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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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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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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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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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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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?

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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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 approximationB_n,f converges uniformly tof on the interval[0,1]asnincreases, for every continuous functionf.

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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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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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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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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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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

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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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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Polynomial Arithmetic

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.

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Polynomials: Vector Space

Theorem 11

The univariate polynomials ofR[x]form an infinitevector spaceoverR. The so-called power basisof this vector space is given by the monomials 1,x,x²,x³, . . ..

The monomials 1,x,x²,x³, . . . ,xⁿform a basis of the vector space of polynomials of degree up tonoverR, for alln∈N0.

Recall: The fact that the monomials 1,x,x²,x³, . . . ,xⁿform a basis of the polynomials of degree up tonoverRmeans that

1 every polynomialp∈R[x]of degree at mostncan be expressed as a linear combination of those monomials: there exista0,a1, . . . ,an∈Rsuch that

p=anxⁿ+an−1xⁿ⁻¹+· · ·+a1x+a0,

2 none of those monomials can be expressed as a linear combination of the other monomials, i.e., the monomials are linearly independent.

The power basis is not the only meaningful basis of the polynomialsR[x]. See, e.g., the Bernstein polynomials that are used to form Bézier curves.

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Polynomials: Roots

Definition 12 (Polynomial equation)

Apolynomial equation(akaalgebraic equation) is an equation in which a polynomial is set equal to another polynomial.

Definition 13 (Root, Dt.: Wurzel)

The polynomialp∈R[x]has aroot(aka zero)r ∈Rif(x−r)dividesp.

Hence, ifr is a root ofpthenp= (x−r)·p1for somep1∈R[x].

Definition 14 (Multiplicity, Dt.: Vielfachheit)

A rootr of a polynomialp∈R[x]is of multiplicitykifk∈Nis the maximum integer such that(x−r)^k dividesp.

Theorem 15 (Fundamental Theorem of Algebra)

The number of complex roots of a polynomial with real coefficients may not exceed its degree. It equals the degree if all roots are counted with their multiplicities.

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Recall the quadratic formula taught in secondary school for solving second-degree polynomial equations: Fora∈R\ {0}andb,c∈R,

x1,2:=−b±√

b²−4ac 2a

yields the two (possibly complex) rootsx1andx2ofax²+bx+c.

Similar (albeit more complex) formulas exist for cubic and quartic polynomials.

Theorem 16 (Abel-Ruffini (1824))

No algebraic solution for the roots of an arbitrary polynomial of degree five or higher exists.

An algebraic solution is a closed-form expressions in terms of the coefficients of the polynomial that relies only on addition, subtraction, multiplication, division, raising to integer powers, and computingk-th roots (square roots, cube roots, and other integer roots).

A closed-form expression is an expression that can be evaluated in a finite number of operations.

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Lemma 17

Fora,b,c∈R, the rootsr1,r2of the quadratic polynomialax²+bx+csatisfy r1+r2=−b

a r1·r2=c a.

Lemma 18

Fora,b,c,d∈R, the rootsr1,r2,r3of the cubic polynomialax³+bx²+cx+dsatisfy r1+r2+r3=−b

a r1·r2+r1·r3+r2·r3= c

a r1·r2·r3=−d a. These two lemmas are special cases of a general theorem by François Viète (Franciscus Vieta, 1540–1603).

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Polynomials: Function

Definition 19 (Polynomial function; Dt.: Polynomfunktion)

A (univariate real) functionf:I→R, for an intervalI⊆R, is apolynomial function overIif there existn∈N⁰anda0,a1, . . . ,an∈Rsuch that

f(x) =anxⁿ+an−1xⁿ⁻¹+· · ·+a1x+a0 for allx∈I.

As usual, two (polynomial) functions over an intervalI⊆Rare identical if their values are identical for all arguments inI.

Note: Two different polynomials may result in the same polynomial function!

(E.g., over finite fields.)

While some fields of mathematics (e.g., abstract algebra) make a clear distinction between polynomials and polynomial functions, we will freely mix these two terms. Also, unless noted explicitly, we will only deal with polynomials overR. Note: Polynomial functions may come in disguise:f(x) := cos(2arccos(x))is a polynomial function over[−1,1], since we havef(x) =2x²−1 for allx∈[−1,1].

