Hello All.
".
I am quite honored that Daniel has invited me to become a regular contributor here at the Excelhero.com/Blog and hope I can maintain the high standard on content that this site has become renowned for. I look forward to your feedback on this and future posts.
I figured I'd give it a go in Excel and this post documents
my approach to a solution.
Below you see a rotating array of blue crosses and 3 yellow dots. Now fixate on the centre (watch the flashing red/green spot). Note that the yellow spots disappear once in a while: singly, in pairs or all three simultaneously, right?
n reality, the 3 yellow spots are continuously present, honest! This is captively called "Motion Induced Blindness" or MIB.
What to see
On the right you see a rotating array of blue crosses and 3 yellow dots.
Now fixate on the centre (watch the flashing red/green spot). Note that the yellow
spots disappear once in a while: singly, in pairs or all three simultaneously, right?
In reality, the 3 yellow spots are continuously present, honest! This is captively called "motion induced blindness" or MIB.
The actual MIB Excel model is much smoother than this animated GIF representation.
The MIB Model
There were three approaches I thought about using for this optical illusion.
1. Use a single series to define all the points
(49) and place a cross at each point.
2. Use 2 series to define each cross, there are 49
crosses.
3. Use a Bitmap for the Background including
crosses and rotate it.
Choice of Attack
The first method wouldn't suit the needs of the illusion as each
marker doesn't rotate as the series is rotated, but instead stays fixed
relative to the ordinal axis.
The second method would require a large number of
co-ordinates for each cross, that is 4 X and 4 Y co-ordinates for each cross
and there are 49 Crosses, for a total of 98 series, and 196 co-ordinates. Time
consuming but at least the crosses will rotate. This is the methodology I choose for the model.
The Third method of rotating a fixed bitmap although very
feasible, wasn't I felt in the spirit of doing it all in an Excel Chart. I
should note that this method would allow for much faster rotation than has been
achieved using Method 2.
You can follow along with the real model and all associated preparatory workings in the attached file: Motion Induced Blindness.xlsm. All preparatory workings described below are on worksheet "2".
Setup
Firstly I setup a table of numbers -3 to +3 in X and Y and
then added/subtracted a small amount to each one to represent the width of the
cross. I settled on 0.15 as it looks about right.
This gave me a table of X and Y values for each point.
|
|
Offset
|
0.15
|
|
|
Pt No
|
X1
|
X2
|
Y
|
Circle
Quadrant
|
|
1
|
-3.15
|
-2.85
|
-3
|
3
|
|
2
|
-2.15
|
-1.85
|
-3
|
3
|
|
3
|
-1.15
|
-0.85
|
-3
|
3
|
|
4
|
-0.15
|
0.15
|
-3
|
3
|
|
5
|
0.85
|
1.15
|
-3
|
4
|
|
6
|
1.85
|
2.15
|
-3
|
4
|
|
7
|
2.85
|
3.15
|
-3
|
4
|
|
8
|
-3.15
|
-2.85
|
-2
|
3
|
|
9
|
-2.15
|
-1.85
|
-2
|
3
|
|
10
|
-1.15
|
-0.85
|
-2
|
3
|
|
etc
|
|
|
|
|
Rotation
The problem with X and Y values is that to rotate them
around a point it is easier to use Polar coordinates, but Excel requires
Cartesian Coordinates to plot.
So the process would be
1.
Setup the 49 points of 4 sets of X, Y pairs
2.
Transform them to polar co-ordinates.
3.
Setup a named formula for each end point
4.
Add a radial increment to the revised polar
coordinates
5.
Use a named formula to convert the polar
coordinates to Cartesian coordinates
6.
Plot
7.
Repeat from Pt 4.
Polar Co-ordinates
Point 2 in the above sequence means converting each set of
coordinates into polar co-ordinates consisting of Radius r and Angle Ø.
Solving the above we have
r =sqrt(x2
+ y2)
= Sqrt(-2.852 + -22)
Ø = Atan(y/x)
=Atan(-2/-2.85)
This is done for every end point of each cross section, 98 pairs
of X, Y Co-ordinates.
Now we have the polar co-ordinates of the end points, we can
setup the rotation equations.
That is the equations to convert the original polar
co-ordinates back to X & Y values, which Excel needs to plot.
This is done by 2 simple equations:
X = r * Cos(Ø)
Y = r * Sin(Ø)
X = 3.731*cos(3.707)
Y = 3.731*sin(3.707)
Now we can add a rotation angle, lets
use t.
So that the new position after
rotation is
X = r * Cos(Ø + t )
Y = r * Sin(Ø + t )
Xrot = 3.731*cos(3.707 + t)
Yrot = 3.731*sin(3.707 + t)
This is done for each point of the
cross for all crosses.
|
x1
|
y1
|
|
4.350*cos(3.903+t)
|
4.350*sin(3.903+t)
|
|
3.691*cos(4.091+t)
|
3.691*sin(4.091+t)
|
|
3.213*cos(4.346+t)
|
3.213*sin(4.346+t)
|
|
3.004*cos(4.662+t)
|
3.004*sin(4.662+t)
|
|
3.118*cos(4.988+t)
|
3.118*sin(4.988+t)
|
|
3.525*cos(5.265+t)
|
3.525*sin(5.265+t)
|
|
4.138*cos(5.472+t)
|
4.138*sin(5.472+t)
|
|
3.731*cos(3.707+t)
|
3.731*sin(3.707+t)
|
|
2.936*cos(3.891+t)
|
2.936*sin(3.891+t)
|
|
2.307*cos(4.191+t)
|
2.307*sin(4.191+t)
|
|
etc.
|
|
Matrix Arithmetic
To draw a line on a scatter chart,
Excel needs 2 X values either in a Range or an Array as well as 2 Y values in a
Range or Array.
