28 July, 2023

Shapes Based On Parametric Equations: Epitrochoids

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 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(1).Values = yValues
    chartObj.Chart.SeriesCollection(1).xValues = xValues
    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
    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
    With Selection.Font
        .Bold = msoTrue
        .Size = 14
    End With
    With Selection.Format.Line
        .Visible = msoTrue
        .Weight = 0.25
    End With
    With Selection.Format.Fill
        .Visible = msoTrue
        .ForeColor.RGB = RGB(255, 255, 200)
    End With
    Selection.MarkerStyle = 8
    Selection.MarkerSize = 5
    Selection.Format.Fill.ForeColor.RGB = RGB(192, 0, 0)
    Selection.Format.Line.Visible = msoFalse
    Selection.MarkerStyle = 8
    Selection.MarkerSize = 5
    Selection.Format.Fill.ForeColor.RGB = RGB(192, 0, 100)
    Selection.Format.Line.Visible = msoFalse
    With Selection.Format.Line
        .Visible = msoTrue
        .ForeColor.RGB = RGB(0, 112, 192)
        .Weight = 1
    End With
End Sub

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