only if their dot product is zero

a1 := Vector([1, 2]); b1 := Vector([-2, 1]); dot_product := DotProduct(a1, b1); is_perpendicular := dot_product = 0

(14)

•	Slope Method

slope_a := a1[2]/a1[1]; slope_b := b1[2]/b1[1]; is_perpendicular := slope_a*slope_b = -1

(15)

•	Special case: If one vector is vertical (undefined slope) and the other is horizontal (zero slope), they are perpendicular.

a2 := Vector([0, 3]); b2 := Vector([4, 0]); is_perpendicular := a2[1] = 0 and b2[2] = 0

(16)

Download vectors.mw

Triggering Command Completion with Enter in Maple...

by: TechnicalSupport 750 Product: Maple 2024

October 16 2024

2 0

The tab key, and the mouse can be used to trigger completion selection in Maple 2024.1 like previous versions.

For interface resposiveness, a new option (disabled by default) has been added to the interface tab of Maple 2024.1 Options (available from the Tools menu). Users that prefer to use enter to trigger completion selection can enable the option.

Change the option here and apply to the session or globally if using enter to trigger completion selection is preferred.

A customised column graph

by: mmcdara 7721 Product: Maple

October 14 2024

5 0

This is an answer to the last question ASHAN recently asked.
As this answer goes beyond the simple example supplied by ASHAN in that it extends the possibilities of Statistics:-ColumnGraph, I thought it would be appropriate to share it with the entire community of Maple users.

Possible improvements:

add controls for parameter types
add consistency controls (numer of colors = number of data rows for instance)
add some custom parameters
improve legend position

>

restart

>

CustomColumnGraph := proc(
                       data
                       , BarColors
                       , GroupTitles
                       , BarWidth
                       , GroupOffset
                       , LegendBarHeight
                       , VerticalShiftFactor
                       , BarFont
                       , GroupFont
                       , LegendFont
                       , BarTitleFormat
                     )

  local Ni, Nj, W, DX, LH, LW, SF, L, DL:

  local SingleBar := (w, h, c) -> rectangle([0, 0], [w, h], color=c):
  local BarGraph, Legends:

  uses plots, plottools:

  Ni := numelems(data[..,1]):
  Nj := numelems(data[1]):
  W := BarWidth:
  DX := GroupOffset:
  LH := LegendBarHeight:
  SF := VerticalShiftFactor;
  L := DX*(Ni-1) + W*(Nj-1) + DX/2 + DX/4:
  LW := L / Ni / Nj / 2;                          # LegendBarWidth
  DL := solve(dl*(Nj+1) + Nj*(3*LW) = L, dl);

  BarGraph := display(
                seq(
                  seq(
                    translate(SingleBar(W, data[i, j], BarColors[j]), DX/4 + DX*(i-1) + W*(j-1), 0)
                    , i=1..Ni
                  )
                  , j=1..Nj
                )
                , axis[1] = [gridlines = [], tickmarks = []]
                , axis[2] = [gridlines = [11, linestyle = dot, thickness = 0, color = "LimeGreen"]]
              ):

  Legends := display(
               seq(
                 translate(SingleBar(LW, LH, BarColors[j]), DL/2 + LW + (3*LW+DL)*(j-1), min(data)*(2*SF-1))
                 , j=1..Nj
               )
               ,
               seq(
                 translate(
                   textplot([0, LH/2, GroupTitles[j]], align=right, font = LegendFont)
                   , DL/2 + 2*LW + (3*LW+DL)*(j-1), min(data)*1.4)
                   , j=1..Nj
               )
             ):

  display(
    BarGraph
    , Legends

    #------------------------------- X base line
    , plot(0, x=0..DX*(Ni-1)+W*(Nj-1)+DX/2 + DX/4)

    #------------------------------- Left bar group boundaries
    ,
    seq(
      line(
        [DX/4 + DX*(i-1) - W/2, min(data)*SF],
        [DX/4 + DX*(i-1) - W/2, max(data)*SF]
        , color="LightGray"
      )
      , i=1..Ni
    )

