Ax+By+C : an affine function of 2 variables.
Ax+By+Cz+D : an affine function of 3 variables.

The function in 2D: Ax+By+C
If c = 0, then the function level line Ax+By is equal constant.
Ax+By = 0
[A,B].[x,y] = 0
[A,B] is perpendicular to [x,y]

The level line is a curve where a function has a single value.
The "curve" with Ax+By=0 is the line through the origional point and perpendicular to [A,b]
Ax+By = C
Axo+Byo = C
(x-xo)+(y-yo) = 0

This means all the level lines of Ax+By are all perpendicular to [A,B],
the distance from the line Ax+By=0 where C>0 is C/(square root of A^2 + B^2).

The way to calculate the distance:
if we have function Ax+By=C1 and Ax+By=C2
the signed distance between them = (C1-C2)/(square root of A^2+B^2)

Wave in Crests:
If time = 0, then the wave height = Acos(Ax+By+Cz)
where (a,b,c) is related to lambda,the wave length vector.
The wave height and lambda are prependicular to each other.