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The evaluate function described in the previous post has been used in writing a UDF that will evaluate the integral of any suitable function over a specified range.

The revised worksheet can be downloaded from:

Eval.zip

EvalInt

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OK, here are some of the promised basics of using VBA to create drawings in Excel.

This and following posts will cover:

• The shape objects and how to create them
• How to select, group, and delete shapes
• How to modify existing shapes
• How to get the properties of existing shapes
• How to scale shapes
• How to create 3D perspective wireframe drawings

There are two main problems with creating code driven drawings in Excel:

• There are several alternative ways to do almost everything, and the ways they work are not always obvious (at least to me), and there are significant difference between versions.
• The documentation is sparse, and what there is is often far from helpful.  To make matters worse, there is little on the subject in the popular Excel books, and even web sites covering the topic are few and far between.

In the rest of this post I will list the available VBA methods that apply to shape objects, then look at some examples of how these methods work in practice.

The methods applicable to shapes are:

• AddCallout
• AddChart
• AddConnector
• AddCurve
• AddFormControl
• AddLabel
• AddLine
• AddOLEObject
• AddPicture
• AddPolyline
• AddShape
• AddTextbox
• AddTextEffect
• BuildFreeform
• SelectAll

The ones I will be concentrating on are:

• AddCurve
• AddLine
• AddPolyline
• AddShape
• AddTextbox

Which will give us more than enough for what we want to do.

The examples given below can be found in: Plot Shapes.zip

The examples are based on drawing similar dodecagons (regular 12 sided polygons) by various methods.  The coordinates defining the shapes are listed on the spreadsheet, we will be looking at how screen coordinates work more closely in a later post.  For now just note that the shapes are defined by 12 x,y coordinates, with the first point repeated at the end of the list, to create a closed shape.  There are also 4 straight lines, defined by the start and end points.


Sub ExShapeAdd()
Dim PointArray() As Single, CoordA As Variant, shp As Shape, i As Long, ffshp As Shape
Dim myBuilder As FreeformBuilder, XNode As Single, YNode As Single

ShapeDelete ‘Routine to delete old shapes, see later post
CoordA = [a1:b13] ‘ Get coordinates array from the worksheet

‘ AddLine
With ActiveSheet.Shapes.AddLine(CoordA(1, 1), CoordA(1, 2), CoordA(7, 1), CoordA(7, 2))
.Name = “straight”
.Line.Weight = 2
.Line.ForeColor.SchemeColor = 8
End With

‘Plot the same line and rotate through 45 degrees
With ActiveSheet.Shapes.AddLine(CoordA(1, 1), CoordA(1, 2), CoordA(7, 1), CoordA(7, 2))
.Name = “straight-a”
.Rotation = 90
End With

‘ Copy CoordA (variant) into PointArray (single)
ReDim PointArray(1 To 13, 1 To 2)
For i = 1 To 13
PointArray(i, 1) = CoordA(i, 1)
PointArray(i, 2) = CoordA(i, 2)
Next i

‘ AddCurve
Set shp = ActiveSheet.Shapes.AddCurve(PointArray)
With shp
.Fill.Visible = False
.Name = “Curve1″
End With
With ActiveSheet.Shapes.AddCurve(PointArray)
.Fill.Visible = False
.Rotation = 45
.Name = “Curve2″
End With

AddCurve


CoordA = [a15:b27]
‘ AddShape; 183 = straight connector
ActiveSheet.Shapes.AddShape(183, CoordA(7, 1), CoordA(1, 2), CoordA(1, 1) - CoordA(7, 1), CoordA(7, 2) - CoordA(1, 2)).Name = “Straight2″
With ActiveSheet.Shapes.AddShape(183, CoordA(7, 1), CoordA(1, 2), CoordA(1, 1) - CoordA(7, 1), CoordA(7, 2) - CoordA(1, 2))
.Name = “Straight2-A”
.Rotation = 90
End With

For i = 1 To 13
PointArray(i, 1) = CoordA(i, 1)
PointArray(i, 2) = CoordA(i, 2)
Next i

‘ AddPolyLine
ActiveSheet.Shapes.AddPolyline(PointArray).Fill.Visible = False

AddPolyLine


CoordA = [a29:b41]
For i = 1 To 13
PointArray(i, 1) = CoordA(i, 1)
PointArray(i, 2) = CoordA(i, 2)
Next i

‘ Shapebuilder
‘ first point
XNode = PointArray(1, 1)
YNode = PointArray(1, 2)
Set myBuilder = ActiveSheet.Shapes.BuildFreeform(msoEditingAuto, XNode, YNode)
‘ remaining points
For i = 2 To 13
XNode = PointArray(i, 1)
YNode = PointArray(i, 2)
myBuilder.AddNodes msoSegmentLine, msoEditingAuto, XNode, YNode
Next

Set ffshp = myBuilder.ConvertToShape
ffshp.Name = “Built-shape”

We are so familiar with the formula for the area enclosed by a circle that we tend not to think much about how it was derived, at least I don’t.