    #------------------------------- Right bar group boundaries
    ,
    seq(
      line(
        [DX/4 + DX*(i-1) + W*Nj + W/2, min(data)*SF],
        [DX/4 + DX*(i-1) + W*Nj + W/2, max(data)*SF]
        , color="LightGray"
      )
      , i=1..Ni
    )

    #------------------------------- Top bar group boundaries
    ,
    seq(
      line(
        [DX/4 + DX*(i-1) - W/2, max(data)*SF],
        [DX/4 + DX*(i-1) + W*Nj + W/2, max(data)*SF]
        , color="LightGray"
      )
      , i=1..Ni
    )

    #------------------------------- Bar titles
    ,
    seq(
      seq(
        textplot(
          # [DX/4 + DX*(i-1) + W*(j-1) +W/2, data[i, j], evalf[3](data[i, j])]
          [DX/4 + DX*(i-1) + W*(j-1) +W/2, data[i, j], sprintf(BarTitleFormat, data[i, j])]
          , font=BarFont
          , align=`if`(data[i, j] > 0, above, below)
        )
        , i=1..Ni
      )
      , j=1..Nj
    )

    #------------------------------- Bar group titles
    ,
    seq(
      textplot(
        [DX/4 + DX*(i-1) + W*(Nj/2), max(data)*SF, cat("B", i)]
        , font=GroupFont
        , align=above
      )
      , i=1..Ni
    )

    , labels=["", ""]
    , size=[800, 400]
    , view=[0..DX*(Ni-1) + W*(Nj-1) + DX/2 + DX/4, default]
  );
end proc:

Data from AHSAN

>

A := [0.1, 0.5, 0.9]:
B := [0.5, 2.5, 4.5]:

Nu_A := [-0.013697, 0.002005, 0.017707, -0.013697, 0.002005, 0.017707, -0.013697, 0.002005, 0.017707]:
Nu_B := [0.010827, 0.011741, 0.012655, 0.013967, 0.014881, 0.015795, 0.017107, 0.018021, 0.018935]:

i:=4:
data := Matrix([[Nu_A[i], Nu_B[i]], [Nu_A[i + 1], Nu_B[i + 1]], [Nu_A[i + 2], Nu_B[i + 2]]]):

Column Graph

>

Ni, Nj := LinearAlgebra:-Dimensions(data):

CustomColumnGraph(
                   data                                        # data
                   , ["Salmon", "LightGreen"]                  # BarColors
                   , ["Sensitivity to A", "Sensitivity to B"] # GroupTitles
                   , 1/(numelems(data[1])+2)                   # BarWidth
                   , 1                                         # GroupOffset
                   , max(abs~(data))/10                        # LegendBarHeight
                   , 1.2                                       # VerticalShiftFactor
                   , [Tahoma, bold, 10]                        # BarFont
                   , [Tahoma, bold, 14]                        # GroupFont
                   , [Tahoma, bold, 12]                        # LegendFont
                   , "%1.5f"                                   # BarTitleFormat
                 )

Column Graph of an extended set of data

>

ExtentedData := < data | Statistics:-Mean(data^+)^+ >:
ExtentedData := < ExtentedData, Statistics:-Mean(ExtentedData) >:

interface(displayprecision=6):
data, ExtentedData;

Ni, Nj := LinearAlgebra:-Dimensions(ExtentedData):

CustomColumnGraph(
                   ExtentedData
                   , ["Salmon", "LightGreen", "Moccasin"]
                   , ["Sensitivity to A", "Sensitivity to B", "Mean Sensitivity to A and B"]
                   , 1/(numelems(data[1])+2)                   # BarWidth
                   , 1                                         # GroupOffset
                   , max(abs~(data))/10                        # LegendBarHeight
                   , 1.2                                       # VerticalShiftFactor
                   , [Tahoma, 10]                              # BarFont
                   , [Tahoma, bold, 14]                        # GroupFont
                   , [Tahoma, 12]                              # LegendFont
                   , "%1.3f"                                   # BarTitleFormat
                 )

>

Download CustomColumnGraph.mw

Throwing a ball into a tube - The Golf Ball Parado...

by: C_R 3347 Product: MapleSim

October 13 2024

10 10

This post is about the visualization of a gyroscopic phenomenon of a rotating body. MapleSim models and a description for those who do not have MapleSim are provided for their own analysis. Implementation with other tools like Maple might give further insight into the phenomenon.