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Horner’s Algorithm for Evaluation of a Polynomial

Consider a polynomialp∈R[x]of 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 parameterx0results ink multiplications for a monomial of degreek, plus a total ofnadditions.

Hence, we would get

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

Can we do better?

Obviously, we can reduce the number of multiplications toO(nlogn)by resorting to exponentiation by squaring:

xⁿ=

(x(x²)ⁿ⁻¹² ifnis odd, (x²)ⁿ² ifnis even.

Can we do even better?

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Horner’s Algorithm: The idea is to rewrite the polynomial such that p(x) =a0+x

a1+x a2+. . .+x(an−2+x(an−1+x an)). . . and compute the resulth0:=p(x0)as follows:

hn:=an

hi:=x0·hi+1+ai fori=0,1,2, ...,n−1

1 /** Evaluates a polynomial of degree n at point x 2 * @param p: array of n+1 coefficients

3 * @param n: the degree of the polynomial 4 * @param x: the point of evaluation 5 * @return the evaluation result

6 */

7 double evaluate(double *p,int n,double x)

8 {

9 double h = p[n];

11 for (inti = n - 1; i >= 0; --i) 12 h = x * h + p[i];

14 return h;

15 }

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Lemma 20

Horner’s Algorithm consumesnmultiplications andnadditions to evaluate a polynomial of degreen.

Caveat

Subtractive cancellation could occur at any time, and there is no easy way to determine a priori whether and for which data it will indeed occur.

Subtractive cancellation: Subtracting two nearly equal numbers (on a

conventional IEEE-754 floating-point arithmetic) may yield a result with few or no meaningful digits. Aka: catastrophic cancellation.

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Forward Differencing

If a polynomial has to be evaluated fork+1 evenly spaced arguments

x0,x1, . . . ,xk, withxi+1=xi+δfor 0≤i<k, thenforward differencingis faster than Horner’s Algorithm.

Consider a polynomial of degree one:

p(x) =a0+a1x

The difference in the function valuesp(xi)andp(xi+1)of two neighboring points xiandxi+1is

∆ :=p(xi+1)−p(xi) =p(xi+δ)−p(xi) =a0+a1(xi+δ)−(a0+a1xi) =a1δ.

Hence, to evaluate the polynomial for several arguments, we may start with the evaluation ofp(x0)and recursively compute

p(xi+1) =p(xi+δ) =p(xi) + ∆, with∆ :=a1δ.

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Forward Differencing

For a quadratic polynomialp(x) =a0+a1x+a2x²the difference of the function values of neighboring arguments is

∆1(xi) :=p(xi+1)−p(xi) =p(xi+δ)−p(xi)

=a0+a1(xi+δ) +a2(xi+δ)²−(a0+a1xi+a2x_i²)

=a1δ+a2δ²+2a2xiδ

As∆1(x)is itself a linear polynomial, it can also be evaluated using forward differencing:

∆2(x) := ∆1(x+δ)−∆1(x) =2a2δ²

This approach can be extended to polynomials of any degree.

One can conclude that for a polynomial of degreeneach successive evaluation requiresnadditions.

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Differentiation of Functions of One Variable

Definition 21 (Derivative, Dt.: Ableitung)

LetS⊆Rbe an open set. A (scalar-valued) functionf:S→Risdifferentiableat an interior pointx0∈Sif

lim

h→0

f(x0+h)−f(x0) h

exists, in which case the limit is called thederivativeoff atx0, denoted byf⁰(x0).

Definition 22

LetS⊆Rbe an open set. A (scalar-valued) functionf:S→Risdifferentiable on Sif it is differentiable at every point ofS.

Iffis differentiable onSandf⁰is continuous onSthenf iscontinuously differentiable on S. In this casef is said to be ofdifferentiability class C¹.

By taking one-sided limits one can also consider one-sided derivatives on the boundary of closed setsS.

By applying differentiation tof⁰, a second derivativef⁰⁰offcan be defined.