Thankfully I've been a member of
Daniel's Excel Hero Academy. In a Module on Matrix Arithmetic we learn
that we can add 2 named formulas together to make an array in a Named Formula.
We need to do this to end up with an
Array representing the X and Y values for each of the 98 segments of the 49
Crosses.
X Values = { X1, X2 }
Y Values = { Y1, Y2 }
As an Excel Named Formula I used:
Named
Formula Formula
sx_08 =
{1,0} * 3.731*cos(3.707+t) + {0,1} *
3.482*cos(3.753+t)
sy_08 =
{1,0} * 3.731*sin(3.707+t) + {0,1} *
3.482*sin(3.753+t)
This is done for all the 98 cross
segments.
To simplify the construction of all
these, the co-ordinates, transformation to polar coordinates and construction
of the rotated transform formulas was done in Excel (Refer Worksheet "2" in the
example file).
This allows errors in co-ordinates to
be checked.
Once all the named formula are ready to
be uploaded, I have used a technique involving a simple VBA Named Formula
upload subroutine. This is described in my post at: http://chandoo.org/wp/2011/06/23/automating-repetitive-tasks.
The VBA routine is available in Module
2 of the attached Sample File, "Load_Named_Ranges()".
Add Chart Series
Once the named formula are constructed
and loaded, it is simply a matter of adding a blank scatter chart to Excel and
setting up a table of Series Names, X value and Y Values:
|
Chart
Series Name
|
X values
|
Y values
|
|
S01
|
=1!sx_01
|
=1!sy_01
|
|
S02
|
=1!sx_02
|
=1!sy_02
|
|
S03
|
=1!sx_03
|
=1!sy_03
|
|
S04
|
=1!sx_04
|
=1!sy_04
|
|
S05
|
=1!sx_05
|
=1!sy_05
|
|
S06
|
=1!sx_06
|
=1!sy_06
|
|
S07
|
=1!sx_07
|
=1!sy_07
|
|
S08
|
=1!sx_08
|
=1!sy_08
|
|
S09
|
=1!sx_09
|
=1!sy_09
|
|
S10
|
=1!sx_10
|
=1!sy_10
|
|
Etc
|
|
|
Once again I have setup a table of
Named Formula name, together with X and Y Named Formula and used a small VBA
routine to add these series to the chart.
The VBA routine to do this is available
in Module 2 of the attached Sample File, as "Add_Cht_Series()".
The 3 Yellow Spots
The 3 yellow spots are a manually loaded
series in the chart using an Array of coordinates.
X Series ={1.5, 0, -1.5}
Y Series ={1.5, -1.8, 1.5}
The Marker was set to Yellow and size
15
The Line Type was set to None
The Centre Spot
The centre spot was a manually loaded
series in the chart
X Series =0
Y Series =0
The Marker was set to Red and size 12.
The Line Type was set to None.
Animation
Animation of the chart is achieved by
adding a simple Named Formula "t" and the changing the value of t and updating
the chart.
This is done through a simple VBA
routine "Rotate()"
This is described below
Sub Rotate()
Dim t As Double 'Dimension
the only variable
t = 361 'Start at 361 Degrees
Do While [AA1] 'Loop while cell AA1 is True
t = t - 1 'Decrease
rotation angle by 1 Deg
If t = 0 Then t = 360 'If Rotation
= 0 go back to 360
ActiveWorkbook.Names.Add Name:="t", RefersToR1C1:=(t * 2 * Pi
/ 360)
' Add a
named Formula t with value = t * 2 * Pi / 360
' t expressed in radians
DoEvents 'Refresh
screen
If (t >= 0 And t < 90) Or (t >= 180 And t < 270) Then 'If t in a range set Centre Marker color
Red or Green
ActiveSheet.ChartObjects("Chart
2").Chart.SeriesCollection(99).Format.Fill.ForeColor.RGB = RGB(255, 0, 0)
Else
ActiveSheet.ChartObjects("Chart
2").Chart.SeriesCollection(99).Format.Fill.ForeColor.RGB = RGB(0, 255, 0)
End If
Loop
End Sub
Download
The above example is attached below:
Worksheet 1, contains the working model.
Worksheet 2, contains the original
source data as well as all transformations of it.
Download here: Motion Induced Blindness.xlsm
FINALLY
This is my second post at ExcelHero.com
and I'd like to thank Daniel for allowing me to
post here again.
I am a member of the inaugural Excel
Hero Academy and MVP of the Excel Hero Academy 2 & 3, where Daniel explains a lot of the techniques you will see throughout this site as well as so much more.
It is one of these techniques that made
this project possible.
I am a regular contributor at
Chandoo.org where I answer questions at the Forums and have contributed over 30 Posts.
For more about my Excel work please visit:
http://chandoo.org/wp/about-hui/
