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January 23, 2018

MAT4170/9170 - Lecture 1 - 17/1-2018

• Me: Martin Reimers, Professor computational mathematics > martinre [snabela] math.uio.no

• You?

• Need a student representative ("Tillitsstudent")> Kristian Haug (krimha [snabela] student.matnat.uio.no)

• The course:

> 10 credits/studiepoeng

> About 17 lectures, wednesdays 1015-1200– http://www.uio.no/studier/emner/matnat/math/MAT4170/v18/timeplan/index.html

> Lecture notes: posted after each lecture

> Curriculum: From the Compendium– http://www.uio.no/studier/emner/matnat/math/MAT4170/v18/pensumliste/index.html

> 4 obligatory exercises (obligs). All must be passed

> Oblig tutorials before and after submission (tuesdays)

> Weekly problems

> 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

MAT4170/9170Spline Methods

Introlecture 17. january 2018

SplinesPiecewise polynomials

SplinesPiecewise polynomials

SplinesPiecewise polynomials

SplinesPiecewise polynomials

Spline-applicationsLots of applications, and interesting research!

Representing functions, curves and surfacesFonts, data-approximation, statistics,…

CAD/CAM

Entertainment: movies, games

Finite element methods – isogeometric analysis

Machine Learning / Neural networks

…..

Spline-applicationsRepresenting functions, curves and surfaces

Spline-applicationsAnimation movies, special effectsGeometrisk 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-applicationsGames

Data-grafikk - kantete silhouetter

Tekstur gir inntrykk av pen og glatt geometri

... men ikke langs silhouettene!

Spline-applicationsCAD/CAM Composing patches ...

Nice shading/reflections reveal continuity

How to render spline patches? Convert to triangles ...

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Spline-applicationsIsogeometric analysis

Spline-applicationsMachine Learning

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Section 1 - Introduction• Basic concepts, how to construct curves in the plane

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Affine and convex combinations

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Convex HullsMengden av alle punkter

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Line Segments

Kan kombinere slike segmenter til en stykkevist lineær kurve

Lineær parametrisk kurve/avbildning

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Functions vs Parametric Curves

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Interpolating Curves by affine combinations

Can combine parametric line segments by affine combination,

to form interpolating quadratic curve:

where

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Examples

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Interpolating curves - cubicsCan combine quadratics, to form interpolating cubics

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Interpolation - repeated affine combinations

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Combining Curves by convex combinations

Combine line segments by convex combinations, to form a quadratic curve

where

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Bezier curves

Explicit form:

=> convex combination

of ci's

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Examples: quadratic Bezier curves