Inductively, we obtainf⁽ⁿ⁾by differentiatingf⁽ⁿ⁻¹⁾.

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Definition 23 (C^k, Dt.:k-mal stetig differenzierbar)

LetS⊆Rbe an open set. A functionf:S→Rthat hasksuccessive derivatives is calledk times differentiable. If, in addition, thek-th derivative is continuous, then the function is said to be ofdifferentiability class C^k.

If thek-th derivative off exists then the continuity off⁽⁰⁾,f⁽¹⁾, . . . ,f^(k−1)is implied.

Definition 24 (Smooth, Dt.: glatt)

LetS⊆Rbe an open set. A functionf:S→Ris calledsmoothif it has infinitely many derivatives, denoted by the classC^∞.

We haveC^∞⊂Cⁱ ⊂C^j, for alli,j∈N0ifi>j.

Notation:

f⁽⁰⁾(x) :=f(x)for convenience purposes.

f⁰(x)=f⁽¹⁾(x) = _dx^df(x) =_dx^df(x).

f⁰⁰(x)=f⁽²⁾(x) = _dx^d²₂f(x) =_dx^d²₂^f(x).

f⁰⁰⁰(x)=f⁽³⁾(x) = ^d³

dx³f(x) = ^d³^f

dx³(x).

f⁽ⁿ⁾(x)=_dx^dⁿnf(x) = ^d_dxⁿn^f(x).

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Definition 25

Forn∈Nconsidernfunctionsfi:S→R(with 1≤i≤n) and definef:S→Rⁿas

f(x) :=





 f1(x) f2(x) ... fn(x)





 .

Then the (vector-valued) functionf isdifferentiableat an interior pointx0∈Sif and only iffi is differentiable atx0, for alli∈ {1,2, . . . ,n}. The derivative offatx0is given by

f⁰(x0) :=





 f₁⁰(x0) f₂⁰(x0)

... fn⁰(x0)





 .

All other definitions related to differentiability carry over from scalar-valued functions to vector-valued functions of one variable in a natural way.

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Differentiation of Functions of Several Variables

Definition 26 (Partial derivative, Dt.: partielle Ableitung)

LetS⊆R^mbe an open set. Thepartial derivativeof a (vector-valued) function f:S→Rⁿat point(a1,a2, . . . ,am)∈Swith respect to thei-th coordinatexiis defined as

∂f

∂xi

(a1,a2, . . . ,am) := lim

h→0

f(a1,a2, . . . ,ai+h, . . . ,am)−f(a1,a2, . . . ,ai, . . . ,am)

h ,

if this limit exists.

Hence, for a partial derivative with respect toxiwe simply differentiatef with respect toxiaccording to the rules for ordinary differentiation, while regarding all other variables as constants.

That is, for the purpose of the partial derivative with respect toxiwe regardf as univariate function inxiand apply standard differentiation rules.

Some authors prefer to writefx instead of _∂x^∂f. We will mix notations as we find it convenient.

Note

A function ofmvariables may have all first-order partial derivatives at a point (a1, . . . ,am)but still need not be continuous at(a1, . . . ,am).

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Higher-order partial derivatives offare obtained by repeated computation of a partial derivative of a (higher-order) partial derivative off.

Does it matter in which sequence we compute higher-order partial derivatives?

Theorem 27

Suppose that ^∂f_∂x,_∂y^∂f,_∂y∂x^∂²^f exist on a neighborhood of(x0,y0), and that _∂y^∂²_∂x^f is continuous at(x0,y0). Then _∂x^∂²_∂y^f (x0,y0)exists, and we have

∂²f

∂x∂y(x0,y0) = ∂²f

∂y∂x(x0,y0).

Note the difference between the Leibniz notation and the subscript notation for higher-order mixed partial derivatives!

