July 22, 2024
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R3 – Rotation Ball

R3 Rotation Prop

Use a Tennis Ball as a Rotation Prop:

A physical prop helps to understand 3d rotation. There are many digital 3d spheres on-line. They limit the user experience as they are only in 2d. It is difficult to find a physical 3d sphere to help with understanding rotation. A major requirement is to be able to hold the 3d sphere in one hand to perform rotations. I took a Wilson Triniti eco-friendly tennis ball from my sports bag to make a 3d sphere, the Wave Number Rotation ball. Make your own as follows or buy one at our shop.

Marking the Poles

The only writing on the ball was the Triniti logo. This is used to represent the North Pole.

  • Here with the logo the correct way around, write Z^ in the centre of the logo and the coordinates (0, 0, j^) below.
  • On the opposite side, mark Zv in the centre and (0, 0, jv) below.

Hold the tennis ball at these 2 points between thumb and index finger. As a result it is held by the z-axis. Rotation of the tennis ball is easy.

  • Holding the ball vertically by the z-axis, mark the X^ pole in the centre of the facing side and (1^, 0, 0) below and then, on the opposite side, mark the Xv pole in the centre, with (1v, 0, 0) below, to get the x-axis.
  • Continuing to hold the ball at the z-axis points, turn it 90o clockwise and mark in the Y^ pole in the centre of the facing side, with (0, i^, 0) below, and then, on the opposite side mark the Yv pole in the centre with (0, iv, 0) below.

Marking Counterclockwise and Clockwise guides


Turning counterclockwise and clockwise on the tennis ball can be confusing as the ball and viewpoint seem to change. To make sure it rotates as required mark the counterclockwise rotation on the ball using the diagram above.

  • With Z^ facing and X^ below, draw the symbol  (Note counterclockwise direction of arrow)   around the Z^. This shows the direction of the counterclockwise rotation, no matter what orientation the ball is held.
  • With Zv facing and Xv below, draw the symbol  (Note clockwise direction of arrow) around the Zv . Even though the symbol is clockwise, it shows the direction of the counterclockwise rotation relative to the standard orientation, no matter what orientation the ball is held.
  • With Y^ facing and Zv below draw the symbol   around the Y^.
  • With Yv facing and Zv below draw the symbol   around the Yv.
  • With X^ facing and Zv below draw the symbol   around the X^.
  • With Xv facing and Zv below draw the symbol  around the Xv.

Adding more arrows to the and  symbols makes the rotation direction even clearer. Check out the video of the Wave Number Rotation Ball above.

To check that the arrows are correct, hold each axis parallel to the ground with the ^ side facing. Rotation clockwise and counterclockwise around the axis is then in the same direction as for a clock.

Optical Illusion

Look down on the Z^ from above and note the directions of the rotation arrow(s). Rotate the ball around the y-axis so that Zv is on top. The rotation arrow(s) appear to be in the opposite direction than for Z^. Why do the rotation arrows appear in a different direction? Hold the ball by the z-axis with the Z^ on top and rotate counterclockwise. The ball is following the direction of the arrows at Z^ and it is also following the direction of the arrows at Zv even though the arrows at Z^ and Zv appear to be in different directions.

Understanding Rotation with the Ball

The Rotation Ball can be used to understand rotation. Instead of trying to form a mental picture of rotation, see and experience rotation with the prop. Check out how to use the Rotation Ball to explore rotation in the next post.


If you don’t want to make your own Wave Number Rotation Ball, then visit our shop where you will find a selection of our rotation balls including the Wave Number prop.

Next: Using the Wave Number Rotation Ball

Previous: Rotation Definition and Quaternions

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