Geometric Modeling (WS 2021/22)

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Geometric Modeling (WS 2021/22)

Martin Held

FB Computerwissenschaften Universität Salzburg A-5020 Salzburg, Austria

held@cs.sbg.ac.at

July 20, 2021

Computational Geometry and Applications Lab UNIVERSIT ¨AT SALZBURG

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Personalia

Instructor (VO+PS): M. Held.

Email: held@cs.sbg.ac.at.

Base-URL: https://www.cosy.sbg.ac.at/˜held.

Office: Universität Salzburg, Computerwissenschaften, Rm. 1.20, Jakob-Haringer Str. 2, 5020 Salzburg-Itzling.

Phone number (office): (0662) 8044-6304.

Phone number (secr.): (0662) 8044-6328.

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Formalia

URL of course (VO+PS): Base-URL/teaching/geom_mod/geom_mod.html.

Lecture times (VO): Friday 12⁴⁵–14⁴⁵.

Venue (VO): T03, Computerwissenschaften, Jakob-Haringer Str. 2.

Lecture times (PS): Friday 11²⁰–12²⁰.

Venue (PS): T03, Computerwissenschaften, Jakob-Haringer Str. 2.

Note — PS is graded according to continuous-assessment mode!

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M. Held (Univ. Salzburg) Geometric Modeling(WS 2021/22) 3/246

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Electronic Slides and Online Material

In addition to these slides, you are encouraged to consult the WWW home-page of this lecture:

https://www.cosy.sbg.ac.at/˜held/teaching/geom_mod/geom_mod.html.

In particular, this WWW page contains up-to-date information on the course, plus links to online notes, slides and (possibly) sample code.

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A Few Words of Warning

I hope that these slides will serve as a practice-minded introduction to various aspects of geometric modeling. I would like to warn you explicitly not to regard these slides as the sole source of information on the topics of my course. It may and will happen that I’ll use the lecture for talking about subtle details that need not be covered in these slides! In particular, the slides won’t contain all sample calculations, proofs of theorems, demonstrations of algorithms, or solutions to problems posed during my lecture. That is, by making these slides available to you I do not intend to encourage you to attend the lecture on an irregular basis.

See alsoIn Praise of Lecturesby T.W. Körner.

Abasic knowledge of calculus, linear algebra, discrete mathematics, and geometric computing, as taught in standard undergraduate CS courses, should suffice to take this course. It is my sincere intention to start at such a hypothetical low level of “typical prior undergrad knowledge”. Still, it is obvious that different educational backgrounds will result in different levels of prior knowledge. Hence, you might realize that you do already know some items covered in this course, while you lack a decent understanding of some other items which I seem to presuppose. In such a case I do expect you to refresh or fill in those missing items on your own!

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A Few Words of Warning

I hope that these slides will serve as a practice-minded introduction to various aspects of geometric modeling. I would like to warn you explicitly not to regard these slides as the sole source of information on the topics of my course. It may and will happen that I’ll use the lecture for talking about subtle details that need not be covered in these slides! In particular, the slides won’t contain all sample calculations, proofs of theorems, demonstrations of algorithms, or solutions to problems posed during my lecture. That is, by making these slides available to you I do not intend to encourage you to attend the lecture on an irregular basis.

See alsoIn Praise of Lecturesby T.W. Körner.

Abasic knowledge of calculus, linear algebra, discrete mathematics, and geometric computing, as taught in standard undergraduate CS courses, should suffice to take this course. It is my sincere intention to start at such a hypothetical low level of “typical prior undergrad knowledge”. Still, it is obvious that different educational backgrounds will result in different levels of prior knowledge. Hence, you might realize that you do already know some items covered in this course, while you lack a decent understanding of some other items which I seem to presuppose. In such a case I do expect you to refresh or fill in those missing items on your own!

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Acknowledgments

A small portion of these slides is based on notes and slides originally prepared by students — most notably Dominik Kaaser, Kamran Safdar, and Marko Šuleji´c — on topics related to geometric modeling. I would like to express my thankfulness to all of them for their help. This revision and extension was carried out by myself, and I am responsible for all errors.

I am also happy to acknowledge that I benefited from material published by colleagues on diverse topics that are partially covered in this lecture. While some of the material used for this lecture was originally presented in traditional-style publications (such as textbooks), some other material has its roots in non-standard publication outlets (such as online documentations, electronic course notes, or user manuals).