∂²f

∂x∂y(x,y) := ∂

∂x(∂f

∂y)(x,y) but fxy(x,y) := (fx)y(x,y) Also, note that Theorem27does not imply that it could not be simpler to compute, say, _∂y^∂²_∂x^f rather than_∂x∂y^∂²^f :

f(x,y) :=xe^2y+ q

e^ysin(ytan(logy)) +p

1+y²cos²y

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Definition 28 (Differentiable, Dt.: total differenzierbar)

LetS⊆R^mbe an open set. A functionf:S→Rⁿofmvariables isdifferentiableat a pointa:= (a1, . . . ,am)∈Sif there exists ann×mmatrixJsuch that

xlim→a

f(x)−f(a)−J(x−a) kx−ak =0.

Theorem 29

LetS⊆R^mbe an open set. If a functionf:S→Rⁿofmvariables is differentiable at a pointa∈Sthen the coefficientsaijof the matrixJof Def.28are given by

aij= ∂fi

∂xj

(a) fori∈ {1,2, . . . ,n}andj∈ {1,2, . . . ,m}.

The matrixJis calledJacobi matrixoffata.

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Theorem 30

If a functionf:S→Rⁿofmvariables is differentiable at a pointa∈Sthen it is continuous ata.

Theorem 31 If _∂x^∂f

1,_∂x^∂f

2, . . . ,_∂x^∂f

m exist for a functionf:S→Rⁿofmvariables on a neighborhood of a pointa∈Sand are continuous atathenfis differentiable ata.

Definition 32 (Continuously differentiable, Dt.: stetig differenzierbar)

We say that a functionf:S→Rⁿofmvariables iscontinuously differentiableon an open subsetSofR^mif_∂x^∂f

1,_∂x^∂f

2, . . . ,_∂x^∂f

m exist and are continuous onS.

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Curves

Intuitively, a curve inR²is generated by a continuous motion of a pencil on a sheet of paper.

A formal mathematical definition is not entirely straightforward, and the term

“curve” is associated with two closely related notions: kinematic and geometric.

In the kinematic setting, a (parameterized) curve is a function of one real variable.

In the geometric setting, a curve, also called an arc, is a 1-dimensional subset of space that is “similar” to a line (albeit it need not be straight).

Both notions are related:

The image of a parameterized curve describes an arc.

Conversely, an arc admits a parametrization.

Since the kinematic setting is easier to introduce, we resort to a kinematic definition of “curve”.

Note that fairly counter-intuitive curves exist: e.g., space-filling curves like the Sierpinski curve.

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Caveat: Sierpinski Curves

Sierpinski curves are a sequence of recursively defined continuous and closed curves inR².

Sierpinski curve of orders 1–3 :

Their limit curve,the Sierpinski curve, is a space-filling curve: It fills the unit square completely! It is a continuous and surjective (but not injective!) mapping of[0,1]onto[0,1]×[0,1].

The Euclidean lengthLnof the order-nSierpinski curve is Ln= 2

3(1+√

2)2ⁿ−1 3(2−√

2)1 2ⁿ.

Hence, its length grows exponentially and unboundedly asngrows.

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Curves inRⁿ

Definition 33 (Curve, Dt.: Kurve)

LetI⊆Rbe an interval of the real line. A continuous (vector-valued) mapping γ:I→Rⁿis called aparametrizationofγ(I)or aparametric curve.

Well-known examples of parameterized curves include a straight-line segment, a circular arc, and a helix.

E.g.,γ: [0,1]→R³with γ(t) :=





px+t·(qx−px) py+t·(qy−py) pz+t·(qz−pz)





maps[0,1]to a straight-line segment from pointptoq.

The intervalIis called thedomainofγ, andγ(I)is calledimage(Dt.: Bild, Spur).

Definition 34 (Plane curve, Dt.: ebene Kurve)

Forγ:I→Rⁿ, the curveγ(I)isplaneifγ(I)⊆R²or ifγ(I)lies within an

affine/projective plane. A non-plane curve is called askew curve(Dt.: Raumkurve).

Analgebraic plane curveis the zero set of a polynomial in two variables.

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Curves inRⁿ

Definition 35 (Start and end point)

IfIis a closed interval[a,b], for somea,b∈R, then we callγ(a)thestart pointand γ(b)theend pointof the curveγ:I→Rⁿ.

