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https://github.com/mmiscool/nurbs-book-algorithms-js

Implementations of NURBS book algorithms from pseudo code to JS.
https://github.com/mmiscool/nurbs-book-algorithms-js

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Implementations of NURBS book algorithms from pseudo code to JS.

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# NURBS-BOOK-ALGORITHMS-JS



Implementations of NURBS book algorithms from pseudo code to JS.



## Overview



This library provides JavaScript implementations of various algorithms from the NURBS book. The algorithms cover a wide range of topics including curve and surface basics, B-spline basis functions, B-spline curves and surfaces, rational B-spline curves and surfaces, fundamental geometric algorithms, advanced geometric algorithms, conics and circles, construction of common surfaces, curve and surface fitting, and advanced surface construction techniques.



## Table of Contents



1. [Curve and Surface Basics](#curve-and-surface-basics)

2. [B-Spline Basis Functions](#b-spline-basis-functions)

3. [B-Spline Curves and Surfaces](#b-spline-curves-and-surfaces)

4. [Rational B-Spline Curves and Surfaces](#rational-b-spline-curves-and-surfaces)

5. [Fundamental Geometric Algorithms](#fundamental-geometric-algorithms)

6. [Advanced Geometric Algorithms](#advanced-geometric-algorithms)

7. [Conics and Circles](#conics-and-circles)

8. [Construction of Common Surfaces](#construction-of-common-surfaces)

9. [Curve and Surface Fitting](#curve-and-surface-fitting)

10. [Advanced Surface Construction Techniques](#advanced-surface-construction-techniques)



## Curve and Surface Basics



### Horner's Algorithm for Power Basis Curve



**Function:** `Horner1(a, n, u0)`



**Description:** Computes a point on a power basis curve using Horner's algorithm.



**Parameters:**

- `a`: Array of coefficients.

- `n`: Degree of the curve.

- `u0`: Parameter value.



**Returns:** Computed point on the curve.



**Example:**

```javascript

let a = [/* coefficients array */];

let n = a.length - 1;

let u0 = /* value of u */;

let point = Horner1(a, n, u0);

console.log(point);

```



### Bernstein's Algorithm for Bernstein Polynomial



**Function:** `Bernstein(i, n, u)`



**Description:** Computes the value of a Bernstein polynomial.



**Parameters:**

- `i`: Index of the polynomial.

- `n`: Degree of the polynomial.

- `u`: Parameter value.



**Returns:** Value of the Bernstein polynomial.



**Example:**

```javascript

let i = /* index */;

let n = /* degree */;

let u = /* value of u */;

let B = Bernstein(i, n, u);

console.log(B);

```



### Algorithm for Computing All nth-Degree Bernstein Polynomials



**Function:** `AllBernstein(n, u)`



**Description:** Computes all nth-degree Bernstein polynomials.



**Parameters:**

- `n`: Degree of the polynomials.

- `u`: Parameter value.



**Returns:** Array of Bernstein polynomial values.



**Example:**

```javascript

let n = /* degree */;

let u = /* value of u */;

let Bs = AllBernstein(n, u);

console.log(Bs);

```



### Point on Bezier Curve Using Bernstein Polynomials



**Function:** `PointOnBezierCurve(P, n, u)`



**Description:** Computes a point on a Bezier curve using Bernstein polynomials.



**Parameters:**

- `P`: Array of control points.

- `n`: Degree of the curve.

- `u`: Parameter value.



**Returns:** Computed point on the Bezier curve.



**Example:**

```javascript

let P = [/* array of control points */];

let n = P.length - 1;

let u = /* value of u */;

let pointOnCurve = PointOnBezierCurve(P, n, u);

console.log(pointOnCurve);

```



### Point on Bezier Curve Using de Casteljau's Algorithm



**Function:** `deCasteljau1(P, n, u)`



**Description:** Computes a point on a Bezier curve using de Casteljau's algorithm.



**Parameters:**

- `P`: Array of control points.

- `n`: Degree of the curve.

- `u`: Parameter value.



**Returns:** Computed point on the Bezier curve.



**Example:**

```javascript

let P = [/* array of control points */];

let n = P.length - 1;

let u = /* value of u */;

let pointOnCurve = deCasteljau1(P, n, u);

console.log(pointOnCurve);

```



### Point on Power Basis Surface



**Function:** `Horner2(a, n, m, u0, v0)`



**Description:** Computes a point on a power basis surface.



**Parameters:**

- `a`: Array of arrays representing the surface control points.

