# Surface¶

Sverchok uses the term Surface mostly in the same way it is used in mathematics.

From the user perspective, a Surface object is just some (more or less smooth) surface laying in 3D space, which has some boundary. It may appear that boundary of the surface from one side coincides with the boundary of the surface from another side; in such case we say the surface is closed, or cyclic in certain direction. Examples of closed surfaces are:

• Cylindrical surface (closed in one direction);
• Toroidal surface (closed in two directions);
• Sphereical surface (closed in two directions).

The simples example of non-closed (open) surface is a unit square.

You will find that the Surface object has a lot in common with Curve object. One may say, that Surface is almost the same as Curve, just it is a 2D object rather than 1D.

Mathematically, a Surface is a set of points in 3D space, which can be defined as a codomain of some function from R^2 to R^3; i.e. the function, which maps each point on 2D plane to some point in 3D space. We will be considering only “good enough” functions - continuous and having at least one derivative at each point.

It is important to understand, that each surface can be defined by more than one function (which is called parametrization of surface). We usually use the one which is most fitting our goals in specific task.

We usually use the letters u and v for surface parameters.

Excercise for the reader: write down several possible parametrization for the unit square surface, which has corners (0, 0, 0), (0, 1, 0), (1, 1, 0), (1, 0, 0).

The range of the surface parameters corresponding to the whole surface within it’s boundaries (in specific parametrization) is called surface domain. The same surface can have different domains in different parametrizations.

Similar to curves, the values of surface parameters have nothing to do with distances or areas, which are covered by points on the surface.

Since Blender has mostly a mesh-based approach to modelling, as well as Sverchok, to “visualize” the Surface object you have to convert it to mesh. It is usually done by use of “Evaluate Surface” node.

It is also possible to “visualize” the surface by use of “Tessellate & Trim” node. This node allows one to tessellate the part of surface, trimmed by some curve.

# Implcit Surfaces¶

Another possible way to define a surface is to say that the surface is a set of all points (X, Y, Z), which are solutions of equation: `F(X, Y, Z) = C`. Here F is some function, which maps each point in 3D space to a numeric value, and C is some numeric constant. Surfaces that are defined in this way are called “implicit surfaces”. Implicit surfaces are, in general, a more wide class of objects than parametric surfaces, defined in the previous topic.

Note that F function, required to define an implicit surface, is what we call “scalar field”. So, implicit surfaces are also known as iso-surfaces of scalar fields.

Sverchok does not have a separate object or a separate socket type for implicit surfaces. Instead, for each node which can work with implicit surface, F function is provided as a scalar field, and C constant is provided in a separate socket.

Implicit surfaces can be “visualized” by use of “Marching Cubes” node.