Salzburg, July 2021 Martin Held

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Legal Fine Print and Disclaimer

To the best of my knowledge, these slides do not violate or infringe upon somebody else’s copyrights. If copyrighted material appears in these slides then it was considered to be available in a non-profit manner and as an educational tool for teaching at an academic institution, within the limits of the “fair use” policy. For copyrighted material we strive to give references to the copyright holders (if known).

Of course, any trademarks mentioned in these slides are properties of their respective owners.

Please note that these slides are copyrighted. The copyright holder grants you the right to download and print the slides for your personal use. Any other use, including instructional use at non-profit academic institutions and re-distribution in electronic or printed form of significant portions, beyond the limits of “fair use”, requires the explicit permission of the copyright holder. All rights reserved.

These slides are made available without warrant of any kind, either express or implied, including but not limited to the implied warranties of merchantability and fitness for a particular purpose. In no event shall the copyright holder and/or his respective employer be liable for any special, indirect or consequential damages or any damages whatsoever resulting from loss of use, data or profits, arising out of or in connection with the use of information provided in these slides.

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Recommended Textbooks I

G. Farin.

Curves and Surfaces for CAGD: A Practical Guide.

Morgan Kaufmann, 5th edition, 2002; ISBN 978-1-55860-737-8.

R.H. Bartels, J.C. Beatty, B.A. Barsky.

An Introduction to Splines for Use in Computer Graphics and Geometric Modeling.

Morgan Kaufmann, 1995; ISBN 978-1558604001.

H. Prautzsch, W. Boehm, M. Paluszny.

Bézier and B-spline Techniques.

Springer, 2002; ISBN 978-3540437611.

J. Gallier.

Curves and Surfaces in Geometric Modeling.

Morgan Kaufmann, 1999; ISBN 978-1558605992.

http://www.cis.upenn.edu/~jean/gbooks/geom1.html R. Goldman.

An Integrated Introduction to Computer Graphics and Geometric Modeling.

CRC Press, 2019; ISBN 978-1138381476.

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Recommended Textbooks II

N.M. Patrikalakis, T. Maekawa, W. Cho.

Shape Interrogation for Computer Aided Design and Manufacturing.

Springer, 2nd corr. edition, 2010; ISBN 978-3642040733.

http://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/

M. Botsch, L. Kobbelt, M. Pauly, P. Alliez, B. Levy.

Polygon Mesh Processing.

A K Peters/CRC Press, 2010; ISBN 978-1568814261.

http://www.pmp-book.org/

G.E. Farin, D. Hansford.

Practical Linear Algebra: A Geometry Toolbox.

A K Peters/CRC Press, 3rd edition, 2013; ISBN 978-1-4665-7956-9.

M.E. Mortenson.

Mathematics for Computer Graphics Applications.

Industrial Press, 2nd rev. edition, 1999; ISBN 978-0831131111.

A. Dickenstein, I.Z. Emiris (eds.).

Solving Polynomial Equations: Foundations, Algorithms, and Applications.

Springer, 2005; ISBN 978-3-540-27357-8.

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Table of Content

1 Introduction

2 Mathematics for Geometric Modeling

3 Bézier Curves and Surfaces

4 B-Spline Curves and Surfaces

5 Approximation and Interpolation

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1 Introduction Motivation Notation

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Motivation: Evaluation of a Polynomial

Assume that we have an intuitive understanding of polynomials and consider a polynomial inxof degreenwith coefficientsa0,a1, . . . ,an∈R, withan6=0:

p(x) :=

n

X

i=0

aixⁱ =a0+a1x+a2x²+. . .+an−1xⁿ⁻¹+anxⁿ.

A straightforward polynomial evaluation ofpfor a given parameterx0— i.e., the computation ofp(x0)— results inkmultiplications for a monomial of degreek, plus a total ofnadditions.

Hence, we would get

0+1+2+. . .+n=n(n+1)

2 =O(n²) multiplications (andnadditions).

Can we do better?

Yes, we can: Horner’s Algorithm consumes onlynmultiplications andnadditions to evaluate a polynomial of degreen!

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p(x) :=

n

X

i=0

Hence, we would get

0+1+2+. . .+n=n(n+1)

Can we do better?

