Modelling Surface Curvature with NURBS

NURBS (Non-Uniform Rational B-Splines) are a powerful tool for representing smooth curves and surfaces in computer graphics and engineering. They are defined by control points and curves, and can be used to create a wide range of shapes and forms. In this article, we’ll explore some of the key concepts and techniques involved in modelling surface curvature with NURBS.

NURBs Patches

One common approach to modelling surface curvature with NURBS is to use NURBS patches. These are defined by a grid of control points, connected by curves. To create a smooth, curved surface, you can adjust the positions of the control points and the shape of the curves.

For example, imagine you want to create a surface that represents the hood of a car. You could start by defining a grid of control points, arranged in a rectangular shape. You could then adjust the positions of these points to create the desired curvature of the surface. By manipulating the control points and curves, you can create a wide range of shapes and forms, such as rounded corners, curved edges, and more.

NURBS Surfaces

Another approach to modelling surface curvature with NURBS is to use NURBS surfaces. These are defined by two sets of control points arranged in a grid, and can be used to create more complex shapes and forms, such as cylinders and spheres.

To create a NURBS surface, you can start by defining a grid of control points and manipulating their positions to create the desired curvature of the surface. You can then use specialized software tools, such as 3D modelling software, to refine and manipulate the surface further.

Surface Continuity

In computer graphics and engineering, G0, G1, G2, and G3 continuity refer to the smoothness of a curve or surface at a particular point. These terms are used to describe how closely a curve or surface follows a smooth mathematical curve at a given point.

G0 continuity, also known as point continuity, refers to a curve or surface that passes through a given point, but does not necessarily have a smooth tangent at that point. This means that the curve or surface may have a sharp corner or discontinuity at the point in question.

G1 continuity, also known as tangent continuity, refers to a curve or surface that has a smooth tangent at a given point. This means that the curve or surface does not have a sharp corner or discontinuity at the point in question, and the slope of the curve or surface is continuous.

G2 continuity, also known as curvature continuity, refers to a curve or surface that has a smooth tangent and curvature at a given point. This means that the curve or surface does not have a sharp corner or discontinuity at the point in question, and the slope and curvature of the curve or surface are continuous.

G3 continuity, also known as higher-order continuity, refers to a curve or surface that has a smooth tangent, curvature, and higher-order derivatives at a given point. This means that the curve or surface does not have a sharp corner or discontinuity at the point in question, and the slope, curvature, and higher-order derivatives of the curve or surface are continuous.

In modelling, achieving G1, G2, or G3 continuity can be important for creating smooth, realistic surfaces and curves. This may involve using specific techniques and algorithms to ensure that the curve or surface follows a smooth mathematical curve at a given point. For example, in computer graphics, NURBS (Non-Uniform Rational B-Splines) are often used to create smooth, continuous surfaces and curves with G1, G2, or G3 continuity.