My next object was a bulge-head solid. This solid lies above the $$xy$$–plane, outside the unit sphere, and inside the cardioid of revolution given by $$\rho=1+\cos\phi$$. Professor Beanland had given us these equations, since he was really curious to see what the solid looked like. He’d nicknamed it the cone-head solid, but after printing we renamed it the bulge-head solid.
Since the outside of the solid was a cardioid of revolution, I decided to create the solid in Cinema 4D by creating two splines (one for the cardioid, the other for the hemisphere) and revolving each around an appropriate axis.  Professor Denne helped me to figure out which parametric equations to place into Cinema 4D’s inputs for a formula spline. These were $$x(t)=1+2\cos(t) + \cos(2t)$$, $$y(t)=2\sin(t)+\sin(2t)$$, and $$z(t)=0$$, where $$t=[0,\pi/2]$$.  For the spline that would later become the hemisphere, I used $$x(t)=\cos(t)$$, $$y(t)=\sin(t)$$, and $$z(t)=0$$, where $$t=[0, \pi/2]$$. I then used the Lathe Tool with an angle of $$360^\circ$$ to make the two boundaries of the solid.  I then put them into a Boole to make a union between the two boundaries.  I printed the bulge-head solid on the FormLabs printer using clear resin. When loading the object into the FormLabs software, we got a warning about the object’s integrity, but we decided to continue the print anyway. Later on we were worried that the object would use up too much resin and that it may have some problems on the surface (like the smooth strange bowl did).  It turned out that added a bit more resin mid-build, just to be on the safe side.  The solid looks pretty good right now because it only has a few pimples on the inside, but no significant lumps. The object is still hardening and once it’s completely dry we’ll remove the outside supports. This will probably leave a few pimples as well.