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p(x) :=

n

X

i=0

Hence, we would get

0+1+2+. . .+n=n(n+1)

Can we do better?

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p(x) :=

n

X

i=0

Hence, we would get

0+1+2+. . .+n=n(n+1)

Can we do better?

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p(x) :=

n

X

i=0

Hence, we would get

0+1+2+. . .+n=n(n+1)

Can we do better?

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Motivation: Smoothness of a Curve

What is a characteristic difference between the three curves shown below?

Answer: The green curve has tangential discontinuities at the vertices, the blue curve consists of straight-line segments and circular arcs and is

tangent-continuous, while the red curve is a cubic B-spline and is curvature-continuous.

By the way, when precisely is a geometric object a “curve”?

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Motivation: Tangent to a Curve

What is a parametrization of the tangent line at a pointγ(t0)of a curveγ?

γ γ(t₀)

Answer: Ifγis differentiable then a parametrization of the tangent line`that passes throughγ(t0)is given by

`(λ) =γ(t0) +λγ⁰(t0) withλ∈R. How can we obtainγ⁰(t)forγ:R→R^d?

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γ γ(t₀)

`(λ) =γ(t0) +λγ⁰(t0) withλ∈R.

How can we obtainγ⁰(t)forγ:R→R^d?

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γ γ(t₀)

`(λ) =γ(t0) +λγ⁰(t0) withλ∈R. How can we obtainγ⁰(t)forγ:R→R^d?

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Motivation: Bézier Curve

How can we model a “smooth” polynomial curve inR²by specifying so-called

“control points”. (E.g., the pointsp0,p1, . . . ,p10in the figure.)

p0

p₁ p₂

p3

p4

p5

p6

p7

p8

p9

p10

One (widely used) option is to generate aBézier curve. (The figure shows a Bézier curve of degree 10 with 11 control points.)

What is the degree of a Bézier curve? Which geometric and mathematical properties do Bézier curves exhibit?

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p0

p₁ p₂

p3

p4

p5

p6

p7

p8

p9

p10

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p0

p₁ p₂

p3

p4

p5

p6

p7

p8

p9

p10

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Motivation: B-Spline Curve

How can we model a (piecewise) polynomial curve inR²by specifying so-called

“control points” such that a modification of one control point affects only a “small”

portion of the curve?

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Answer: Use B-spline curves.

Which geometric and mathematical properties do B-spline curves exhibit?

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Motivation: NURBS

Is it possible to parameterize a circular arc by means of a polynomial term? Or by a rational term?

Yes, this is possible by means of a rational term: 1−t²

1+t², 2t 1+t²

fort∈R.

More generally, NURBS can be used to model all types of conics by means of rational parametrizations.

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Yes, this is possible by means of a rational term:

1−t² 1+t², 2t

1+t²

fort∈R.

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Yes, this is possible by means of a rational term:

1−t² 1+t², 2t

1+t²

fort∈R.

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Motivation: Approximation of a Continuous Function How can we approximate a continuous function by a polynomial?

Answer: We can use a Bernstein approximation.

SampleBernstein approximationsof acontinuous function: f: [0,1]→R f(x) := sin (πx) +1

5sin

6πx+πx²

0.2 0.4 0.6 0.8 1.0

One can prove that the Bernstein approximationBn,f converges uniformly tof on the interval[0,1]asnincreases, for every continuous functionf.

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SampleBernstein approximationsof acontinuous function:

f: [0,1]→R f(x) := sin (πx) +1 5sin

6πx+πx²

0.2 0.4 0.6 0.8 1.0

One can prove that the Bernstein approximationBn,f converges uniformly tof on the interval[0,1]asnincreases, for every continuous functionf.

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SampleBernstein approximationsof acontinuous function:

f: [0,1]→R f(x) := sin (πx) +1 5sin

6πx+πx²

0.2 0.4 0.6 0.8 1.0

One can prove that the Bernstein approximationB converges uniformly tof on

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Notation: Numbers and Sets Numbers:

The set{1,2,3, . . .}of natural numbers is denoted byN, withN⁰:=N∪ {0}.

The set{2,3,5,7,11,13, . . .} ⊂Nof prime numbers is denoted byP. The (positive and negative) integers are denoted byZ.