- `n`: Degree in the u direction.

- `m`: Degree in the v direction.

- `u0`: Parameter value in the u direction.

- `v0`: Parameter value in the v direction.



**Returns:** Computed point on the power basis surface.



**Example:**

```javascript

let a = [/* array of arrays representing the surface control points */];

let n = /* the degree in the u direction */;

let m = /* the degree in the v direction */;

let u0 = /* value of u */;

let v0 = /* value of v */;

let surfacePoint = Horner2(a, n, m, u0, v0);

console.log(surfacePoint);

```



### Point on Bezier Surface Using de Casteljau's Algorithm



**Function:** `deCasteljau2(P, n, m, u0, v0)`



**Description:** Computes a point on a Bezier surface using de Casteljau's algorithm.



**Parameters:**

- `P`: Array of arrays of control points.

- `n`: Degree in the u direction.

- `m`: Degree in the v direction.

- `u0`: Parameter value in the u direction.

- `v0`: Parameter value in the v direction.



**Returns:** Computed point on the Bezier surface.



**Example:**

```javascript

let P = [/* array of arrays representing the surface control points */];

let n = /* the degree in the u direction */;

let m = /* the degree in the v direction */;

let u0 = /* value of u */;

let v0 = /* value of v */;

let surfacePoint = deCasteljau2(P, n, m, u0, v0);

console.log(surfacePoint);

```



## B-Spline Basis Functions



### Knot Span Index in NURBS Calculations



**Function:** `FindSpan(n, p, u, U)`



**Description:** Determines the knot span index in NURBS calculations.



**Parameters:**

- `n`: Number of knots minus one.

- `p`: Degree of the B-spline.

- `u`: Parameter value.

- `U`: Knot vector.



**Returns:** Knot span index.



**Example:**

```javascript

let n = /* the number of knots minus one */;

let p = /* the degree of the B-spline */;

let u = /* the parameter value */;

let U = [/* the knot vector */];

let span = FindSpan(n, p, u, U);

console.log(span);

```



### Nonvanishing Basis Functions



**Function:** `BasisFuns(i, p, u, U)`



**Description:** Computes the nonvanishing basis functions.



**Parameters:**

- `i`: Knot span.

- `p`: Degree of the B-spline.

- `u`: Parameter value.

- `U`: Knot vector.



**Returns:** Array of basis functions.



**Example:**

```javascript

let i = /* the knot span */;

let p = /* the degree of the B-spline */;

let u = /* the parameter value */;

let U = [/* the knot vector */];

let basisFuns = BasisFuns(i, p, u, U);

console.log(basisFuns);

```



### Span and Multiplicity of a Knot



**Function:** `findSpanMult(n, p, u, UP)`



**Description:** Finds the span and multiplicity of a knot in a knot vector.



**Parameters:**

- `n`: Number of control points minus one.

- `p`: Degree of the curve.

- `u`: Parameter value.

- `UP`: Knot vector.



**Returns:** Array containing the span and multiplicity of the knot.



**Example:**

```javascript

let n = /* number of control points minus 1 */;

let p = /* degree of curve */;

let u = /* parameter value */;

let UP = [/* knot vector */];

let [span, mult] = findSpanMult(n, p, u, UP);

console.log('Span:', span, 'Multiplicity:', mult);

```



### Nonvanishing Basis Functions and Their Derivatives



**Function:** `DersBasisFuns(i, p, u, U, n)`



**Description:** Computes the nonvanishing basis functions and their derivatives.



**Parameters:**

- `i`: Knot span.

- `p`: Degree of the B-spline.

- `u`: Parameter value.

- `U`: Knot vector.

- `n`: Order of the derivative.



**Returns:** Array of derivatives of the basis functions.



**Example:**

```javascript

let i = /* the knot span */;

let p = /* the degree of the B-spline */;

let u = /* the parameter value */;

let U = [/* the knot vector */];

let n = /* the order of the derivative */;

let ders = DersBasisFuns(i, p, u, U, n);

console.log(ders);

```



### Basis Function Nip



**Function:** `OneBasisFun(p, m, U, i, u)`



**Description:** Computes the basis function Nip.



**Parameters:**

- `p`: Degree of the B-spline.

- `m`: Upper index of the knot vector.

- `U`: Knot vector.

- `i`: Knot span.

- `u`: Parameter value.



