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