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The next most simple spline has degree 1. It is also called a linear spline. A closed linear spline (i.e, the first knot and the last are the same) in the plane is just a polygon. A common spline is the natural cubic spline. A cubic spline has degree 3 with continuity C 2, i.e. the values and first and second derivatives are continuous. Natural ...
Originally, spline was a term for elastic rulers that were bent to pass through a number of predefined points, or knots. These were used to make technical drawings for shipbuilding and construction by hand, as illustrated in the figure. We wish to model similar kinds of curves using a set of mathematical equations.
Spline curve drawn as a weighted sum of B-splines with control points/control polygon, and marked component curves. In the mathematical subfield of numerical analysis, a B-spline or basis spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition.
The curve is named after Edwin Catmull and Raphael Rom. The principal advantage of this technique is that the points along the original set of points also make up the control points for the spline curve. [7] Two additional points are required on either end of the curve. The uniform Catmull–Rom implementation can produce loops and self ...
Spline interpolation uses low-degree polynomials in each of the intervals, and chooses the polynomial pieces such that they fit smoothly together. The resulting function is called a spline. For instance, the natural cubic spline is piecewise cubic and twice continuously differentiable. Furthermore, its second derivative is zero at the end points.
The Bézier curve is named after French engineer Pierre Bézier (1910–1999), who used it in the 1960s for designing curves for the bodywork of Renault cars. [3] Other uses include the design of computer fonts and animation. [3] Bézier curves can be combined to form a Bézier spline, or generalized to higher dimensions to form Bézier ...
Example showing non-monotone cubic interpolation (in red) and monotone cubic interpolation (in blue) of a monotone data set. Monotone interpolation can be accomplished using cubic Hermite spline with the tangents modified to ensure the monotonicity of the resulting Hermite spline.
Angle and curvature constraints are most often added to the ends of a curve, and in such cases are called end conditions. Identical end conditions are frequently used to ensure a smooth transition between polynomial curves contained within a single spline. Higher-order constraints, such as "the change in the rate of curvature", could also be added.