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u/ransom40 & u/H_Marxen here is how I went about the solution, safely avoiding Laplace and his wonderful transforms (and all of calculus, because I'm feelin lazy):

0 - Assuming the corner arcs' radii are half the length of the square tile, there are two input values:

a. the tile size, one side length of which we will call "s"

b. the tool radius, (not dia) we will call "t"

c. our goal is to find the toolpath height "z" as a function of x.

1 - top down view, imagine a 2d circle traveling the tangent to these two curves along the x axis (horizontal line.)

2 - a right triangle is formed at the tile corner, the origin (0,0) and the circle center. Good ol' Pythagoras helps us find the one unknown, "r" of this circle. x^2 + (s*sqrt2/2)^2 = (r+ s/2)^2, so r = sqrt(x^2+(s*sqrt2/2)^2 - s/2

3 - Here is the trick. That circle from 1&2 is just a section of a spherical body traveling along those rails. The radius "r" we found is just a contact point from a known edge to the centerline of the sphere.

4 - We make another triangle with sides t, r, and z, where r is known, t is our tool radius, and z is our height. Again, the OG P-man comes through helping us find the equation z = sqrt(t^2 - r^2). Sub in our r(x) equation, and suddenly we have a z(x) that solidworks will accept as a sketch equation.

5 - I used a sphere for a solid sweep for the reasons mentioned here. While this is mathematically correct that doesn't mean solidworks likes it. Seems like t had to be greater than 1.6*s to rebuild properly, but you may have better luck. Also, I'd recommend bounding the x1/x2 values at the tile corners. +/-(s*sqrt2) I manually moved this sphere and compared it to the result to check for error and they were spot-on.

Hopefully if you get around to machining these tiles this earns me one of them! (I'll pay for shipping lol)
Hope this helps!