Download presentation

Presentation is loading. Please wait.

Published byUlysses Conte Modified over 2 years ago

1
Anupam Saxena Associate Professor Indian Institute of Technology KANPUR 208016

2
Partition of Unity: The sum of all non-zero order p basis functions over the span [t i, t i+1 ) is 1 titi t i+1 t i+2 t i+3 t i+4 t i-1 t i-2 t i-3 t i-4 t N 4,i+1 (t) N 4,i+2 (t)N 4,i+3 (t) N 4,i+4 (t) B-spline basis functions within a subgroup add to unity Not all B-spline basis functions add to one as opposed to Bernstein polynomials

3
Partition of Unity: The sum of all non-zero order k basis functions over the span [t i, t i+1 ) is 1 GENERAL PROOF

5
Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1- 2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1- 5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite NURBS Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformati ons and Projections 1-2 Modeling of Curves Representati on, Differential Geometry Ferguson Segments Bezier Segments 1- 2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite NURBS

6
Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformati ons and Projections 1-2 Modeling of Curves Representati on, Differential Geometry Ferguson Segments Bezier Segments 1- 2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite NURBS

7
For number of knots as m+1 and the number of degree p–1 basis functions as n+1, m = n + p The first normalized spline on the knot set [t 0, t m ) is N p,p (t) the last spline on this set is N p,m (t) m p+1 basis splinesn+1 = m p+1

8
If a knot t i appears k times (i.e., t i k+1 = t i k+2 =... = t i ), where k > 1, t i is termed as a multiple knot or knot of multiplicity k for k = 1, t i is termed as a simple knot Multiple knots can significantly change the properties of basis functions and are useful in the design of B-spline curves

9
01234 1 0.8 0.6 0.4 0.2 N 3,i (t) At a knot i of multiplicity k, the basis function N p i (t) is C p 1 k continuous at that knot

10
01234 1 0.8 0.6 0.4 0.2 N 3,i (t) The effect is similar when knots are moved to the left

11
At each internal knot of multiplicity k, the number of non-zero order p basis functions is at most p k N 4,i-1 t i-4 t i-1 t i-5 t i-2 t i-6 t i-3 t i-7 N 4,i-2 N 4,i-3 non-zero splines over a simple knot t i 4 p k = 4 1 = 3 non-zero splines over a double knot t i 4 p k = 4 2 = 2 non-zero splines over a triple knot t i 4 p k = 4 3 = 1 non-zero splines over a quadruple knot t i 4 p k = 4 4 = 0

Similar presentations

OK

Anupam Saxena Associate Professor Indian Institute of Technology KANPUR 208016.

Anupam Saxena Associate Professor Indian Institute of Technology KANPUR 208016.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on review of literature apa Ppt on cyclone in india Ppt on dry cell and wet cellulose Ppt on solar power satellites sps Attractive backgrounds for ppt on social media Ppt on agile project management Ppt on democratic rights class 9 Ppt on area of parallelogram formula Ppt on power quality problems Ppt on forward rate agreement explain