The proofs of the formula are in fact many and varied; the first one found by Google is at:
http://www.artofproblemsolving.com/LaTeX/Examples/AreaOfACircle.pdf

Don’t worry, that’s not the elegant one.

There are many proofs that don’t (directly) involve the use of calculus, and Wikipedia gives a good sample of them:

http://en.wikipedia.org/wiki/Area_of_a_disk

of which the rearrangement proof is perhaps the most elegant.  Another presentation of this proof is given here (along with Archimedes’ equally elegant derivation of the volume of a sphere):

http://www.mathreference.com/geo,circle.html

Yesterday I came across an approach that to me seems even simpler, based on a post at:

http://foxmath.wordpress.com/2008/06/24/perimeter-area/

This shows that for any regular polygon with an area equal to its circumference, the length of the apothem (the red line in the diagram to the left) is 2.  This is immediately obvious from the fact that the area of each individual triangle is equal to the base length, when the height equals 2.

In the limit as the number of sides of a regular polygon tends to infinity the polygon approaches a circle, and the length of the apothem approaches the radius of the enclosing circle.  It therefore follows that the area of a circle of radius 2 is equal to its circumference; i.e. 4.pi.

A circle of radius R may be scaled to radius 2 by multiplying the radius by 2/R.  The radius of this circle is then 4pi x (R/2)^2 = pi.R^2.

Finally a “wordless” proof provided by the people at SSSF:

http://www.maa.org/pubs/Calc_articles/ma018.pdf

Images based on Pythagorean tiling, Penrose tiling, and projections onto the Riemann Sphere by PM2ring, a regular contributor to the ABC Self-Service Science Forum (words by the artist):

 Various renderings of a Pythagorean tiling. Mostly using the 3,4,5 triangle. This tiling has been called a “wordless proof” of Pythagoras’ Theorem. It shows by dissection that the area of the “shadow” square on the hypotenuse is equal to the sum of the areas of the squares on the two other sides. IMHO, it’s not quite a wordless proof, since you still have to prove that the large squares are really squares & not just rhombuses.

The floor shows two versions of Penrose tiles, which are intimately related to root(5) and the Golden ratio, so it’s kind of on topic.

A 16 colour checker pattern, displayed on the Riemann Sphere.

More Riemann Sphere pics. These images look a bit flatter than the first one because they use uniform lighting.

Pythagorean tessellation on the Riemann Sphere.

A checker pattern, with half the squares clear. The other squares are yellow on the outside of the sphere & red on the inside.

The next two pictures use various raytracing techniques to achieve a more photorealistic image.

A pair of spheres in black & white checks, one sitting on its North pole, the other sitting on its South pole.

A pentagonal Penrose tiling.

The text import file has been modified to split large files into one or more ranges.

The maximum number of rows for each range is specified, together with a sufficient number of valid range names, which may be on the same sheet or different sheets. The modified file has been tested with a text file with about 1.4 million lines (43 MB). In Excel 2007 this is imported into two ranges in about 10 seconds, or about 75 seconds if it is split into 7 columns. In Excel 2000 the import stopped with a memory error after about 800,000 lines, so for very large files Excel 2007 does seem to have a clear advantage. On the down side, clearing large ranges in Excel 2007 is often excrutiatingly slow. Clearing the 1.4 million lines of data imported in 10 seconds took several minutes!

The ReadText UDF has also been modified to work with large files. It is now possible to specify a column with a list of line numbers, and only the specified lines will be imported. In this way the complete file may be imported into separate ranges, or parts of the file may be conveniently extracted. Note that if the UDF is used with a file exceeding the number of available lines in the spreadsheet the UDF will return an error if the line numbers have not been specified. For smaller files the line number list is optional.

The final refinement in this version is that the SplitText routines now allow the maximum number of columns to be specified. This is illustrated with a directory listing, with the file date and size details split into columns, but the name kept in one column, even if it includes spaces.

Download Text-in2.zip Ver1.30

1.4 million line text file imported into Excel 2007 in two ranges (about 10 seconds)

Bottom of range 1 (just to prove it’s all there!)

1.4 million line text file split into columns (about 75 seconds)

ReadText function reading every 40th line from the 1.4 million line text file

Reading a directory listing into 5 columns

On the 16th October 1843 the Irish mathematician William Hamilton was taking a walk with his wife, alongside the Royal Canal in Dublin, when the answer to a problem that he had been puzzling over came to him, and he was so excited by this discovery that he carved the equation:

I² = j² = k² = ijk = -1

on a stone of the nearest bridge.

History does not record the reaction of his wife to this act, but judging by his recollection of his conversation with his sons:

Every morning in the early part of the above-cited month, on my coming down to breakfast, your (then) little brother William Edwin, and yourself, used to ask me: “Well, Papa, can you multiply triplets?” Whereto I was always obliged to reply, with a sad shake of the head: `No, I can only add and subtract them”.

I suspect that she was not unduly surprised.

