Implicit 3D Plots
Visualizing parametrized space curves or surfaces is not complicated: We simply need to iterate the parametrization's domain (usually ) and get a reasonably smooth plot as a result. Implicitly defined surfaces and curves pose a problem however: If we were to proceed in a similar manner, we would have to iterate through and check, whether the implicit equations are 0 in any of the points. This approach is problematic in two ways. Firstly, numerics and potentially instable implicit functions prohibit this method: It is unlikely, that we will find rational points that actually lie on the curve or surface in this way. Generally, this method will only yield points that approximately lie on the implicit set. However, this we could cope with: the grid can be chosen arbitrarily small, so thoretically we could reach machine presision with this method - at the expense of a reasonable runtime. More problematically however, there is no natural notion of what point comes next on the curve or surface. In the case of parametrized curves, we always know the next point on the curve, because there is a natural total order on . Finding the next point on a parametrized curve is harder, as the proximity of points is an insufficient criterion in this setting, as the closest point might not be unique.
Drawing inspiration from the Julia package ImplicitPlots.jl there is a way out from this dilemma though. Using the library Meshing.jl immediately enables us to sample an implicit surface and form a mesh between the points that covers the surface. For implicit space curves, let us recall that if the space curve can be represented by two equations, we can simply intersect the two corresponding surfaces to obtain the space curve. For this reason, we can utilize the previously mentioned meshes and intersect them.
using Pkg Pkg.add(url="https://github.com/matthiashimmelmann/Implicit3DPlotting.jl.git")
There are two main methods in this package:
plot_implicit_curve. Let us first consider an example of the former:
julia> f(x) = x^2+x^2-x^2-1 julia> scene = plot_implicit_surface(f; transparency=false)
The result of this can be seen in the following image:
Notice that the standard options are that the plot's color is
:steelblue, the plot is transparent, the surface's shading is activated and the standard meshing method is
MarchingCubes, because it is slightly faster than
As an example of the latter method, let us consider the input
julia> f(x)=x^2+x^2-x^2-1 julia> g(x)=x^2+x^2+x^2-2 julia> plot_implicit_curve(f, g)
which produces the following picture:
In this context, it is important to notice that the two implicit surfaces generated by the implicit equations need to lie in relative general position. If they are only kissing, the mesh might not be able to reflect their intersection properly.