MAT-INF 4170/9170 - Lecture 1 - 18/1-2017
• Me: Martin Reimers, Professor computational math. > Room 1015, martinre [snabela] math.uio.no
• You?
• Need a student representative ("Tillitsstudent")> Stein Dahl sadahl [snabela] student.matnat.uio.no was chosen
• The course:
> 10 credits/studiepoeng
> About 17 lectures, wednesdays 1015-1200 (alternatively tuesdays)– http://www.uio.no/studier/emner/matnat/math/MAT-INF4170/v17/timeplan/index.html
> Lecture notes: posted after each lecture
> Curriculum: From the Compendium– http://www.uio.no/studier/emner/matnat/math/MAT-INF4170/v17/pensumliste/index.html
> 4 obligs. All must be passed
> Oblig tutorials before and after submission (tuesdays)
> Oral exam may/june
> Recommended Prerequisites– MAT1100, MAT1110, MAT1120, MAT-INF1100, INF1100– Programming: C++, matlab, python, ...
> Additional reading (not required):– G. Farin. "Curves and surfaces for CAGD: a practical guide"– C. Deboor "A practical Guide to Splines" – L. Schumaker "Spline Functions: basic theory"
> Learning outcomes: spline methods for modelling curves and surfaces, with emphasis both on the mathematical theory and practical methods
> Intro slides
MAT-INF4170/9170 Spline Methods
Introlecture 18. january 2017
Splines Piecewise polynomials
Splines Piecewise polynomials
Splines Piecewise polynomials
Splines Piecewise polynomials
Spline-applications Lots of applications, and interesting research!
Representing functions, curves and surfaces Fonts, data-approximation, statistics,…
Finite element methods – isogeometric analysis
Entertainment: movies, games
CAD/CAM
…..
Spline-applications Animation movies, special effects Geometrisk modellering - glattere flater
I Mer avanserte matematiske formater for glatt/krum geometri,f.eks splines
I tilsvarende vektor format for bilder
I Enklere a editere/animere/komprimere/transportere
I ... mer beregninger for a tegne (konvertere til trekanter)
Spline-applications Games
Data-grafikk - kantete silhouetter
Tekstur gir inntrykk av pen og glatt geometri
... men ikke langs silhouettene!
Spline-applications CAD/CAM Composing patches ...
Nice shading/reflections reveal continuity
How to render spline patches? Convert to triangles ...
Page 7
Further studies MAT-INF42160 Topics in geometric modeling (fall)
Section 1 - Introduction• Basic concepts, curves in the plane
Affine and convex combinations
Convex Hulls
Line Segments
convex combination
Curves
Interpolating Curves
Combine line segments by affine combination
... to form a quadratic curve
Repeated affine combinations
Quadratic case
Examples
Interpolation of four points (cubics)Can generalize to four points, combining quadratic curves:
PROPERTIES
• Get interpolating polynomial curves for any number of points• Can show that q0,p(t)=∑ci li,p(t) where li,p(tj)=1 if i=j and 0 otherwise • Impractical for many points• Not convex combinations => unstable for large p
General interpolation
q0,p(t)
Look for alternatives!
Bezier curvesRepeated convex combinations
Quadratic case
Explicit form:
Examples