Definition 36 (Closed, Dt.: geschlossen)

A parametrizationγ:I→Rⁿis said to beclosed(or aloop) ifIis a closed interval [a,b], for somea,b∈R, andγ(a) =γ(b).

Definition 37 (Simple, Dt.: einfach)

A parametrizationγ:I→Rⁿis said to besimpleifγ(t1) =γ(t2)fort16=t2∈Iimplies {t1,t2}={a,b}andI= [a,b], for somea,b∈R.

Hence, ifγ:I→Rⁿis simple then it is injective onint(I).

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Curves inRⁿ

Many properties of curves can also be stated independently of a specific parametrization. E.g., we can regard a curveCto be simple if there exists one parametrization ofCthat is simple.

In daily math, the standard meaning of a “curve” is the image of the equivalence class of all paths under a certain equivalence relation. (Roughly, two paths are equivalent if they are identical up to re-parametrization.)

Hence, the distinction between a curve and (one of) its parametrizations is often blurred.

For the sake of simplicity, we will not distinguish between a curveCand one of its parametrizationsγif the meaning is clear.

Similarly, we will frequently callγa curve.

For instance, we will frequently speak about a closed curve rather than about a closed parametrization of a curve.

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Jordan Curve inR²

Definition 38 (Jordan curve)

A setC ⊂R²(which is not a single point) is called aJordan curveif there exists a simple and closed parametrizationγ:I→R²that parameterizesC.

Theorem 39 (Jordan 1887)

Every Jordan curveCpartitionsR²\ Cinto two disjoint open regions, a (bounded)

“interior” region and an (unbounded) “exterior” region, withCas the (topological) boundary of both of them.

Although this theorem — the so-called Jordan Curve Theorem (Dt.: Jordanscher Kurvensatz) — seems obvious, a proof is not entirely trivial.

Theorem 40 (Schönflies 1906)

For every Jordan curveCthere exists a homeomorphism from the plane to itself that mapsCto the unit sphereS¹.

Roughly, a homeomorphism is a bijective continuous stretching and bending of one space into another space such that the inverse function also is continuous.

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Differentiable Curves

Definition 41 (C^r-parametrization)

Ifγ:I→Rⁿisr times continuously differentiable thenγis called a parametric curve of classC^r, or aC^r-parametrizationofγ(I), or simply aC^r-curve.

IfI= [a,b], thenγis called aclosed C^r-parametrizationifγ^(k⁾(a) =γ^(k⁾(b)for all 0≤k≤r.

One-sided differentiability is meant at the endpoints ofIifIis a closed interval.

Definition 42 (Smooth curve, Dt.: glatte Kurve)

Ifγ:I→Rⁿhas derivatives of all orders thenγis (the parametrization of) asmooth curve(or of classC^∞).

Definition 43 (Piecewise smooth curve, Dt.: stückweise glatte Kurve) IfIis the union of a finite number of sub-intervals over each of whichγ:I→Rⁿis smooth thenγispiecewise smooth.

Note: Smoothness depends on the parametrization!

There do exist curves which are continuous everywhere but differentiable nowhere [Weierstrass 1872, Koch 1904].

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Differentiable Curves

Definition 44 (Regular, Dt.: regulär)

AC^r-curveγ:I→Rⁿis calledregular of order k, for some 0<k≤r, if the vectors {γ⁰(t), γ⁰⁰(t), . . . , γ^(k)(t)}are linearly independent for everyt∈I.

In particular,γis calledregularifγ⁰(t)6=0∈Rⁿfor everyt∈I.

Definition 45 (Singular, Dt.: singulär)

For aC¹-curveγ:I→Rⁿandt0∈I, the pointγ(t0)is called asingular pointofγif γ⁰(t0) =0.

Note: Regularity and singularity depend on the parametrization!

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Equivalence of Parametrizations inRⁿ

Note that parametrizations of a curve (regarded as a setC ⊂Rⁿ) need not be unique: Two different parametrizationsγ:I→Rⁿandβ:J→Rⁿmay exist such thatC=γ(I) =β(J).