**Returns:** Value of the basis function.



**Example:**

```javascript

let p = /* the degree of the B-spline */;

let m = /* the upper index of U */;

let U = [/* the knot vector */];

let i = /* the knot span */;

let u = /* the parameter value */;

let Nip = OneBasisFun(p, m, U, i, u);

console.log(Nip);

```



### Derivatives of Basis Function Nip



**Function:** `DersOneBasisFun(p, m, U, i, u, n)`



**Description:** Computes the derivatives of the basis function Nip.



**Parameters:**

- `p`: Degree of the B-spline.

- `m`: Upper index of the knot vector.

- `U`: Knot vector.

- `i`: Knot span.

- `u`: Parameter value.

- `n`: Derivative order.



**Returns:** Array of derivatives of the basis function.



**Example:**

```javascript

let p = /* the degree of the B-spline */;

let m = /* the upper index of U */;

let U = [/* the knot vector */];

let i = /* the knot span */;

let u = /* the parameter value */;

let n = /* the derivative order */;

let derivatives = DersOneBasisFun(p, m, U, i, u, n);

console.log(derivatives);

```



## B-Spline Curves and Surfaces



### Curve Point



**Function:** `CurvePoint(n, p, U, P, u)`



**Description:** Computes a point on a B-spline curve.



**Parameters:**

- `n`: Number of control points minus one.

- `p`: Degree of the curve.

- `U`: Knot vector.

- `P`: Array of control points.

- `u`: Parameter value.



**Returns:** Computed point on the curve.



**Example:**

```javascript

let n = /* the number of control points minus 1 */;

let p = /* the degree of the curve */;

let U = [/* the knot vector */];

let P = [/* the array of control points */];

let u = /* the parameter value */;

let curvePoint = CurvePoint(n, p, U, P, u);

console.log(curvePoint);

```



### Curve Derivatives



**Function:** `CurveDerivsAlg1(n, p, U, P, u, d)`



**Description:** Computes the derivatives of a B-spline curve.



**Parameters:**

- `n`: Number of control points minus one.

- `p`: Degree of the curve.

- `U`: Knot vector.

- `P`: Array of control points.

- `u`: Parameter value.

- `d`: Derivative order.



**Returns:** Array of derivatives of the curve.



**Example:**

```javascript

let n = /* the number of control points minus 1 */;

let p = /* the degree of the curve */;

let U = [/* the knot vector */];

let P = [/* the array of control points */];

let u = /* the parameter value */;

let d = /* the derivative order */;

let curveDerivatives = CurveDerivsAlg1(n, p, U, P, u, d);

console.log(curveDerivatives);

```



### Control Points of Curve Derivatives



**Function:** `CurveDerivCpts(n, p, U, P, d, r1, r2)`



**Description:** Computes the control points of the derivatives of a B-spline curve.



**Parameters:**

- `n`: Number of control points minus one.

- `p`: Degree of the curve.

- `U`: Knot vector.

- `P`: Array of control points.

- `d`: Derivative order.

- `r1`: Lower index of the range of control points.

- `r2`: Upper index of the range of control points.



**Returns:** Array of control points for the curve derivatives.



**Example:**

```javascript

let n = /* the number of control points minus 1 */;

let p = /* the degree of the curve */;

let U = [/* the knot vector */];

let P = [/* the control points as an array of [x, y, z] points */];

let d = /* the derivative order */;

let r1 = /* the lower index of the range of control points */;

let r2 = /* the upper index of the range of control points */;

let derivativeControlPoints = CurveDerivCpts(n, p, U, P, d, r1, r2);

console.log(derivativeControlPoints);

```



### Curve Derivatives (Alternative Algorithm)



**Function:** `CurveDerivsAlg2(n, p, U, P, u, d)`



**Description:** Computes the derivatives of a B-spline curve using an alternative algorithm.



**Parameters:**

- `n`: Number of control points minus one.

- `p`: Degree of the curve.

- `U`: Knot vector.

- `P`: Array of control points.

- `u`: Parameter value.

- `d`: Derivative order.



**Returns:** Array of derivatives of the curve.



**Example:**

```javascript

let n = /* the number of control points minus 1 */;

let p = /* the degree of the curve */;

let U = [/* the knot vector */];

let P = [/* the array of control points as [x, y, z] */];

let u = /* the parameter value */;

let d = /* the derivative order */;

let curveDerivatives = CurveDerivsAlg2(n, p, U, P, u, d);

console.log(curveDerivatives);

```



### Surface Point



**Function:** `SurfacePoint(n, p, U, m, q, V, P, u, v)`



**Description:** Computes a point on a B-spline surface.



**Parameters:**

- `n`: Number of control points in the u direction minus one.

