Common How-to's

binomialcoeff(n, i)

Calculate the Binomial Coefficient defined as:

\[\binom{n}{i} = \frac{n!}{i!(n-1)!}\]

bernsteincoeff(u, n, i)

Calculate Bernstein Coefficient (Bezier Basis Function) defined as:

\[B_{i, n}(u) = \binom{n}{i} u^i (1-u)^{n-1}\]

at parametric point, $u$, where $0\leq u\leq1$. $u$ may either be a single value or an array.

(see NURBS, eqn 1.8)

simple_bezier1D(P, u)

Calculate a point along a Bezier curve at the parametric point, $u$, based on the control points, $\mathbf{P}$, where the Bezier curve, $\mathbf{C}(u)$, is defined as:

\[\mathbf{C}(u) = \sum_{i=0}^n B_{i, n}(u) \mathbf{P}_i~, ~~~~ 0 \leq u \leq 1\]

where $B$ is the basis (Bernstein Coefficient) at parametric point, $u$, as calculated from bernsteincoeff, and n is the number of control points in vector $\mathbf{P}$. Again, $u$ may either be a single value or an array.

(see NURBS eqn 1.7)

basisfunctions(span, deg, knots, u)

(private function) Compute nonvanishing basis functions (NURBS A2.2)

Arguments

• deg::Integer: degree
• knots::Vector{Float64}:: a knot vector (u0, ... un+1)
• u::Float64`: nondimensional location we are searching for
• span::Integer: corresponding index i for u between knotsi and knotsi+1 (computed from getspanindex)

Returns

• N::Vector{Float64}: vector of length N0 ... Ndeg

basisfunctionsderivatives(span, deg, knots, u, n)

(private function) Calculate the non-vanishing basis functions and derivatives of the B-Spline of order $p$, defined by knots U at parametric point,u`. (NURBS A 2.3)

Arguments

• span::Integer: knot span containing u
• deg::Integer: the curve order
• knots::Vector{Float}: the knot vector
• u::Float: parametric point of interest
• n::Integer : the max derivative order (n ≦ p)

Returns

• ders::Matrix{Float}: [0..n, 0..p] ders[0, :] function values, ders[1: :], first derivatives, etc.

nurbsbasis(u,p,d,U,w)

Get rational basis functions and derivatives.

\[R_{i,p}(u) = \frac{N_{i,p}(u)w_i}{\sum_{j=0}^n N_{j,p}(u)w_j}\]

where $N_{i,p}(u )$ are B-Spline Basis Functions and $w_i$ are weights associated with the NURBS control points.

(see NURBS eqn 4.2)

Inputs:

• u : parametric point of interest
• p : the curve order
• d : the max derivative order (n ≦ p)
• U : the knot vector
• w : control point weights

Evaluate point on B-spline curve (NURBS, A3.1)

Arguments

• bspline::BSpline: bspline object
• u::Float: point on spline to evaluate at

Returns

• C::Vector{Float}: point in ND space

Evaluate point on rational b-spline curve (NURBS, A4.1)

Arguments

• nurbs::NURBS: NURBS object
• u::Float: point on spline to evaluate at

Returns

• C::Vector{Float}: point in ND space

curveknotinsertion(np, p, UP, Pw, u, k, s, r)

Compute a new curve from knot insertion. Using the formula:

\[\mathbf{Q}_{i, r}^w = \alpha_{i, r} \mathbf{Q}_{i, r-1}^w + (1-\alpha_{i, r}) \mathbf{Q}_{i-1, r-1}^w\]

where

\[\alpha_{i, r} = \begin{cases} 1 & i \leq k-p+r-1 \\ \frac{\bar{u} - u_i}{u_{i+p-r+1} - \bar{u}_i} & k-p+r \leq i\leq k-s \\ 0 & i \geq k-s+1 \end{cases}\]

(see NURBS eqn 5.15 and A5.1)

Inputs:

• np : the number of control points minus 1 (the index of the last control point) before insertion
• p : the curve order
• UP : the knot vector before insertion
• Pw : the set of weighted control points and weights before insertion
• u : the knot to be added
• k : the span index at which the knot is to be inserted.
• s : numer of instances of the new knot alrady present in the knot vector, UP
• r : number of times the new knot is inserted (it is assumed that $r+s \leq p$ )

Outputs:

• nq : the number of control points minus 1 (the index of the last control point) after insertion
• UQ : the knot vector after insertion
• Qw : the set of weighted control points and weights after insertion

refineknotvectorcurve(n, p, U, Pw, X, r)

Refine curve knot vector using NURBS A5.4.

This algorithm is simply a knot insertion algorithm that allows for multiple knots to be added simulataneously, i.e., a knot refinement procedure.

Inputs:

• n : the number of control points minus 1 (the index of the last control point) before insertion
• p : the curve order
• U : the knot vector before insertion
• Pw : the set of weighted control points and weights before insertion
• X : elements, in ascending order, to be inserted into U (elements should be repeated according to their multiplicities, e.g., if x and y have multiplicites 2 and 3, X = [x,x,y,y,y])
• r : length of X vector - 1

Outputs:

• Ubar : the knot vector after insertion
• Qw : the set of weighted control points and weights after insertion

degreeelevatecurve(n,p,U,Pw,t)

Raise degree of spline from p to p $+t$, $t \geq 1$ by computing the new control point vector and knot vector.

Knots are inserted to divide the spline into equivalent Bezier Curves. These curves are then degree elevated using the following equation.

\[\mathbf{P}^t_i = \sum^{\textrm{min}(p,i)}_{j=\textrm{max}(0,i-t)} \frac{\binom{p}{j} \binom{t}{i-j} \mathbf{P}_j}{\binom{p+t}{i}}~,~~~~~i=0,...,p+t\]

where $\mathbf{P}^t_i$ are the degree elevated control points after $t$ -degree elevations

Finally, the excess knots are removed and the degree elevated spline is returned.

(see NURBS eqn 5.36, A5.9)

Inputs:

• n : the number of control points minus 1 (the index of the last control point) before degree elevation
• p : the curve order
• U : the knot vector before degree elevation
• Pw : the set of weighted control points and weights before degree elevation
• t : the number of degrees to elevate, i.e. the new curve degree is p+t

Outputs:

• nh : the number of control points minus 1 (the index of the last control point) after degree elevation
• Uh : the knot vector after degree elevation
• Qw : the set of weighted control points and weights after degree elevation