Does f represent the equations below each example? If so which variable is substituted by p? I'm looking back and forth between this and the previous slide and can't tell how p is involved with these equations.

Good question! Yes, $f$ is the equation. So in the case of the sphere, $f(x,y,z) = x^2 + y^2 + z^2 - 1$, and the surface contains all points $(a,b,c)$ such that $f(a,b,c) = 0$. Remember that a point $\mathbf{p}$ or $\mathbf{r}(t)$ is a vector, so it contains an $x$, $y$, and $z$ component. For example, if you have a ray $\mathbf{r}(t) = (-1, 1, 0) + t(1, 2, 3)$, then when you plug in $f(\mathbf{r}(t))$ you get $f(-1 + t , 1 + 2 t, 3 t)$, which is now a (univariate!) polynomial in $t$. The $t$'s which satisfy this polynomial give you the intersection points of the ray with the implicit surface.

Are planes and spheres the only non-implicit surfaces we can do a ray-intersection test with? What if I wanted to test intersection with an ellipsoid? I can't necessarily use the intersection with a sphere, and I may not have the implicit surface equation either.

@Carpetfizz

Hi! I did some researching online and it seems we can in fact do a ray-intersection test with an ellipsoid, by scaling it to a sphere.

Here is a resource you can look at (page 18): https://cs.oberlin.edu/~bob/cs357.08/VectorGeometry/VectorGeometry.pdf

I Googled for the heart and found a whole bunch of other cool shapes: https://imaginary.org/sites/default/files/imaginary-posterset-english-version1_0.pdf

I'd imagine that it'd be possible to do ray intersections with all of these right?