Zⁿ:={0,1,2, . . . ,n−1}andZ⁺n :={1,2, . . . ,n−1}forn∈N.

The reals are denoted byR; the non-negative reals are denoted byR⁺0, and the positive reals byR⁺.

Open or closed intervalsI⊂Rare denoted using square brackets: e.g., I1= [a1,b1]orI2= [a2,b2[, witha1,a2,b1,b2∈R, where the right-hand “[” indicates that the valueb2is not included inI2.

The set of all elementsa∈Awith propertyP(a), for some setAand some predicateP, is denoted by

{x∈A: P(x)} or {x: x∈A∧ P(x)} or

{x∈A|P(x)} or {x|x∈A∧ P(x)}.

Quantifiers: The universal quantifier is denoted by∀, and∃denotes the existential quantifier.

Bold capital letters, such asM, are used for matrices. The set of all (real)m×nmatrices is denoted byMm×n.

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Zⁿ:={0,1,2, . . . ,n−1}andZ⁺n :={1,2, . . . ,n−1}forn∈N.

Open or closed intervalsI⊂Rare denoted using square brackets: e.g., I1= [a1,b1]orI2= [a2,b2[, witha1,a2,b1,b2∈R, where the right-hand “[”

indicates that the valueb2is not included inI2.

{x∈A: P(x)} or {x: x∈A∧ P(x)} or

{x∈A|P(x)} or {x|x∈A∧ P(x)}.

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Zⁿ:={0,1,2, . . . ,n−1}andZ⁺n :={1,2, . . . ,n−1}forn∈N.

{x∈A: P(x)} or {x: x∈A∧ P(x)}

or

{x∈A|P(x)} or {x|x∈A ∧ P(x)}.

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Zⁿ:={0,1,2, . . . ,n−1}andZ⁺n :={1,2, . . . ,n−1}forn∈N.

{x∈A: P(x)} or {x: x∈A∧ P(x)}

or

{x∈A|P(x)} or {x|x∈A ∧ P(x)}.

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Zⁿ:={0,1,2, . . . ,n−1}andZ⁺n :={1,2, . . . ,n−1}forn∈N.

{x∈A: P(x)} or {x: x∈A∧ P(x)}

or

{x∈A|P(x)} or {x|x∈A ∧ P(x)}.

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Notation: Vectors

Points are denoted by letters written in italics:p,qor, occasionally,P,Q. We do not distinguish between a point and its position vector.

The coordinates of a vector are denoted by using indices (or numbers): e.g., v= (vx,vy)forv∈R², orv= (v1,v2, . . . ,vn)forv∈Rⁿ.

In order to statev∈Rⁿin vector form we will mix column and row vectors freely unless a specific form is required, such as for matrix multiplication.

The vector dot product of two vectorsv,w∈Rⁿis denoted byhv,wi. That is, hv,wi=Pn

i=1vi·wiforv,w∈Rⁿ.

The vector cross-product (inR³) is denoted by a cross:v×w. The length of a vectorvis denoted bykvk.

The straight-line segment between the pointspandqis denoted bypq. The supporting line of the pointspandqis denoted by`(p,q).

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The vector cross-product (inR³) is denoted by a cross:v×w.

The length of a vectorvis denoted bykvk.

The straight-line segment between the pointspandqis denoted bypq. The supporting line of the pointspandqis denoted by`(p,q).

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The vector cross-product (inR³) is denoted by a cross:v×w.

The length of a vectorvis denoted bykvk.

The straight-line segment between the pointspandqis denoted bypq.

The supporting line of the pointspandqis denoted by`(p,q).

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Notation: Sum and Product

Considerkreal numbersa1,a2, . . . ,ak ∈R, together with somem,n∈Nsuch that 1≤m,n≤k.

n

X

i=m

ai :=







0 if n<m

am if n=m

(Pn−1

i=mai) +an if n>m

n

Y

i=m

ai :=







1 if n<m

am if n=m

(Qn−1

i=mai)·an if n>m

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Notation: Sum and Product

Considerkreal numbersa1,a2, . . . ,ak ∈R, together with somem,n∈Nsuch that 1≤m,n≤k.

n

X

i=m

ai :=







0 if n<m

am if n=m

(Pn−1

i=mai) +an if n>m

n

Y

i=m

ai :=







1 if n<m

am if n=m

(Qn−1

i=mai)·an if n>m

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2 Mathematics for Geometric Modeling Factorial and Binomial Coefficient Polynomials

Elementary Differential Calculus

Elementary Differential Geometry of Curves Elementary Differential Geometry of Surfaces

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Factorial and Binomial Coefficient

Definition 1 (Factorial, Dt.: Fakultät, Faktorielle) Forn∈N⁰,

n! :=

1 if n≤1, n·(n−1)! if n>1.

