Epitrochoids are curves generated by a point selected on a circle of smaller radius rolling around the **outside**
of a fixed circle of larger radius, and that selected point can be
chosen at some distance from the center of the smaller circle. These are
then 3-parametric curves. They can be helpful in creating decorative designs and other art applications, and can serve also as pastime and recreation for children. Since 1965 there is available a toy (geometric drawing device) known as **spirograph**, combining mathematics and art. It allows producing physically numerous curve designs based exactly on epitrochoid and hypotrochoid parametric equations.

However, nowadays the **epitrochoids** (as well as **hypotrochoids** - see another post at https://draft.blogger.com/blog/post/edit/3323809043368251287/9189797010242374274?hl=en-GB) can be generated quite easily in Excel by using VBA
macro like the one provided at the end of this post.

Here are some examples of charts showing epitrochoid curves generated with the macro. Some of them present combined double curves.

You can try to use the following macro to plot your own attractive decorative epitrochoid graphics. Just copy it to one of the modules in your workbook sheets and experiment with different settings and parameters.

**Sub CreateEpitrochoidChart()**

*‘Creates a scatter chart object, sets its type and assigns calculated X and Y values to the chart series.*

*'The ReDim statements are used to size the arrays for X and Y values based on the desired*

*'number of data points.*

Dim i As Integer, numPoints As Integer

Dim a As Double, b As Double, c As Double

Dim a1 As Double, b1 As Double, c1 As Double

Dim chartObj As ChartObject

Dim oChart As Chart

Dim xValues() As Double, yValues() As Double

Dim xvValues() As Double, yvValues() As Double

Dim tit As String

*'Define the number of data points and calculate X and Y values*

numPoints = 5000

*'Change as needed*

ReDim xValues(1 To numPoints)

ReDim yValues(1 To numPoints)

ReDim xvValues(1 To numPoints)

ReDim yvValues(1 To numPoints)

a = WorksheetFunction.RandBetween(1, 30): b = WorksheetFunction.RandBetween(1, 20): c = WorksheetFunction.RandBetween(1, 20)

a1 = WorksheetFunction.RandBetween(1, 30): b1 = WorksheetFunction.RandBetween(1, 20): c1 = WorksheetFunction.RandBetween(1, 20)

*‘To set and use your own parameters for the curves, remove the apostrophe in the next line*

‘a = 26: b = 2: c = 4: a1 = 26: b1 = 13: c1 = 4

*'Calculate X and Y values using your equations*

For i = 1 To numPoints Step 2

*'First series*

xValues(i) = (a + b) * Cos(WorksheetFunction.Radians(i)) - c * Cos(WorksheetFunction.Radians((a + b)) / b * i)

yValues(i) = (a + b) * Sin(WorksheetFunction.Radians(i)) - c * Sin(WorksheetFunction.Radians((a + b)) / b * i)

Next i

For i = 1 To numPoints Step 2

*'Second series*

xvValues(i) = (a1 + b1) * Cos(WorksheetFunction.Radians(i)) - c * Cos(WorksheetFunction.Radians((a1 + b1)) / b1 * i)

yvValues(i) = (a1 + b1) * Sin(WorksheetFunction.Radians(i)) - c * Sin(WorksheetFunction.Radians((a1 + b1)) / b1 * i)

Next i

*'Create a new chart object*

Set chartObj = ActiveSheet.ChartObjects.Add(Top:=10, Left:=100, Width:=600, Height:=600)

Set oChart = chartObj.Chart

oChart.ChartType = xlXYScatter

*'Set the chart type to scatter chart*

*'Set X and Y values for the chart series*

chartObj.Chart.SeriesCollection.NewSeries

chartObj.Chart.SeriesCollection(1).Values = yValues

chartObj.Chart.SeriesCollection(1).xValues = xValues

chartObj.Chart.SeriesCollection.NewSeries

chartObj.Chart.SeriesCollection(2).Values = yvValues

chartObj.Chart.SeriesCollection(2).xValues = xvValues

*'The code below formats the chart and plot area; can be modified as needed*

chartObj.Activate

ActiveChart.SetElement (msoElementPrimaryCategoryAxisTitleAdjacentToAxis)

ActiveChart.SetElement (msoElementPrimaryValueAxisTitleAdjacentToAxis)

ActiveChart.SetElement (msoElementPrimaryValueGridLinesNone)

ActiveChart.SetElement (msoElementPrimaryCategoryGridLinesNone)

*'Add chart title and title of the axes*

oChart.Axes(xlCategory).HasTitle = True

oChart.Axes(xlCategory).AxisTitle.Caption = "X values"

oChart.Axes(xlValue).HasTitle = True

oChart.Axes(xlValue).AxisTitle.Caption = "Y values"

oChart.HasTitle = True

tit = "Epitrochoid " & a & "-" & b & "-" & c & ", " & a1 & "-" & b1 & "-" & c1

oChart.ChartTitle.Text = tit

oChart.ChartTitle.Select

With Selection.Font

.Bold = msoTrue

.Size = 14

End With

ActiveChart.ChartArea.Select

With Selection.Format.Line

.Visible = msoTrue

.Weight = 0.25

End With

ActiveChart.PlotArea.Select

With Selection.Format.Fill

.Visible = msoTrue

.ForeColor.RGB = RGB(255, 255, 200)

End With

ActiveChart.FullSeriesCollection(1).Select

Selection.MarkerStyle = 8

Selection.MarkerSize = 5

Selection.Format.Fill.ForeColor.RGB = RGB(192, 0, 0)

Selection.Format.Line.Visible = msoFalse

ActiveChart.FullSeriesCollection(2).Select

Selection.MarkerStyle = 8

Selection.MarkerSize = 5

Selection.Format.Fill.ForeColor.RGB = RGB(192, 0, 100)

Selection.Format.Line.Visible = msoFalse

ActiveChart.PlotArea.Select

With Selection.Format.Line

.Visible = msoTrue

.ForeColor.RGB = RGB(0, 112, 192)

.Weight = 1

End With

ActiveSheet.Range("A1").Select

**End Sub**

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