1 / 21

CS32310 - Lecture 03

CS32310 - Lecture 03. Reyer Zwiggelaar rrz@aber.ac.uk. In This Lecture. Three-dimensional objects B-spline curves and patches. B-splines. B-splines pre-date Bezier curves B-splines based on French curves Thin metal strip Positioned by lead weights (ducks) Ducks are like control points.

consuela-arcelia
Télécharger la présentation

CS32310 - Lecture 03

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. CS32310 - Lecture 03 Reyer Zwiggelaar rrz@aber.ac.uk

  2. In This Lecture • Three-dimensional objects • B-spline curves and patches

  3. B-splines • B-splines pre-date Bezier curves • B-splines based on French curves • Thin metal strip • Positioned by lead weights (ducks) • Ducks are like control points

  4. Bezier Versus B-splines • B-splines are more local • Control points only have local influence • Degree of B-splines • Less related to number of control points • One B-spline can approximate n points • B-splines typically use four control points • Influence of control point covers four segments • C0, C1 and C2 continuous

  5. ith Uniform B-spline Segment

  6. Three-Segment B-splines • 6 Control Points: P0 .. P5 • 3 Line Segments: Q3, Q4, Q5 • Q3 is defined by P0 .. P3 • Scaled by B0 .. B3 • Q4 is defined by P1 .. P4 • Scaled by B1 .. B4 • Q5 is defined by P2 .. P5 • Scaled by B2 .. B5

  7. Basis Functions Uniform B-splines • Identical for each segment • Knots (join points in u) at equal distance

  8. Three-Segment B-splines • Knot values: u=3,4,5,6 • Knot vector: [0,1,2,3,4,5,6,7] B0 B1 B2 B3 B4 B5

  9. Uniform B-splines • Start and end points • Not determined by single control point • Tangent vectors at start and end points • More complicated than Bezier curves

  10. Uniform B-splines

  11. Non-Uniform B-splines • Basis functions not identical for segments • Intervals between knots reduced to zero • Control points becomes part of curve • Computational more expensive • Knot vector: [0,0,0,0,1,2,3,3,3,3] • First and last control point on curve • Basis functions can be determined recursively

  12. Rational Curves • Non-uniform rational B-splines (NURBS) • 4D instead of 3D control points • 4D curves projected onto 3D space • CAD applications • Interaction with control points • Interaction with knots • Interaction with weights

  13. NURBS

  14. B-spline Surface Patches

  15. Surface Fitting • Interpolating curves • Modelling: fitting a surface through points • Interpolation of animation path • Key positions are known • C2 (second order derivative) continuous

  16. Surface Fitting • Interpolating curves • Data points are knot values • System of equations giving control points • Interpolating surfaces • Data points • B-spline curves (x,y knot vector can be equal) • Bezier patches • Only for uniform B-splines, else knot insertion

  17. B-splines to Bezier Conversion Bezier matrix B-spline matrix Bezier control points B-spline control points

  18. Basis Functions Fundamentals d equals degree of curves k equals the position in the knot vector

  19. Which Approach is Best? • Bezier • Single curves/patches • User controls points • B-spline • Large continuous curves/surfaces • NURBS • General approach • Specific modelling: cones

  20. In This Lecture • Three-dimensional objects • B-spline curves and patches

  21. In The Next Lecture • Workshop on some maths aspects

More Related