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Three Dimensional (3D) Rotation via Matrix Multiplication

Keith Greiner

August 9, 2020

This essay is about how to rotate data in 3D. It is intended to be a follow-up to the document at this address that is about rotating data in a 2D space.

The words “range”, “array”, and “matrix” all mean the same thing.

Here, I will denote the ranges, and arrays in capitol letters.

D is the original data in the range C10:E59

R with, numbers appended, denotes the four rotation ranges

R1 holds the x axis constant and is at G12:I14

R2 holds the y axis constant and is at G17:I19

R3 holds the z axis constant and is at G22:I24

RA rotates all axes and is at G27:I29

Results are in output matrices…

O1 holds the results of x axis constant and is at K10:M59

O2 holds the results of y axis constant and is at O10:Q59

O3 holds the results of z axis constant and is at S10:U59

OA holds the results when all axes are rotated and is at W10:Y59

The degrees of rotation are in cell H5 and the radians are in cell H6

The contents of R1 through RA are shown below, when the degrees of rotation are set to 30.

The following Excel Image shows the contents of the rotation matrices. Be sure to note the cells that have minus signs before the = -SIN(…) function. They may not be fully visible.

The following image shows a screen print of the first 53 lines of the setup. Double-click on the image for a closer look.

Using the matrix notation described above,

O1 = D * R1

O2 = D * R2

O3 = D * R3

OA = D * RA

Where the multiplication character indicates use of the Excel =MMULT(Range1, Range2) function.

Graphing the 3D Image

Unfortunately, Excel does not have the ability to graph points of a scatter gram in three dimensions (as of August, 2017) so I used the next best tool available, and that was Apple Grapher.app. The image shown below shows a set of points rotated with the z-axis held constant, while data points for x and y were rotated 30 degrees.

To accomplish this, I followed these steps.

1. I first created a set of x, y, and z points with values that ranged from 0 to .999
2. Saved the points to their own Excel worksheet with no headers.
3. Save (export) the points as a .csv file.
4. Open Grapher
5. Create a new data set with the following steps
6. Open an existing file or create a new one.
7. On the top menu select Equation and then New Point Set.
8. Select the new point set
9. Edit and then Import by following steps to import the .csv file.
10. If the new points do not appear check the graph sizes or check the .csv file.
11. Delete the points at the top of the list that Grapher inserted to start the point set.
12. The imported points should be visible.

To ensure that the x, y, and z axes can always be seen, create three points sets. One has sequential numbers from -1 to +1 in the x column and zeros in the other columns. Do the same for the y and z points. Import these three data sets and they will show up on the x, y, and z axes. Color code them to make sure you always know which axis is which.

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