The graffiti was soon worn away, but the event was later recorded more permenantly with a commemerative stone with the words:

Here as he walked by
on the 16th of October 1843
Sir William Rowan Hamilton
in a flash of genius discovered
the fundamental formula for
quaternion multiplication
i² = j² = k² = ijk = -1
& cut it on a stone of this bridge

For more information look here: http://math.ucr.edu/home/baez/dublin/ for an account of a trip to the bridge by mathematician John Baez and friends, including some nice photographs of the structure itself.

Another site with more on the mathematical significance of the quaternion, and a quote from Hamilton himself: http://adaptivecomplexity.blogspot.com/2007/06/science-in-against-day-vectors-and.html

Before getting down to basics, the attached file:

Animation demo

Illustrates how Excel shapes can be simply animated.

Screen shot:

The code for the animation (having previously drawn a circle and a rectangle with no fill, and noted their names):

Sub Animate()
Dim Start As Single, xInc As Single, yInc As Single, OvlWidth As Single, OvlHeight As Single
Dim OvlX As Single, OvlY As Single
Dim TopBox As Single, BottBox As Single, LeftBox As Single, RightBox As Single
Dim Pi As Double, TimeStep As Double, XV As Double, YV As Double
Pi = Atn(1) * 4

With ActiveSheet.Shapes(”oval 2″)
Do
.IncrementLeft xInc
.IncrementTop yInc
Start = Timer
Do While Timer < Start + TimeStep
DoEvents
Loop
OvlX = .Left + OvlWidth / 2
OvlY = .Top + OvlHeight / 2
If OvlX &LT LeftBox Or OvlX &GT RightBox Then xInc = -xInc
If OvlY &LT TopBox Or OvlY &GT BottBox Then yInc = -yInc
Loop
End With
End Sub


Where &LT and &GT indicate the “Less Than” and “Greater Than” symbols respectively.

Previous post

The spreadsheet presented in the previous post in this series uses the basic equation given below to find the location of the neutral axis:

For a cross section with any applied axial load at eccentricity e, measured from the compression face, and depth of the neutral axis X below the compression face:

INA = QNA(X + e)

Where QNA and INA are the first and second moments of area about the neutral axis.

To illustrate this relationship the value of INA has been plotted for a range of neutral axis positions for a typical precast bridge beam, shown below, and compared with the value QNA(X + e) for 5 different values of e.

The intersection of the INA and QNA(X + e) lines gives the depth of the neutral axis for the corresponding value of load eccentricity, e.  This is illustrated in the second graph, where the value of (INA - QNA(X + e)) is plotted against the depth of the neutral axis, X, for a load eccentricity of 1 metre above the compression face.  The intersection of this line with the X axis gives the depth of the neutral axis for the specified load eccentricity.

Previous post

The text file import routines and functions presented in the previous post have been modified with the following additions:

Lines of text can be split into separate cells, at any designated character (e.g. a coma or a space)

The SplitText function can also be used as a UDF, to split lines of text previously imported.  In this case the most efficent way is to enter the UDF as an array function, operating on the complete range of text.  In this way the UDF is only called once, returning a 2D array, and the whole operation is much quicker.

Download Textin2.zip

Screen shot:

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Following the previous post, I have compared the stresses in the Taq-i-Kisra assuming either a catenary or parabolic profile. I have also compared constant depth sections with sections increasing in depth towards the supports. The arch span was taken as 24 metres, with an arch height of 16.6 metres, supported on vertical faced buttress walls with a height of 11.1 metres. The section depth was taken as 0.85 metres, increasing to 1.7 metres at the supports for the variable depth profiles. The span was taken from the Wikipedia article, and all other dimensions were estimated by scaling from photographs. Four analyses were carried out:

Run 1: Catenary with varying section depth
Run 2: Catenary with constant section depth
Run 3: Parabola with varying section depth
Run 4: Parabola with constant section depth

Bending moments and axial forces from these four runs are shown below:

It can be seen that axial forces are similar, with the constant depth sections having a significantly smaller force, as would be expected.

Bending moments for the constant depth catenary (Run 2) are close to zero throughout. Bending moments for the constant depth parabola (Run 4) and the variable depth catenary (Run 1) are similar, with a maximum of 50 kNm/m. The variable depth parabola (Run 3) had bending moments more than twice as high.

Axial stresses for these four runs are shown below:

It can be seen that stresses were compressive throughout for Run 2, and tensile stresses increased progressively through Runs 1, 4 and 3 respectively. The maximum tensile stress from Run 3 was 400 kPa, which would leave little reserve capacity for wind or earthquake loads.

Since Runs 1 and 3 were intended to be the closest models to the actual structure the analyses show that the maximum tensile stress was about four times higher for the parabolic profile than the catenary profile.

Finally, many sources on the ‘net suggest that the arch was constructed without centering, that is with no temporary supports before the closure of the arch. The analysis below shows that this results in tensile stresses of up to 7.5 MPa, which would be far in excess of the strength of the material, and construction of the arch without some form of temporary support would not be possible.