Airless Tennis Ball: Hyperbolic Mapping to a Sphere

Hi, I'm Joseph Maloney a 16 year old who loves 3D printing, aeronautics, and CAD design. I spend much of my free time working on robotics and other STEM based projects.

This instructible will show you how to CAD and print an airless tennis ball. You will learn how to use hyperbolic scaling, project sketches onto spheres, and properly 3D print round objects in one piece. If you just want to download mine and print it I included the STL file as well as the STEP and f3d files on step 8.

This tennis ball is made up of only hexagons which is mathematically impossible. However, through a combination of taking advantage of the seam and utilizing hyperbolic scaling a hexagon only sphere is achievable. These principles can be used in other cad applications and other projects.

You will need the following supplies to design and print an airless tennis ball.

• ~70 grams of PLA, PC, or TPU filament
• 3D printer
• Computer with Fusion 360
• Slicing software

The first step is to create a sphere. The simplest way to do this is to use the sphere tool; However, a revolve will work just as well. The standard diameter for a tennis ball is 67mm but feel free to try out different sizes. You will also need to make the sphere hollow. I chose a wall thickness of 9mm which made the weight work out to be similar to a real tennis ball.

The next step is creating the "seam" of the tennis ball. You can do this by slicing the ball with the sheet profile shown above. After that, sweep a profile like the one shown in the third picture, along the split line. You can also sweep a line to evenly split the two halves of the sphere as to make them identical.

This step is relatively simple. You just need to create a hexagonal pattern across the diameter of your sphere. Create this sketch on a plane tangent to the sphere. Make sure that the pattern can be made periodic along the y-axis as we will be patterning it in a later step. This just means that you need to have two layers of hexagons vertically. Finally create a construction rectangle that extends past the the patten in the y-axis and that has a length equal to the diameter of your sphere in the x-axis. This will be used as an endpoint for a rail for the sweep feature needed later to hyperbolically scale the sketch.

In this step we will hyperbolically scale the sketch created in the previous sketch so that when we project it onto the sphere it doesn't get super distorted. The first part of this step is to create the conic section used for the distortion. Make the sketch on the a plane perpendicular to the plane used for the previous step. to define the conic, set the anchor point to the origin. Then set ρ, which is how you define the curvature of your conic, equal to (x-y)/x where x is the distance between the origin and the top of the conic, and y is the radius of the sphere. In the example above, y = 57.188 and x = 33.5. Make the conic tangent, at its endpoints, to construction lines which are 45 degrees from the y-axis. Finally chose a conic shape by changing the distance between the origin and the top of the conic or x. This website explains ρ, and how it affects the shape of your conic. If you have a ρ value equal to the sqrt(2) - 1 then your conic will be an arc which is the case in the example above. I do not know what the optimal conic shape is but any x value between 50 and 100 will work well enough to have minimal distortion on the surface of the sphere. Finally, symmetrically extrude the conic to be the same thickness as the rectangle created in the previous step.

The first part of this step is to create the guide rails for the sweep. Create a 3D sketch and connect the the points at two of the corners of the construction rectangle to the top of the extruded conic. Now sweep the sketch created in step 3 along the rails that you just created in this step. Make sure that this is a cut operation and that the sweep is set to stretch not scale. Finally create a sketch on the same plane as the original hexagon sketch. Project the face of the conic onto the sketch plane using the project tool located under the "create" tab. Now finish the sketch. You now know how to hyperbolically scale sketches.

In this step we are going to finally cut the pattern into the sphere. Loft the distorted hexagons in the sketch created in step 5 back to the origin. Now create a circular pattern with the main goal of keeping an even spacing between the hexagons. Finally preform a Boolean cut operation on one half of the tennis ball.

To make the final sphere, copy and move the half of the sphere to create a full sphere. Finally combine this sphere with the seam that you created in step 2.

To slice the tennis ball, or any spherical object, you need to support the bottom. For the tennis ball in particular, you need to use a support blocker so that supports don't generate inside the sphere. I recommend that you support the bottom third of the ball but if your printer has really good cooling you can do less. For the print settings, my recommendations are:

• .24 layer height
• .46 line width
• 75mm/s speed
• 2 line skirt
• 100% infill

Settings vary for different filaments. If you just want a ball to look at, and bounce occasionally, PLA works fine and is quite strong. If you want something stronger I recommend polycarbonate filament as it still bounces quite well. TPU works for a tough ball but it doesn't bounce as well as the harder filaments.

For printing, I recommend glue stick on the bed especially if you are using TPU as it can easily stick too hard. The ball takes about 6 hours to print depending on settings. When removing supports be careful not to break the ball.

Hyperbolic scaling sketches can have many uses. The most obvious one is simply cosmetic. Hyperbolically scaled embosses and debosses can be very visually appealing. You can easily change the patten and how many axis are scaled by changing how you make the conic, either with a revolve of an extrude. In my opinion, the most practical use case of this is projecting sketches onto spheres. This method works better than the typical emboss feature for things that need to cover larger areas of the sphere. It also gives you more control over the way in which things are distorted. I hope that you can apply this new knowledge in your CAD projects.

Thanks, and good luck in future projects!