- `p`: Degree of the surface in the u direction.

- `U`: Knot vector in the u direction.

- `m`: Number of control points in the v direction minus one.

- `q`: Degree of the surface in the v direction.

- `V`: Knot vector in the v direction.

- `P`: Array of arrays of control points.

- `u`: Parameter value in the u direction.

- `v`: Parameter value in the v direction.



**Returns:** Computed point on the surface.



**Example:**

```javascript

let n = /* number of control points in u direction minus 1 */;

let p = /* degree of the surface in u direction */;

let U = [/* knot vector in u direction */];

let m = /* number of control points in v direction minus 1 */;

let q = /* degree of the surface in v direction */;

let V = [/* knot vector in v direction */];

let P = [/* control point grid as an array of arrays of [x, y, z] points */];

let u = /* parameter value in u direction */;

let v = /* parameter value in v direction */;

let surfacePoint = SurfacePoint(n, p, U, m, q, V, P, u, v);

console.log(surfacePoint);

```



### Surface Derivatives



**Function:** `SurfaceDerivsAlg1(n, p, U, m, q, V, P, u, v, d)`



**Description:** Computes the derivatives of a B-spline surface.



**Parameters:**

- `n`: Number of control points in the u direction minus one.

- `p`: Degree of the surface in the u direction.

- `U`: Knot vector in the u direction.

- `m`: Number of control points in the v direction minus one.

- `q`: Degree of the surface in the v direction.

- `V`: Knot vector in the v direction.

- `P`: Array of arrays of control points.

- `u`: Parameter value in the u direction.

- `v`: Parameter value in the v direction.

- `d`: Order of derivatives.



**Returns:** Array of derivatives of the surface.



**Example:**

```javascript

let n = /* number of control points in u direction minus 1 */;

let p = /* degree of the surface in u direction */;

let U = [/* knot vector in u direction */];

let m = /* number of control points in v direction minus 1 */;

let q = /* degree of the surface in v direction */;

let V = [/* knot vector in v direction */];

let P = [/* control point grid as an array of arrays of [x, y, z] points */];

let u = /* parameter value in u direction */;

let v = /* parameter value in v direction */;

let d = /* order of derivatives */;

let surfaceDerivatives = SurfaceDerivsAlg1(n, p, U, m, q, V, P, u, v, d);

console.log(surfaceDerivatives);

```



### Control Points of Derivative Surfaces



**Function:** `SurfaceDerivCpts(n, p, U, m, q, V, P, d, r1, r2, s1, s2)`



**Description:** Computes the control points of the derivatives of a B-spline surface.



**Parameters:**

- `n`: Number of control points in the u direction minus one.

- `p`: Degree of the surface in the u direction.

- `U`: Knot vector in the u direction.

- `m`: Number of control points in the v direction minus one.

- `q`: Degree of the surface in the v direction.

- `V`: Knot vector in the v direction.

- `P`: Array of arrays of control points.

- `d`: Order of derivatives.

- `r1`: Lower index of the range of control points in the u direction.

- `r2`: Upper index of the range of control points in the u direction.

- `s1`: Lower index of the range of control points in the v direction.

- `s2`: Upper index of the range of control points in the v direction.



**Returns:** Array of control points for the surface derivatives.



**Example:**

```javascript

let n = /* the number of control points in u direction minus 1 */;

let p = /* the degree in u direction */;

let U = [/* the knot vector in u direction */];

let m = /* the number of control points in v direction minus 1 */;

let q = /* the degree in v direction */;

let V = [/* the knot vector in v direction */];

let P = [/* the control point grid */];

let d = /* the derivative order */;

let r1 = /* the start index in u direction */;

let r2 = /* the end index in u direction */;

let s1 = /* the start index in v direction */;

let s2 = /* the end index in v direction */;

let derivativeControlPoints = SurfaceDerivCpts(n, p, U, m, q, V, P, d, r1, r2, s1, s2);

console.log(derivativeControlPoints);

```



### Surface Derivatives (Alternative Algorithm)



**Function:** `SurfaceDerivsAlg2(n, p, U, m, q, V, P, u, v, d)`



**Description:** Computes the derivatives of a B-spline surface using an alternative algorithm.



**Parameters:**

- `n`: Number of control points in the u direction minus one.

