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How to Rotate an Ellipse around the Left-Hand Focal Point in Excel

Keith Greiner

August 8, 2020

This essay gives highlights on how to rotate an ellipse around the focal point. This essay builds on the other essays on plotting and rotating an Ellipse that are available at:

1. How to Draw an Ellipse in Excel
2. Data Rotation via Matrix Multiplication
3. Three-Dimensional Rotation via Matrix Multiplication

Below are the steps:

1. Start with an ellipse that is horizontal and centered on (0,0)
2. Recall that the c value is calculated as c = (a^2 – b^2)^0.5. That is, it is the square root of "a" squared minus "b" squared.
3. Since the "c" value is the difference between the focal point (F1) and the (0,0) center point, we subtract the negative of "c" from all "x" values. Leave the "y" values as is.
4. Now, you can multiply the right-shifted matrix of "x" and "y" values by the two dimensional rotation matrix to obtain a new rotated matrix that is rotated around the F1 (left hand) focal point.

When graphing a rotated matrix around the focal point in Excel you may find that the graph appears to be a little off from the focal point. By transferring that same data to the Apple Grapher.app I found that it was correct.

Below is an image of the same set of "x" and "y" points in Excel and then in Apple Grapher.app.

First, the Excel image.

Second, the Apple Grapher.app Image