Implicit Surface

It’s always R = 0 where R is a function of one or more variables.

Like the unit circle equation:
x^2 + y^2 -1 = 0.

Parametric Surface

The components of the output are based on some parameter or parameters

Like the quadratic bezier curve (which A,B,C and CurvePoint are points in N dimensions):
CurvePoint = f(t) = A*(1-t)^2 + B*2t(1-t) + C*t^2

Or the unit circle:
x = cos(t)
y = sin(t)

Or surfaces like this:
SurfacePoint3D = f(u,v)

Explicit Surface

The more usual looking type functions where you have one variable on the left side (dependent variable), and another variable on the right side (independent variable).

Like lines:
y = mx + b

or height fields:
height = f(x,y)

More Info

Here’s a cool set of slides that explain this stuff in more detail (and beyond), and the pros and cons of using various forms.

Representing Smooth Surfaces

Bezier Surface Properties

Here’s a couple pretty cool properties of Bezier surfaces that I learned recently.

The first one is that if you consider a “convex hull” being made up of the control points (connect all the control points into a convex shape), the curve will lie entirely inside that shape. That means you can use the shape of the control points as a “quick test” for rendering or collision detection. Note though, you could also just make a sphere that enclosed all the control points and do a sphere test instead, if you would rather have a simpler/quicker test at the cost of some wasted space (more false positives).

The second interesting property is that you can do back face culling of a Bezier surface if all the control points face away from the camera. while it’s true this isn’t EXACTLY proper back face culling, the odds are good it’s good enough for your needs, especially given how quick a test it is.

The third interesting property is that if you want to transform a bezier surface with something like a translation, rotation, or scale, you can apply the transform to the control points, and the curve will be transformed by the same transformation.”A Bézier surface will transform in the same way as its control points under all linear transformations and translations.” (from Wikipedia: Bezier Surface)

… but unfortunately, as promising as these properties are, it still seems infeasible to render a decent number of bezier surfaces via real time raytracing (something i was planning on) and it seems to only get worse when moving to b-splines and nurbs surfaces, so it seems like this may not be the way to go. It’s still possible though that raymarching these surfaces could be doable, but I haven’t explored too much in that direction yet.