- `p`: Degree of the surface in the u direction.

- `U`: Knot vector in the u direction.

- `m`: Number of control points in the v direction minus one.

- `q`: Degree of the surface in the v direction.

- `V`: Knot vector in the v direction.

- `P`: Array of arrays of control points.

- `u`: Parameter value in the u direction.

- `v`: Parameter value in the v direction.

- `d`: Order of derivatives.



**Returns:** Array of derivatives of the surface.



**Example:**

```javascript

let n = /* the number of control points in u direction minus 1 */;

let p = /* the degree in u direction */;

let U = [/* the knot vector in u direction */];

let m = /* the number of control points in v direction minus 1 */;

let q = /* the degree in v direction */;

let V = [/* the knot vector in v direction */];

let P = [/* the control point grid */];

let u = /* parameter value in u direction */;

let v = /* parameter value in v direction */;

let d = /* the order of derivatives */;

let surfaceDerivatives = SurfaceDerivsAlg2(n, p, U, m, q, V, P, u, v, d);

console.log(surfaceDerivatives);

```



## Rational B-Spline Curves and Surfaces



### Point on Rational B-Spline Curve



**Function:** `CurvePoint(n, p, U, Pw, u)`



**Description:** Computes a point on a rational B-spline curve.



**Parameters:**

- `n`: Number of control points minus one.

- `p`: Degree of the curve.

- `U`: Knot vector.

- `Pw`: Array of control points with weights.

- `u`: Parameter value.



**Returns:** Computed point on the curve.



**Example:**

```javascript

let n = /* the number of control points minus 1 */;

let p = /* the degree of the curve */;

let U = [/* the knot vector */];

let Pw = [/* the control points with weights as an array of [x, y, z, w] */];

let u = /* the parameter value */;

let curvePoint = CurvePoint(n, p, U, Pw, u);

console.log(curvePoint);

```



### Derivatives of Rational B-Spline Curve



**Function:** `RatCurveDerivs(Aders, wders, d)`



**Description:** Computes the derivatives of a rational B-spline curve from the derivatives of the weighted points.



**Parameters:**

- `Aders`: Array of derivatives of the weighted points.

- `wders`: Array of derivatives of the weights.

- `d`: Derivative order.



**Returns:** Array of derivatives of the curve points in Cartesian coordinates.



**Example:**

```javascript

let Aders = [/* array of derivatives of the weighted points */];

let wders = [/* array of derivatives of the weights */];

let d = /* the derivative order */;

let curveDerivatives = RatCurveDerivs(Aders, wders, d);

console.log(curveDerivatives);

```



### Point on Rational B-Spline Surface



**Function:** `SurfacePoint(n, p, U, m, q, V, Pw, u, v)`



**Description:** Computes a point on a rational B-spline surface.



**Parameters:**

- `n`: Number of control points in the u direction minus one.

- `p`: Degree of the surface in the u direction.

- `U`: Knot vector in the u direction.

- `m`: Number of control points in the v direction minus one.

- `q`: Degree of the surface in the v direction.

- `V`: Knot vector in the v direction.

- `Pw`: Array of arrays of control points with weights.

- `u`: Parameter value in the u direction.

- `v`: Parameter value in the v direction.



**Returns:** Computed point on the surface.



**Example:**

```javascript

let n = /* the number of control points in u direction minus 1 */;

let p = /* the degree of the surface in u direction */;

let U = [/* the knot vector in u direction */];

let m = /* the number of control points in v direction minus 1 */;

let q = /* the degree of the surface in v direction */;

let V = [/* the knot vector in v direction */];

let Pw = [/* the weighted control points as an array of [x, y, z, w] */];

let u = /* the parameter value in u direction */;

let v = /* the parameter value in v direction */;

let surfacePoint = SurfacePoint(n, p, U, m, q, V, Pw, u, v);

console.log(surfacePoint);

```



### Derivatives of Rational B-Spline Surface



**Function:** `RatSurfaceDerivs(Aders, wders, d)`



**Description:** Computes the derivatives of a rational B-spline surface from the derivatives of the weighted points.



**Parameters:**

- `Aders`: Array of derivatives of the weighted points.

- `wders`: Array of derivatives of the weights.

- `d`: Derivative order.