With appropriate initial conditions, a ball thrown into a tube can pop out of the tube. This can be reproduced with a MapleSim model

Throwing_a_ball_into_a_tube_A.msim

To hit a perfect shot without trial and error, time reversal was applied for the model (reversed calculation results of a ball exiting the tube are used as initial conditions for the shot). This worked straight away and shows that this model is sufficiently conservative.

This phenomenon has recently attracted attention on YouTube. For example, Steve Mold demonstrates the effect and provides an intuitive explanation which he considers incomplete because the resulting vertical oscillation of the ball does not match theory and his experiments. He suspects that the assumption of a constant axis of rotation of the ball is responsible for this discrepancy.

However, he cannot demonstrate a change of the axis of rotation. In general, the visualization of the rotation axis of a ball is difficult to achieve in an experiment. On the contrary, visualization is much easier in a simulation experiment with this model:

Throwing_a_ball_into_a_tube_B.msim

The following can be observed for a trajectroy that does not exit the tube:

At the apex (the top) of the trajectory, the vector of rotation (red bold in the following images) points downwards and is essentially parallel to the axis of the cylinder. The graph to the left shows the vertical (in green) position and one horizontal position (in red). The model applies gravity in negative y direction.

Ein Bild, das Text, Diagramm, Screenshot, Reihe enthält.

Automatisch generierte Beschreibung

On the way down, the axis of rotation points away from the direction of travel (the ball orbits counterclockwise in the top view).

Ein Bild, das Text, Diagramm, Screenshot, Reihe enthält.

Automatisch generierte Beschreibung

At the bottom, the vector of rotation points towards the axis of the cylinder.

Ein Bild, das Text, Diagramm, Screenshot, Reihe enthält.

Automatisch generierte Beschreibung

On the way up, the axis of rotation points in the direction of travel.

Ein Bild, das Text, Diagramm, Screenshot, Reihe enthält.

Automatisch generierte Beschreibung

These observations confirm that the assumption of a constant axis of rotation is too simplified. Effectively the ball performs a precession movement know from gyroscopes. More specifically, the precession movement of the rotation axis rotates in the opposite direction of the rotation of the ball.

However, the knowledge and the visualization of this precession movement do not provide more insight for a better intuitive explanation of the effect. As the ball acts like a gyroscope, a second attempt is to visualize forces that perturb the motion of the ball. Besides gravity, there are contact forces exerted by the tube. The normal force at the contact as well as the gravitational force cannot generate a perturbing momentum since they point to the center of the ball. Only frictional forces at the contact can cause a perturbing momentum.

Contrary to the visualization of the axis of rotation, visualization of contact forces is not straight forward in MapleSim, because neither the contact point nor the contact forces are directly provided by components of the MapleSim library. Only for a single contact point, a work-around is possible by measuring the reactive forces on the tube and then displaying these forces in a moving reference frame at the contact point. The location and the orientation of this frame are calculated with built-in mathematical components. To illustrate the additional effort, the image below highlights in yellow the components only needed for the visualization of the above images, all other components were required to visualize the contact forces and frictional moments.

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MaplePrimes Posts

MaplePrimes Posts are for sharing your experiences, techniques and opinions about Maple, MapleSim and related products, as well as general interests in math and computing.

Latest Post
Latest Posts Feed

Maple and MapleSim 2024.2 updates

Announcing the MaplePrimes Art Gallery

Vector Fundamentals and Visualization with Maple

Triggering Command Completion with Enter in Maple...

A customised column graph

Throwing a ball into a tube - The Golf Ball Parado...

Ask a Question

Create a Post

MaplePrimes Posts

MaplePrimes Posts are for sharing your experiences, techniques and opinions about Maple, MapleSim and related products, as well as general interests in math and computing.

Latest Post Latest Posts Feed

Maple and MapleSim 2024.2 updates

Announcing the MaplePrimes Art Gallery

Vector Fundamentals and Visualization with Maple

Triggering Command Completion with Enter in Maple...

A customised column graph

Throwing a ball into a tube - The Golf Ball Parado...

Latest Post
Latest Posts Feed