# Curve¶

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

From the user perspective, a Curve object is just a (more or less smooth) curve laying in 3D space, which goes from one point (start) to another point (end). The start and the end point of the curve may coincide, in this case we say the curve is closed, or cyclic.

Let’s state some properties of the Curve object, which will define our understanding of this term in context of Sverchok:

• A curve is a finite, one-dimensional object, existing in 3D space.
• Every curve must have exactly two endpoints. No more, no less.
• If the endpoints coincide, the curve is considered to be closed.
• There may be no gaps on the interior of a curve, as that would result in more than two endpoints.
• There may be no branching points on the curve, except where an endpoint is coincident with some interior point of the curve.

Mathematically, a Curve is a set of points in 3D space, which can be defined as a codomain of some function from R to R^3; i.e. the function, which maps some real number to a vector in 3D space. We will be considering only “good enough” functions; more exactly, such function must be continuous, and have at least 3 derivatives at (mostly) each point.

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

Usually we use the letter t for curve parameter; in some case the letter u is used.

For example, let’s consider a straight line segment, which is beginning at (0, 0, 0) and ending at (1, 1, 1). The following parameterizations all define the same line:

1. x(t) = t

y(t) = t

z(t) = t

2. x(t) = t^2

y(t) = t^2

z(t) = t^2

3. x(t) = t^3

y(t) = t^3

z(t) = t^3

As you understand, we can write down as many equations for it as we want.

Different parametrizations of the same curve can have different values of t parameter corresponding to the beginning and the end of the curve. For example,

1. x(t) = t - 1

y(t) = t - 1

z(t) = t - 1

defines the parametrization, for which t = 1 corresponds to the beginning of the segment, and t = 2 corresponds to the end of the segment. The range of the curve parameter which corresponds to the curve from it’s beginning to the end is called curve domain.

Another important thing to understand is that the value of curve parameter at some point has nothing to do with the length of the curve. With our straight line example, if we consider A) parametrization at the point of t = 0.5, it will be (0.5, 0.5, 0.5), i.e. the middle of the segment. But, if we take the same segment with B) parametrization, t = 0.5 will give us (0.25, 0.25, 0.25), which is not the middle of the segment at all.

Among all possible parametrizations of a curve, one is distinguished. It is called natural parametrization. The natural parametrization of the curve has the property: certain change in t parameter corresponds to exactly the same change in curve length. Each curve has exactly one natural parametrization.

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