γ(t):=

cos2πt sin2πt

x y

Figure:γ(t)fort∈[0,0.9]

β(t):=

cos2πt

−sin2πt

x y

Figure:β(t)fort∈[0,0.9]

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Equivalence of Parametrizations inRⁿ

Definition 46 (Reparametrization, Dt.: Umparameterisierung)

Letγ:I→Rⁿandβ:J→Rⁿboth beC^r-curves, for somer ∈N0. We considerγand βasequivalentif a functionφ:I→Jexists, such that

β(φ(t)) =γ(t) ∀t∈I,

and

1 φis continuous, strictly monotonously increasing and bijective,

2 bothφandφ⁻¹arer times continuously differentiable.

In this case the parametric curveβis called areparametrizationofγ.

γ

β φ

I J

γ(I) =β(J)

t

Caveat

There is no universally accepted definition of a reparametrization! Some authors drop the monotonicity or the differentiability ofφ, while others even requireφto be smooth.

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Arc Length

Definition 47 (Decomposition, Dt.: Unterteilung)

Considerγ:I→Rⁿ, withI:= [a,b]. Adecomposition,P, of the closed intervalIis a sequence ofm+1 numberst0,t1,t2, . . . ,tm, for somem∈N, such that

a=t0<t1<t2<· · ·<tm =b.

The lengthLP(γ)of the polygonal chain(γ(t0), γ(t1), γ(t2), . . . , γ(tm))that corresponds to the decompositiont0,t1,t2, . . . ,tmis given by

LP(γ) :=

m−1

X

j=0

kγ(tj+1)−γ(tj)k

=kγ(t1)−γ(t0)k+kγ(t2)−γ(t1)k+· · ·+kγ(tm)−γ(tm−1)k.

We denote the set of all decompositions of[a,b]byP[a,b].

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Arc Length

Definition 48 (Arc length, Dt.: Bogenlänge)

Considerγ:I→Rⁿ, withI:= [a,b]. Thearc lengthofγ(I)is given by sup{LP(γ) :P∈ P[a,b]},

i.e., by the supremum (over all decompositionst0,t1,t2, . . . ,tmofI) of the length of the polygonal chain defined byγ(t0), γ(t1), γ(t2), . . . , γ(tm).

Definition 49 (Rectifiable, Dt.: rektifizierbar)

If the arc length ofγ:I→Rⁿis a finite number thenγ(I)is calledrectifiable.

Lemma 50

The arc length of a curve does not change for equivalent parametrizations.

Sketch of Proof : Suppose thatγ(t) =β(φ(t))for allt∈I, forβ:J→Rⁿ. Every decompositiont0,t1,t2, . . . ,tm ofImaps to a decompositionφ(t0),φ(t1),φ(t2), . . . , φ(tm) ofJsuch thatγ(ti) =β(φ(ti))for all 1≤i≤m. Hence, there is a bijection from the set of decompositions ofIto the set of decompositions ofJ, and it does not matter which set is used for determining the supremum of all possible chain lengths.

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Arc Length: Non-Rectifiable Curve

Curves exist that are non-rectifiable, i.e., for which there is no upper bound on the length of their polygonal approximations.

Example of a non-rectifiable curve: The graph of the function defined byf(0) :=0 andf(x) :=xsin ¹_x

for 0<x≤a, for somea∈R⁺. It defines a curve γ(t) :=

t

f(t)

.

0.5 1.0 1.5 2.0

-1.0 -0.5 0.5 1.0

The graph off(x) :=xsin ¹_x

forx∈[0,2]

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t

f(t)

.

0.1 0.2 0.3 0.4 0.5

-0.2 -0.1 0.1 0.2

The graph off(x) :=xsin _x¹

forx∈[0,¹₂]

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t

f(t)

.

0.005 0.010 0.015

-0.006 -0.004 -0.002 0.002 0.004 0.006

The graph off(x) :=xsin ¹_x

forx∈[0,₃₂¹]

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Example of a non-rectifiable closed curve: TheKoch snowflake[Koch 1904].

Koch snowflake, iteration 2

The length of the curve after then-th iteration is(⁴/3)ⁿtimes the original triangle perimeter. (Its fractal dimension is^log⁴/log3≈1.261.)

