Chip and Bowl Curvature in Guac Scrapping

We are all faced with a problem, we order some chips and guac or some chips and salsa and once we get towards the end of the dip we must resort to a sort of scrapping to most efficiently get at what remains. Now the dynamics of this vary, most often in my experience dips are served in bowls. The shape of chips varies too from the standard triangles, to the dip bucket sorts to discs.

If you have a flat bottom to the bowl the triangles are able to scrape on that pretty well. The bucket chips for all their effectiveness at holding dip fall quite short on the scraping dimension.

For the disc-shaped chips, assuming the bowl is hemispheric (or at least that the walls have constant curvature). If the bowl has radius $S$ and the chip has radius $R$, and the chip meets the bowl at angle $\theta$ relative to the bowl tangent, then at the contact point: - The bowl bends with curvature $\frac{1}{S}$ - The chip edge bends with curvature $\frac{1}{R}$ - The curvature relevant for scraping is the normal component $\kappa_{\text{chip, normal}} = \frac{\sin\theta}{R}$ To match curvatures for optimal scraping, set $\frac{\sin\theta}{R} = \frac{1}{S}$ which gives $\theta = \arcsin\left(\frac{R}{S}\right)$.

So then you have locally matched curvature, but unfortunately for us as you move away from that point the $\theta$ is no longer tangent and the bowl has curvature in the perpendicular direction along the surface. This the chip moves away from the surface as you move along the chip edge from the contact point. This is mitigated by the chip flexing, and the guac/salsa protrudes at least a bit from the surface.

Now can we do better? Well we could entirely eliminate the problem by having chips that have the same curvature as the bowl, then you just have to keep it tangent for perfect scraping. For discs that would entail very big chips and so probably wouldn't be practical but for a rounded triangle sort of chip this could be reasonably workable.