**Returns:** Array of derivatives of the surface points in Cartesian coordinates.



**Example:**

```javascript

let Aders = [/* array of derivatives of the weighted points */];

let wders = [/* array of derivatives of the weights */];

let d = /* the derivative order */;

let surfaceDerivatives = RatSurfaceDerivs(Aders, wders, d);

console.log(surfaceDerivatives);

```



## Fundamental Geometric Algorithms



### Knot Insertion on a NURBS Curve



**Function:** `CurveKnotIns(n, p, UP, Pw, u, k, s, r)`



**Description:** Performs knot insertion on a NURBS curve.



**Parameters:**

- `n`: Number of control points minus one.

- `p`: Degree of the curve.

- `UP`: Original knot vector.

- `Pw`: Original control points.

- `u`: Knot to insert.

- `k`: Knot span.

- `s`: Multiplicity of the knot.

- `r`: Number of times to insert the knot.



**Returns:** Object containing the new number of control points, new knot vector, and new control points.



**Example:**

```javascript

let n = /* number of control points minus 1 */;

let p = /* degree of curve */;

let UP = [/* original knot vector */];

let Pw = [/* original weighted control points as an array of {x, y, z, w} */];

let u = /* knot to insert */;

let k = /* knot span */;

let s = /* multiplicity of knot u */;

let r = /* number of times to insert knot u */;

let { nq, UQ, Qw } = CurveKnotIns(n, p, UP, Pw, u, k, s, r);

console.log(UQ); // New knot vector

console.log(Qw); // New control points

```



### Point on Rational B-Spline Curve by Corner Cutting



**Function:** `curvePntByCornerCut(np, UP, w, u)`



**Description:** Computes a point on a rational B-spline curve using corner cutting.



**Parameters:**

- `np`: Array of control points.

- `UP`: Knot vector.

- `w`: Weight.

- `u`: Parameter value.



**Returns:** Computed point on the curve.



**Example:**

```javascript

let np = [/* control points as an array of {x, y, z, w} */];

let UP = [/* knot vector */];

let w = /* weight */;

let u = /* parameter value */;

let C = curvePntByCornerCut(np, UP, w, u);

console.log(C); // Computed point on the curve

```



### Surface Knot Insertion in NURBS



**Function:** `SurfaceKnotIns(np, p, UP, mp, q, VP, Pw, dir, uv, k, s, r, nq, UQ, mq, VQ, Qw)`



**Description:** Inserts a knot into a NURBS surface in either the U or V direction.



**Parameters:**

- `np`: Number of control points in the U direction minus one.

- `p`: Degree in the U direction.

- `UP`: Knot vector in the U direction.

- `mp`: Number of control points in the V direction minus one.

- `q`: Degree in the V direction.

- `VP`: Knot vector in the V direction.

- `Pw`: Array of control points.

- `dir`: Direction for knot insertion (0 for U, 1 for V).

- `uv`: Knot value to insert.

- `k`: Span where the knot is to be inserted.

- `s`: Multiplicity of the knot.

- `r`: Number of times the knot is to be inserted.

- `nq`: New number of control points in the U (or V) direction minus one.

- `UQ`: New knot vector in the U (or V) direction.

- `mq`: New number of control points in the V (or U) direction minus one.

- `VQ`: New knot vector in the V (or U) direction.

- `Qw`: New control points array after insertion.



**Returns:** Object containing the new control points and knot vectors.



**Example:**

```javascript

let np = /* number of control points in U minus one */;

let p = /* degree in U direction */;

let UP = /* knot vector in U */;

let mp = /* number of control points in V minus one */;

let q = /* degree in V direction */;

let VP = /* knot vector in V */;

let Pw = /* control points array */;

let dir = /* direction for knot insertion (0 for U, 1 for V) */;

let uv = /* knot value to insert */;

let k = /* span where knot is to be inserted */;

let s = /* multiplicity of knot */;

let r = /* number of times knot is to be inserted */;

let nq = /* new number of control points in U (or V) minus one */;

let UQ = /* new knot vector in U (or V) */;

let mq = /* new number of control points in V (or U) minus one */;

let VQ = /* new knot vector in V (or U) */;

let Qw = /* new control points array after insertion */;

let result = SurfaceKnotIns(np, p, UP, mp, q, VP, Pw, dir, uv, k, s, r, nq, UQ, mq, VQ, Qw);

console.log('New Knot Vector:', result.UQ || result.VQ);

console.log('New Control Points:', result.Qw);

```



### Knot Refinement in NURBS Curve



**Function:** `RefineKnotVectCurve(n, p, U, Pw, X, r, Ubar, Qw)`



**Description:** Refines a NURBS curve by inserting a given set of knots.



**Parameters:**

- `n`: Number of control points minus one.

