A spline curve is a smooth curve that passes through a series of given points. Firstly, spline curves are obtained with the help of physical splines. The lofting personnel fix the elastic thin wood strip (or plexiglass strip) at the given set point where the curve will pass, and the curve drawn by spline natural bending is spline curve. Spline curve not only passes through each ordered point, but also the first and second derivatives at each point are continuous, that is, the curve has the characteristics of continuity and uniform curvature change.
Non-uniform rational B-spline curve (NURBS) is a widely used spline curve, which can not only describe free curves and surfaces, but also provide a unified expression for various geometric shapes including quadrics and surfaces. Since 1983, SDRC has successfully applied NURBS model to its solid modeling software. NURBS has become the basis of geometric modeling of computer-aided design and computer-aided manufacturing, and has been widely used.
AutoCAD uses this NURBS mathematical model to create spline curves, which is also the basis of surface modeling and solid modeling in MDT.
Before explaining in detail the function of AutoCAD to construct and modify NURBS curves (hereinafter referred to as spline curves), it is necessary to understand several terms about NURBS curves from the perspective of mathematics and geometry. Only the terms related to understanding NURBS curves in AutoCAD are explained here. For other detailed mathematical knowledge, please refer to relevant materials.
Related terms of NURBS curve [/color]
Type point or fitting point
The known given point through which the spline curve should pass.
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Feature polygon or control polygon
Spline curves are polygons composed of some broken lines. Simply put, spline curves are constructed by approximating polygonal lines with smooth parametric curves through numerical calculation. Changing the vertices and number of polygons will affect the shape of curves. The polyline polygon mentioned here is the characteristic polygon or control polygon of spline curve.
Spline curve segment
Spline curve is composed of a group of smooth parametric curve segments that approximate the control polygon, and these curve segments are spline curve segments.
Vertices/control points of feature polygons/control polygons
The endpoint of the dotted line that constitutes the characteristic polygon is the vertex of the characteristic polygon, which is also called the control point of the control polygon. Only under special circumstances can spline curves pass through control points.
Degree of spline curve
The number of spline curves is determined by the basis function in the mathematical definition of spline curves. Intuitively speaking, a smooth parametric curve segment of a spline curve is determined by several adjacent continuous polylines of the control polygon, that is, several splines, and the most commonly used ones are quadratic and cubic splines. A curve segment of a quadratic spline curve is only related to two corresponding polylines and three control polygon vertices. Changing one of the vertices will affect the three-segment spline curve segment. Similarly, for cubic spline, a curve segment is determined by the corresponding three-segment broken line segment and four control points.
Order of spline curve
The order is related to the number of times, and the order of spline curve is the number of times that spline curve adds one. The higher the order of spline curve, the more control points. The order of quadratic spline is three, which is determined by spline curve segment and three control points. The order of cubic spline is four, which is determined by spline curve segment and four control points.
Weight of spline curve
Weight can control the spline curve segment to adjust locally in the control polygon, which reflects the degree of curve approaching the control polygon. The greater the weight, the closer the curve segment is to the control polygon. On the contrary, stay away. . When the weight is 1, NURBS curve degenerates into irrational B-spline curve, which shows that irrational B-spline curve is a subset of NURBS.
Tolerance of spline curve
Tolerance refers to the accuracy of spline curves passing through text points. The smaller the tolerance, the closer the spline curve is to the text point. If the tolerance is zero, the spline curve will pass through the typed point.
Method of generating spline curve by AutoCAD
AutoCAD uses SPLINE command to create spline curves, that is, NURBS curves. PEDIT command is also used to generate approximate spline curves by smooth polyline fitting, hereinafter referred to as "spline fitting polyline". This kind of curve is not a real spline curve, but a multi-segment line composed of many straight lines (curves), which is close to spline curve. However, the SPLINE command can be used to convert 2D and 3D spline-fitted polylines into spline curves.
Compared with SPLINE fitting polyline generated by editing smooth polyline, spline curve created by spline command has the following differences:
Spline curve is obviously much more accurate than spline fitting polyline. In engineering applications, spline fitting of polylines can not be used as the basis of mathematical analysis, nor can it generate tangents, normals or extract point data on curves.