Note that 0! =1 by definition!

Definition 2 (Binomial coefficient, Dt.: Binomialkoeffizient) Letn∈N0andk∈Z. Thebinomial coefficient ⁿ_k

ofnandkis defined as follows:

n k

! :=











0 if k<0,

n!

k!·(n−k)! if 0≤k≤n,

0 if k>n.

The binomial coefficient ⁿ_k

is pronounced as “nchoosek”; Dt.: “nüberk”.

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n! :=

1 if n≤1, n·(n−1)! if n>1.

n k

! :=











0 if k<0,

n!

k!·(n−k)! if 0≤k≤n,

0 if k>n.

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n! :=

1 if n≤1, n·(n−1)! if n>1.

n k

! :=











0 if k<0,

n!

k!·(n−k)! if 0≤k≤n,

0 if k>n.

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n! :=

1 if n≤1, n·(n−1)! if n>1.

n k

! :=











0 if k<0,

n!

k!·(n−k)! if 0≤k≤n,

0 if k>n.

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Lemma 3

Letn∈N0andk∈Z. n

0

!

= n

n

!

=1 n

1

!

= n

n−1

!

=n n

k

!

= n

n−k

!

Theorem 4 (Khayyam, Yang Hui, Tartaglia, Pascal) Forn∈Nandk∈Z,

n k

!

= n−1 k−1

!

+ n−1 k

! .

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Lemma 3

Letn∈N0andk∈Z. n

0

!

= n

n

!

=1 n

1

!

= n

n−1

!

=n n

k

!

= n

n−k

!

Theorem 4 (Khayyam, Yang Hui, Tartaglia, Pascal) Forn∈Nandk∈Z,

n k

!

= n−1 k−1

!

+ n−1 k

! .

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Theorem 5 (Binomial Theorem, Dt.: Binomischer Lehrsatz) For alln∈N0anda,b∈R,

(a+b)ⁿ= n 0

! aⁿ+ n

1

!

aⁿ⁻¹b+· · ·+ n n

! bⁿ

or, equivalently, (a+b)ⁿ=

n

X

i=0

n i

! aⁿ⁻ⁱbⁱ.

In particular, for alla,b∈R,

(a+b)²=a²+2ab+b² (a+b)³=a³+3a²b+3ab²+b³.

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Theorem 5 (Binomial Theorem, Dt.: Binomischer Lehrsatz) For alln∈N0anda,b∈R,

(a+b)ⁿ= n 0

! aⁿ+ n

1

!

aⁿ⁻¹b+· · ·+ n n

! bⁿ

or, equivalently, (a+b)ⁿ=

n

X

i=0

n i

! aⁿ⁻ⁱbⁱ.

In particular, for alla,b∈R,

(a+b)²=a²+2ab+b² (a+b)³=a³+3a²b+3ab²+b³.

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Definition Arithmetic Roots Evaluation

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Polynomials

Definition 6 (Monomial, Dt.: Monom)

A (real)monomialinmvariablesx1,x2, . . . ,xmis a product of a coefficientc∈Rand powers of the variablesxiwith exponentski∈N⁰:

c

m

Y

i=1

x_i^kⁱ=c·x₁^k¹·x₂^k²·. . .·xm^k^m.

Thedegree of the monomialis given byPm i=1ki. Definition 7 (Polynomial, Dt.: Polynom)

A (real)polynomialinmvariablesx1,x2, . . . ,xmis a finite sum of monomials in x1,x2, . . . ,xm.

A polynomial isunivariateifm=1,bivariateifm=2, andmultivariateotherwise. Definition 8 (Degree, Dt.: Grad)

Thedegree of a polynomialis the maximum degree of its monomials.

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Polynomials

c

m

Y

i=1

x_i^kⁱ=c·x₁^k¹·x₂^k²·. . .·xm^k^m.