- `p`: Degree of the curve.

- `U`: Original knot vector.

- `Pw`: Array of control points.

- `X`: Array of new knots to insert.

- `r`: Number of new knots minus one.

- `Ubar`: New knot vector.

- `Qw`: New control points.



**Example:**

```javascript

let n = /* number of control points minus one */;

let p = /* degree of the curve */;

let U = /* original knot vector */;

let Pw = /* control points */;

let X = /* new knots to insert */;

let r = X.length - 1;

let Ubar = new Array(U.length + r + 1); // New knot vector

let Qw = new Array(Pw.length + r + 1); // New control points

RefineKnotVectCurve(n, p, U, Pw, X, r, Ubar, Qw);

console.log('New Knot Vector:', Ubar);

console.log('New Control Points:', Qw);

```



### Knot Refinement in NURBS Curve (Alternative Algorithm)



**Function:** `RefineKnotVectCurve(n, p, U, Pw, X, r)`



**Description:** Refines a NURBS curve by inserting a set of new knots into the original knot vector.



**Parameters:**

- `n`: Number of control points minus one.

- `p`: Degree of the curve.

- `U`: Original knot vector.

- `Pw`: Array of control points.

- `X`: Array of new knots to insert.

- `r`: Number of new knots minus one.



**Returns:** Object containing the new knot vector and new control points.



**Example:**

```javascript

let n = 3; // Example value for number of control points minus one

let p = 2; // Example value for degree of the curve

let U = [0, 0, 0, 1, 1, 1]; // Example original knot vector

let Pw = [/* Control points array */];

let X = [0.5, 0.75]; // Example array of new knots to be inserted

let r = X.length - 1;

let result = RefineKnotVectCurve(n, p, U, Pw, X, r);

console.log('New Knot Vector:', result.Ubar);

console.log('New Control Points:', result.Qw);

```



### Decomposing a NURBS Curve into Bézier Segments



**Function:** `DecomposeCurve(n, p, U, Pw)`



**Description:** Decomposes a NURBS curve into its constituent Bézier segments.



**Parameters:**

- `n`: Number of control points minus one.

- `p`: Degree of the curve.

- `U`: Knot vector.

- `Pw`: Array of control points.



**Returns:** Array of Bézier segments.



**Example:**

```javascript

let n = /* number of control points minus one */;

let p = /* degree of the curve */;

let U = /* knot vector */;

let Pw = /* control points array */;

let bezierSegments = DecomposeCurve(n, p, U, Pw);

console.log('Bézier Segments:', bezierSegments);

```



### Decomposing a NURBS Surface into Bézier Patches



**Function:** `DecomposeSurface(n, p, U, m, q, V, Pw, dir)`



**Description:** Decomposes a NURBS surface into its constituent Bézier patches in either the U or V direction.



**Parameters:**

- `n`: Number of control points in the U direction minus one.

- `p`: Degree in the U direction.

- `U`: Knot vector in the U direction.

- `m`: Number of control points in the V direction minus one.

- `q`: Degree in the V direction.

- `V`: Knot vector in the V direction.

- `Pw`: Array of control points.

- `dir`: Direction for decomposition (0 for U, 1 for V).



**Returns:** Array of Bézier patches.



**Example:**

```javascript

let n = /* number of control points in U minus one */;

let p = /* degree in U direction */;

let U = /* knot vector in U */;

let m = /* number of control points in V minus one */;

let q = /* degree in V direction */;

let V = /* knot vector in V */;

let Pw = /* control points array */;

let dir = /* direction for decomposition (0 for U, 1 for V) */;

let bezierPatches = DecomposeSurface(n, p, U, m, q, V, Pw, dir);

console.log('Bézier Patches:', bezierPatches);

```



### Removing a Knot from a NURBS Curve



**Function:** `RemoveCurveKnot(n, p, U, Pw, u, r, s, num)`



**Description:** Attempts to remove a knot from a NURBS curve a specified number of times.



**Parameters:**

- `n`: Number of control points minus one.

- `p`: Degree of the curve.

- `U`: Knot vector.

- `Pw`: Array of control points.

- `u`: Knot to be removed.

- `r`: Knot span index.

- `s`: Multiplicity of the knot.

- `num`: Number of times to remove the knot.



