4" Wooden Mecanum Wheel

[TheOtherGuy]

From eq.#1, it looks like you used the diameter of the mecanum wheel as the radius instead of the radius. The equation for the ellipse without translation should be y^2/4 + x^2/8 = 1. The second ellipse needs to be translated up 3 units in order to give the roller a diameter of 1 in the middle, so the second equation is (y-3)^2/4 + x^2/8 = 1. Solving for y, you should get 3/2, and plugging that back into the first equation gives roots of x as +/- sqrt(7/2).

Intersection of two ellipses

Ellipse shifted down 1.5

Roots of ellipse shifted down

I did a quick check on the wheel I CADed, and the rollers do indeed follow these ellipses (and have the correct side profile on the wheel itself).

Heh, just checked back on the parabola equation, seems I made a small error in setting the radius of the wheel R equal to 4" instead of 2"! That would do it. Here is the fixed parabolic equation with the ellipse above. Sorry about the confusion.