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Computer Aided Engineering Design

Computer Aided Engineering Design. Anupam Saxena Associate Professor Indian Institute of Technology KANPUR 208016. Implementation and Coding Parameterization and Knot Vector generation. Examples. Lecture #32 Interpolation with B- spline curves NURBS. Interpolation with B- spline curves.

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Computer Aided Engineering Design

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  1. Computer Aided Engineering Design AnupamSaxena Associate Professor Indian Institute of Technology KANPUR 208016

  2. Implementation and CodingParameterization and Knot Vector generation

  3. Examples

  4. Lecture #32Interpolation with B-spline curvesNURBS

  5. Interpolation with B-spline curves Given n+1 data points p0, p1, ..., pn n + 1 conditions fit them with a B-spline curve of given order pn a set of parameters u0, u1, ..., un may be generated the number of knots m + 1 may be computed knot vector [t0, t1, …, tm] may then be computed Basis functions known

  6. p0 p1 p2 … pn b0 b1 b2 … bn Interpolation with B-spline curves Required to find the interpolating B-splinecurve Control points bi’s are (n+1) unknowns pk = b(uk) = k = 0, …, n Consider Np,p(u0) Np,p+1(u0) Np,p+2(u0) … Np,n+p(u0) Np,p(u1) Np,p+1(u1) Np,p+2(u1) … Np,n+p(u1) Np,p(u2) Np,p+1(u2) Np,p+2(u2) … Np,n+p(u2) … … … … … Np,p(un) Np,p+1(un) Np,p+2(un) … Np,n+p(un) P = = = NB

  7. Examples

  8. NURBS Short for Non-Uniform Rational B-Splines Recall from Rational Bézier curves that Likewise, NURBS can be computed as weights wispecified by the user to gain additional design freedom Possess local shape control & all other Properties of B-spline curves Offer great flexibility in design non-uniform: knots are not placed at regular intervals wi= 0: location of bi does not affect the curve’s shape Widely used in freeform curve design For larger values of wi, the curve gets pushed towards bi Can also model analytical curves

  9. Examples

  10. B-spline and Bernstein polynomials For an order p curve … Repeat the first knot ‘0’ p times Repeat the last knot ‘1’ p times Consider n + 1 = p B-spline basis functions/ Control points  number of knots (m + 1); m = n + p = p + p = 2p The b-spline basis functions degenerate to bernstein polynomials

  11. Examples

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