Thedegree of the monomialis given byPm i=1ki.

Definition 7 (Polynomial, Dt.: Polynom)

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Polynomials

c

m

Y

i=1

x_i^kⁱ=c·x₁^k¹·x₂^k²·. . .·xm^k^m.

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Polynomials

c

m

Y

i=1

x_i^kⁱ=c·x₁^k¹·x₂^k²·. . .·xm^k^m.

A polynomial isunivariateifm=1,bivariateifm=2, andmultivariateotherwise.

Definition 8 (Degree, Dt.: Grad)

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Polynomials

c

m

Y

i=1

x_i^kⁱ=c·x₁^k¹·x₂^k²·. . .·xm^k^m.

A polynomial isunivariateifm=1,bivariateifm=2, andmultivariateotherwise.

Definition 8 (Degree, Dt.: Grad)

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Polynomials

Hence, a univariate polynomial overRwith variablexis a term of the form anxⁿ+an−1xⁿ⁻¹+· · ·+a1x+a0,

with coefficientsa0, . . . ,an∈Randan6=0.

It is a convention to drop all monomials whose coefficients are zero.

Univariate polynomials are usually written according to a decreasing order of exponents of their monomials.

In that case, the first term is theleading termwhich indicates the degree of the polynomial; its coefficient is theleading coefficient.

Univariate polynomials of degree

0 are called constant polynomials,

1 are called linear polynomials,

2 are called quadratic polynomials,

3 are called cubic polynomials,

4 are called quartic polynomials,

5 are called quintic polynomials.

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Polynomials

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Polynomials

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Polynomial Arithmetic

We define the addition of (univariate) polynomials based on the pairwise addition of corresponding coefficients:

n

X

i=0

aixⁱ

! +

n

X

i=0

bixⁱ

! :=

n

X

i=0

(ai+bi)xⁱ

The multiplication of polynomials is based on the multiplication withinR, distributivity, and the rules

ax=xa and x^mx^k=x^m+k for alla∈Randm,k∈N:

n

X

i=0

aixⁱ

!

·





m

X

j=0

bjx^j



:=

n

X

i=0 m

X

j=0

(aibj)x^i+j

Elementary properties of polynomials: One can prove easily that the addition, multiplication and composition of two polynomials as well as their derivative and antiderivative (indefinite integral) again yield a polynomial.

Same for multivariate polynomials.

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n

X

i=0

aixⁱ

! +

n

X

i=0

bixⁱ

! :=

n

X

i=0

(ai+bi)xⁱ

ax=xa and x^mx^k =x^m+k for alla∈Randm,k∈N:

n

X

i=0

aixⁱ

!

·





m

X

j=0

bjx^j



:=

n

X

i=0 m

X

j=0

(aibj)x^i+j

(69)

n

X

i=0

aixⁱ

! +

n

X

i=0

bixⁱ

! :=

n

X

i=0

(ai+bi)xⁱ

n

X

i=0

aixⁱ

!

·





m

X

j=0

bjx^j



:=

n

X

i=0 m

X

j=0

(aibj)x^i+j

c

(70)

n

X

i=0

aixⁱ

! +

n

X

i=0

bixⁱ

! :=

n

X

i=0

(ai+bi)xⁱ

n

X

i=0

aixⁱ

!

·





m

X

j=0

bjx^j



:=

n

X

i=0 m

X

j=0

(aibj)x^i+j

(71)

Instead ofRany commutative ring(R,+,·)and symbolsx,y, . . .that are not contained inRwould do. E.g.,

a2,3x²y³+a1,1xy+a0,1y+a0,0 witha2,3,a1,1,a0,1,a0,0∈R.

Lemma 9

The set of all polynomials with coefficients in the commutative ring(R,+,·)and a symbol (variable)x6∈Rforms a commutative ring, thering of polynomials over R, commonly denoted byR[x].

Multivariate polynomials can also be seen as univariate polynomials with coefficients out of a ring of polynomials. E.g.,

a2,3x²y³+a1,1xy+a0,1y+a0,0= (a2,3x²)y³+ (a1,1x+a0,1)y+a0,0

is an element ofR[x,y] := (R[x])[y]. Definition 10

Two polynomials are equal if and only if the sequences of their coefficients (arranged in some specific order) are equal.

c