**Example:**

```javascript

let n, p, U, Pw, u, r, s, num;

// Initialize these variables as per your curve specifications

RemoveCurveKnot(n, p, U, Pw, u, r, s, num);

// The control points Pw and knot vector U are modified in place

```



### Degree Elevation of a NURBS Curve



**Function:** `DegreeElevateCurve(n, p, U, Pw, t)`



**Description:** Increases the degree of a NURBS curve by a specified factor.



**Parameters:**

- `n`: Number of control points minus one.

- `p`: Degree of the curve.

- `U`: Knot vector.

- `Pw`: Array of control points.

- `t`: Degree elevation factor.



**Returns:** Object containing the new degree, new knot vector, and new control points.



**Example:**

```javascript

let n = /* number of control points minus one */;

let p = /* degree of the curve */;

let U = /* original knot vector */;

let Pw = /* control points */;

let t = /* number of times to elevate degree */;

let { nh, Uh, Qw } = DegreeElevateCurve(n, p, U, Pw, t);

console.log('New degree:', nh);

console.log('New knot vector:', Uh);

console.log('New control points:', Qw);

```



### Degree Elevation of a NURBS Surface



**Function:** `DegreeElevateSurface(n, p, U, m, q, V, Pw, dir, t)`



**Description:** Increases the degree of a NURBS surface in either the U or V direction by a specified factor.



**Parameters:**

- `n`: Number of control points in the U direction minus one.

- `p`: Degree in the U direction.

- `U`: Knot vector in the U direction.

- `m`: Number of control points in the V direction minus one.

- `q`: Degree in the V direction.

- `V`: Knot vector in the V direction.

- `Pw`: Array of control points.

- `dir`: Direction for degree elevation (0 for U, 1 for V).

- `t`: Degree elevation factor.



**Returns:** Object containing the new degree, new knot vectors, and new control points.



**Example:**

```javascript

let n, p, U, m, q, V, Pw, dir, t;

// Initialize these variables as per your surface specifications

let result = DegreeElevateSurface(n, p, U, m, q, V, Pw, dir, t);

console.log('New degree in U direction:', result.nh);

console.log('New knot vector in U direction:', result.Uh);

console.log('New degree in V direction:', result.mh);

console.log('New knot vector in V direction:', result.Vh);

console.log('New control points:', result.Qw);

```



### Degree Reduction of a NURBS Curve



**Function:** `DegreeReduceCurve(n, p, U, Qw)`



**Description:** Attempts to reduce the degree of a NURBS curve from `p` to `p-1`.



**Parameters:**

- `n`: Number of control points minus one.

- `p`: Degree of the curve.

- `U`: Original knot vector.

- `Qw`: Array of control points.



**Returns:** Object containing the new degree, new knot vector, and new control points.



**Example:**

```javascript

let n = /* number of control points minus one */;

let p = /* degree of the curve */;

let U = /* original knot vector */;

let Qw = /* control points */;

let { nh, Uh, Pw } = DegreeReduceCurve(n, p, U, Qw);

console.log('New degree:', nh);

console.log('New knot vector:', Uh);

console.log('New control points:', Pw);

```



## Advanced Geometric Algorithms



### Matrix to Convert Bézier Form to Power Form



**Function:** `BezierToPowerMatrix(p)`



**Description:** Computes the matrix to convert a Bézier curve to its power form.



**Parameters:**

- `p`: Degree of the Bézier curve.



**Returns:** Matrix for converting Bézier form to power form.



**Example:**

```javascript

let p = /* degree of the Bézier curve */;

let matrix = BezierToPowerMatrix(p);

console.log(matrix);

```



### Matrix to Convert Power Form to Bézier Form



**Function:** `PowerToBezierMatrix(p, M)`



**Description:** Computes the matrix to convert a curve from its power form to the Bézier form.



**Parameters:**

- `p`: Degree of the curve.

- `M`: Matrix to be inverted (from Bézier to power form).



**Returns:** Inverse matrix for converting power form to Bézier form.



**Example:**

```javascript

let p = /* degree of the curve */;

let M = /* matrix from Bézier to power form (from A6.1) */;

let MI = PowerToBezierMatrix(p, M);

console.log(MI);

```



## Conics and Circles



### NURBS Circular Arc



**Function:** `MakeNurbsCircle(O, X, Y, r, ths, the)`



**Description:** Creates an arbitrary NURBS circular arc.



**Parameters:**

- `O`: Center point.

- `X`: Vector defining the x