Wednesday, March 12, 2008

Spline Terminology

Spline Terminology

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Overview: Splines created in CATIA are similar to NURBS, but within the Generative Shape Design workbench, they do not have full NURB functionality. Splines are created using defining data, poles, or as the result of another operation. Defining data can include points, slopes, and curvatures that describe the spline's shape. You also can generate a spline from an operation, such as Intersect Faces.

Splines

Splines are flexible and powerful tools in the creation of geometric data within the computer graphics area. There are many advantages for using splines; they can represent virtually any desired shape, from points to straight lines to conic sections, they give you control over the shape of the curve. Control points guide the curve's shape and can be directly manipulated to control its smoothness and curvature.

Control Points

The shape of a spline is controlled by a set of points known as 'control points'. Moving one of these points modifies the shape of the spline. However only the region around the control point is modified.

Weights

If you investigate the mathematical representation of the control points, you would find that four coordinates define each one. Three coordinates represent the standard x, y, z points in space and the fourth is the weight. The weight is the influence a control point has over the shape of the spline. Control points with higher weights insert a higher influence over the spline. Curves that are defined in this way with a weight for each control point are called 'rational curves'.

Continuity

The term 'continuity' refers to the change in flow of a curve. Continuity applies when creating and simplifying curves, which requires manipulation of the curve to ensure that you create the desired type of continuity.
A curve can be described as having C0, C1, and C2 continuity. These define the three curve definition types. The continuity types reflect the geometry combinations used in the creation of a curve.
C0 is continuous with respect to position. The curve contains a corner. This condition can be defined by a group of linear geometry joined at their endpoints. With C0 continuity, the slope is non-continuous and there is no curvature.

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C1 is continuous with respect to tangency. The curve contains a tangent point. Linear segments joined by a circular tangent curve can define this condition. With C1 continuity, the slope is continuous but the curvature is non-continuous because of the linear sections.

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C2 is continuous with respect to curvature. The curve is continuous in position, slope, and curvature. This is defined by a

curve in which no linear sections exist. Consequently, its shape is constantly changing.

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Splines in Wireframe and Surface Design

Having read the paragraphs above, you will notice that with the Wireframe and Surface Design and Generative Shape Design workbenches you cannot control the spline's shape by defining the control points. You can only modify the control points using the FreeStyle workbench. Within Wireframe and Surface Design and Generative Shape Design workbenches, you define the points that the spline passes through and the tangency and curvature values for each point. You cannot define the weight of the control points, but you can modify the Tangency Tension value, which works in a similar manner.
When creating wireframe splines, the passing points must pre-exist. In Sketcher, you use more of a freeform method when defining points. The sketched spline, however, must remain planar.

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