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Arc Length

Lemma 51

Ifγ:I→Rⁿis aC¹-curve thenγ(I)is rectifiable.

Theorem 52

Letγ:I→Rⁿbe aC¹curve, withI:= [a,b]. Then the arc length ofγ(I)is given by Z b

a

γ⁰(t) dt.

Corollary 53

Letγ:I→Rⁿbe aC¹curve, and[a,b]⊆I. Then the arc length ofγ([a,b])is given by Z b

a

γ⁰(t) dt.

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Arc Length: Unit Speed

Definition 54 (Speed, Dt.: Geschwindigkeit)

Ifγ:I→Rⁿis aC¹-curve then the vectorγ⁰(t)is thevelocity vectorat parametert, andkγ⁰(t)kgives thespeedat parametert, for allt∈I.

Definition 55 (Natural parametrization)

AC¹-curveγ:I→Rⁿis callednatural(or atunit speed) ifkγ⁰(t)k=1 for allt∈I.

Theorem 56

Ifγ:I→Rⁿ, withI:= [a,b], is a regular curve then there exists an equivalent reparametrizationγ˜that has unit speed.

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Tangent Vector

γ

γ(t₀) γ(t₁)

Ifγ(t0)is a fixed point on the curveγ, andγ(t1), witht1>t0, is another point, then the vector fromγ(t0)toγ(t1)approaches thetangent vectortoγatγ(t0)as the distance betweent1andt0is decreased.

The infinite line throughγ(t0)that is parallel to this vector is known as thetangent lineto the curveγat pointγ(t0).

If we disregard the orientation of the tangent vector then we would like to obtain the same result for the tangent line by considering a pointγ(t1)witht1<t0.

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Tangent Vector

Definition 57 (Tangent vector)

Letγ:I→Rⁿbe aC¹-curve. Ifγ⁰(t)6=0 fort∈Ithenγ⁰(t)forms thetangent vector at the pointγ(t)ofγ.

The tangent vector indicates the forward direction ofγrelative to increasing parameter values.

Ifγis at unit speed thenγ⁰(t)forms a unit vector.

A parametrization of the tangent line`that passes throughγ(t)is given by

`(λ) =γ(t) +λγ⁰(t) withλ∈R. If

γ(t) = x(t)

y(t)

is a curve inR²then the vector −y⁰(t)

x⁰(t)

is normal on the tangent line atγ(t).

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Frenet Frame for Curves inR³

Definition 58 (Frenet frame, Dt.: begleitendes Dreibein)

Letγ:I→R³be aC²curve that is regular of order two. Then theFrenet frame(aka moving trihedron) atγ(t)is defined as an orthonormal basis of vectors

T(t),N(t),B(t)as follows:

T(t) := _kγ^γ⁰0^(t)(t)k unit tangent;

N(t) :=_kT^T⁰0^(t)(t)k unit (principal) normal;

B(t) :=T(t)×N(t) unit binormal.

Lemma 59

Letγ:I→R³be aC²curve that is regular of order two, and defineT,N,Bas in Def.58. We get for allt∈I:

N(t)is normal toT(t), and B(t)is a unit vector.

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Osculating Plane for a Curve inR³

Letγ(t0)be a fixed point onγ, and two other pointsγ(t1)andγ(t2)that move alongγ.

Obviously,γ(t0),γ(t1)andγ(t2)define a plane.

As bothγ(t1)andγ(t2)approachγ(t0), the plane determined by them approaches a limiting position.

This limiting plane is known as theosculating planeto the curveγat pointγ(t0).

Definition 60 (Osculating plane, Dt.: Schmiegeebene)

Letγ:I→R³be aC²curve that is regular of order two. Theosculating planeat the pointγ(t)is the plane spanned byT(t)andN(t), as defined in Def.58.

Of course, the osculating plane contains the tangent line toγatγ(t0).

The osculating plane at the pointγ(t0)contains alsoγ⁰⁰(t0).

Thus, the binormal is the normal vector of the osculating plane.