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Episode 61 Understanding the Ellipsograph

The standard equation for an ellipse.

Being a nerd is a high price to pay. A simple little toy has lead me into lots of charts and graphs and math and … it goes on and on.  There is always more to discover, even about simple things. I want to point out one thing as you read though this episode. Some of the pictures come out small to fit into the web page and are hard to read. Clicking on a picture will expand it to full size. To return back to the text, use your browser “back” or previous screen button.

From some researching I found that there actually are “professional tools” called the ellipsograph used as drawing aids for hand drafting and as guides for routers to cut both inside and outside ellipses.  The question was: “How do I change the size of the ellipse by adjusting the ellipsograph?”   I started with the standard equation for an ellipse.  That ellipse is defined with the major axis being on the X-axis and the center of the ellipse being located at the origin (0,0) for the coordinate system.

Setting the Y shuttle to zero

I then defined my ellipsograph with with the distance between the two shuttles as D1 and the distance between the shuttle in the middle of the shaft and the handle to be D2.    The first step in the analysis was to place the shuttle that moves up and down right in the center.  This makes y in our equation equal to zero, allowing us to simplify the equation giving us:  x = a.

Y = 0 – Major Axis Calculation

I then rotate the mechanism so the Y shuttle is in the center. Since the handle must be at the furthest distance from the center, I now know two things. First, the shuttle at the end of the shaft, is aligned with the minor axis and the shuttle in the middle of the shaft is aligned with the major axis.  I also know variable “a” in  the equation = D1+D2 on our device.

Setting X = 0 in the equation

I do the same procedures by setting the X axis shuttle to 0 and quickly determine variable “b” in the equation = D2.

The X-axis shuttle at 0 on the ellipsograph.

Now that we know that the major axis is a=D1+D2, and minor axis b = D2.  We can set these dimensions to determine how “squeezed” the ellipse looks and the size of the ellipse.   The larger D1, the distance between the shuttles, the more long and skinny the ellipse.  The distance from the handle to the middle shuttle determines the size of the ellipse.

It is always good to look at extreme cases to really understand things.  If we set D1 = 0, then the two shuttles would be stuck at the center and the ellipsograph would break down to become a simple compass and would draw a circle of radius = D2.

Calculations used on the spreadsheet.

After doing all this work, I wanted to ” see” results, but I am not yet set up to actually build one.   Also, when build one, I will be interested in lots of movement of the shuttles (a large D1) and not necessarily in the shape of the ellipse the handle will draw. I created a spreadsheet with the calculations showing the movement of the Y shuttle as the X shuttle is moved.

Ellipse X-Y plot from a spreadsheet.

My interest that this point was seeing if the values D1 and D2 do what I expected. I used the equations above to create a spreadsheet and moved the X shuttle in small steps and plotted both the X and Y values.  I did get results, it is just not the visual results I hoped for because the software automatically sets the X and Y scales so the plot almost always looks the same.    The devil is in the details, and it is necessary to look at the scale values to get an idea of the shape.   The spreadsheet is available for your download should you decide to play with the numbers.  D1 and D2 values are highlighted in the in the spreadsheet, the rest of the numbers calculate from those.  The graph is on the 2nd tab.

I do want to also give credit to the wikipedia page where I got some of the initial information.  Wikipedia Ellipse.

We will come back to this.   There is no use in doing all this work unless we actually build something from it.

Gary

 

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