BSplineCurve3dBase Class

Base class for BSplineCurve3d and BSplineCurve3dH.

• A bspline curve consists of a set of knots and a set of poles.
• The bspline curve is a function of the independent "knot axis" variable
• The curve "follows" the poles loosely.
• The is a set of polynomial spans.
• The polynomial spans all have same degree.
• Within each span, the polynomial of that degree is controlled by order = degree + 1 contiguous points called poles.
• The is a strict relationship between knot and poles counts: `numPoles + order = numKnots + 2'
• The number of spans is numSpan = numPoles - degree
• For a given spanIndex:
  • The order poles begin at index spanIndex.
  • The 2*order knots begin as span index
  • The knot interval for this span is from knot[degree+span-1] to knot[degree+span]
• The active part of the knot axis is knot[degree-1] < knot < knot[degree-1 + numSpan] i.e. `knot[degree-1] < knot < knot[numPoles]

Nearly all bsplines are "clamped ".

• Clamping make the curve pass through its first and last poles, with tangents directed along the first and last edges of the control polygon.

• The knots for a clamped bspline have degree copies of the lowest knot value and degree copies of the highest knot value.

• For instance, the knot vector `[0,0,0,1,2,3,3,3]

  • can be evaluated from 0<=knot<=3
  • has 3 spans: 0 to 1, 1 to 2, 2 to 3
  • has 6 poles
  • passes through its first and last poles.

• create methods may allow classic convention that has an extra knot at the beginning and end of the knot vector.

  • The extra knots (first and last) were never referenced by the bspline recurrence relations.
  • When the create methods recognize the classic setup (numPoles + order = numKnots), the extra knot is not saved with the BSplineCurve3dBase knots.

• The weighted variant has the problem that CurvePrimitive 3d typing does not allow undefined result where Point4d has zero weight.

• The convention for these is to return 000 in such places.

• Note the class relationships:

  • BSpline1dNd knows the bspline recurrence relations for control points (poles) with no physical meaning.
  • BsplineCurve3dBase owns a protected BSpline1dNd
  • BsplineCurve3dBase is derived from CurvePrimitive, which creates obligation to act as a 3D curve, such as
    • evaluate fraction to point and derivatives wrt fraction
    • compute intersection with plane
  • BSplineCurve3d and BSplineCurve3dH have variant logic driven by whether or not there are "weights" on the poles.
    • For BSplineCurve3d, the xyz value of pole calculations are "final" values for 3d evaluation
    • For BSplineCurve3dH, various BSpline1dNd results with xyzw have to be normalized back to xyz.

• These classes do not support "periodic" variants.

  • Periodic curves need to have certain leading knots and poles replicated at the end

Extends

• CurvePrimitive

Extended by

• BSplineCurve3d
• BSplineCurve3dH

Methods

Inherited methods

Properties

Inherited properties

Defined in

• bspline/BSplineCurve.ts Line 86

Last Updated: 20 June, 2023