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What is a knot in spline?
A spline of order is a piecewise polynomial function of degree in a variable . The values of where the pieces of polynomial meet are known as knots, denoted and sorted into nondecreasing order. When the knots are distinct, the first derivatives of the polynomial pieces are continuous across each knot.
What does Nurbs stand for?
Non-Uniform Rational B-Splines
NURBS, Non-Uniform Rational B-Splines, are mathematical representations of 3D geometry that can accurately describe any shape from a simple 2D line, circle, arc, or curve to the most complex 3D organic free-form surface or solid.
When are derivatives of the spline continuous across a knot?
When the knots are distinct, the first derivatives of the polynomial pieces are continuous across each knot. When derivatives of the spline are continuous across that knot.
How is continuity loss defined in a spline curve?
The continuity loss at any point can be considered to be the result of multiple knots located at that point, and a spline type can be completely characterized by its degree n and its extended knot vector . is a spline curve if both X and Y are spline functions of the same degree with the same extended knot vectors on that interval.
Specifically, the curve is times continuously differentiable at a knot with multiplicity , and thus has continuity. Therefore, the control polygon will coincide with the curve at a knot of multiplicity , and a knot with multiplicity indicates continuity, or a discontinuous curve.
How are B-spline functions of the same order defined?
B-splines of order n {\\displaystyle n} are basis functions for spline functions of the same order defined over the same knots, meaning that all possible spline functions can be built from a linear combination of B-splines, and there is only one unique combination for each spline function.