[PDF] 18 Isometries Every isometry is an affine

View - Download 18 Isometries

Previous PDF

Next PDF

18 Isometries

Basic Properties

Recall that the distance between two pointsx;y2Rnis given by dist(x;y) =jjxyjj=p(xy)(xy): Denition.A functionF2Trans(Rn) is anisometryif everyx;y2Rnsatisfy dist(F(x);F(y)) = dist(x;y) In words, we say that a functionFis an isometry if it preserves distances. Isometries are also calledrigidtransformations and we view them as the natural family of transformations that preserve the \structure" of Euclidean space. Example:InR2(as we will prove) every rotationRx;and every mirrorMLis an isometry. Our goal is to develop an understanding of isometries. The next two lemmas take a couple of steps toward this goal.

Lemma 18.1.Every translation is an isometry

Proof.For the translationTzandx;y2Rn. We have

dist(Tz(x);Tz(y)) = dist(z+x;z+y) =jj(z+x)(z+y)jj=jjxyjj= dist(x;y)Lemma 18.2.The set of all isometries ofRnis a subgroup ofTrans(Rn)

Proof.We need to show identity containment and closure under multiplication and inverses. (identity)It is immediate from the denition that the identity is an isometry. (mult. closure)IfF;Gare isometries ofRnandx;y2Rnthen dist(FG(x);FG(y)) = dist(F(G(x));F(G(y))) = dist(G(x));G(y)) = dist(x;y): It follows thatFGis an isometry, thus establishing closure under multiplication. (inverses)LetFbe an isometry and letx;y2Rn. SinceFis an isometry dist(F1(x);F1(y)) = dist(F(F1(x));F(F1(y))) = dist(x;y) and it follows thatF1is also an isometry. This establishes closure under inverses. 2 Lemma 18.3.LetF2Trans(Rn)be an isometry. IfLis a line inRn, thenF(L)is a line. Proof.To prove this lemma, it suces to show that wheneverx;y;zlie on a line, then F(x);F(y);F(z) lie on a line. So assumex;y;zlie on a line withybetweenxandz. Then dist(x;z) = dist(x;y) + dist(y;z) )dist(F(x);F(z)) = dist(F(x);F(y)) + dist(F(y);F(z)) )F(x);F(y);F(z) lie on a line withybetweenxandz giving the desired conclusion.Symmetry Denition.Asymmetryof a setSRnis an isometryF2Trans(Rn) so thatF(S) =S. Proposition 18.4.For everySRn, the set of symmetries ofSis a subgroup ofTrans(S). Proof.LetGbe the set of symmetries ofS. We need to prove thatGcontains the identity, is closed under products, and closed under inverses. (identity)The identityIsatisesI(S) =SsoI2 G. (mult. closure)IfF;G2 GthenF(S) =SandG(S) =Swe nd thatFG(S) =F(G(S)) =

F(S) =Sand thusFG2 G.

(inverses)Finally, ifF2 GthenF(S) =SsoFmapsSbijectively toS. It follows that F

1also mapsSbijectively toS, soF12 G.Linearity

Denition.A functionF:Rn!Rmis calledlinearif it satises the following properties: (1) For everyx2Rnand everyt2Rwe haveF(tx) =tF(x). (2) For everyx;y2Rnwe haveF(x+y) =F(x) +F(y). Note:IfFis linear, then there exists anmnmatrixAso that the functionFis given by the ruleF(x) =Ax.

Denition.We say that a functionFxesxifF(x) =x.

3

Lemma 18.5.Every isometry that xes0is linear.

Proof.LetF2Trans(Rn) be an isometry that satisesF(0) =0. We will show thatF satises (1) and (2) in the denition of linear. To prove (1) letx2Rnand lett2R. Ifx=0thenF(tx) =F(0) =0=t0. So, we may assumex6=0. Lety=F(x) and observe that sinceFis an isometry xing0we must have jjxjj= dist(x;0) = dist(y;0) =jjyjj DeneLxto be the line Span(x) andLyto be the line Span(y) and note that Lemma 18.3 shows thatF(Lx) =Ly. Now,txis the unique point onLxthat has distancejjtxjjto0and distancejj(t1)xjjtox. Similarly,tyis the unique point onLythat has distancejjtyjjto

0and distancejj(t1)yjjtoy. It follows thatF(tx) =ty=tF(x) as desired.

To prove (2) letx;x02Rn. SinceFis an isometry, it must map the midpoint betweenx andx0to the midpoint between their imagesF(x) andF(x0), so F(12 x+12 x0) =12

F(x) +12

F(x0):

It follows from (1) that

12

F(x+x0) =F(12

x+12 x0):

Combining these equations givesF(x+x0) =F(x) +F(x0) as desired.Lemma 18.6.Every isometry is an ane transformation.

Proof.LetF2Trans(Rn) be an isometry and lety=F(0). Now we may dene the transformationG=TyFand we have

G(0) =TyF(0) =Ty(y) =0

It follows from Lemma 18.5 thatGis linear, so we may choose a matrixAso thatG(x) =Ax. NowF= (Ty)1G=TyGsoFis given by the ruleF(x) =Ax+y, soFis an ane transformation.quotesdbs_dbs1.pdfusesText_1

[PDF] isométrie du plan cours

[PDF] isométrie du plan exercices corrigés

[PDF] isométrie du plan exercices corrigés pdf

[PDF] isométrie vectorielle dimension 3

[PDF] isométries du plan exercices corrigés

[PDF] isométries vectorielles exercices corrigés

[PDF] isover laine de verre

[PDF] isover par 45 fiche technique

[PDF] isover par duo

[PDF] isp

[PDF] ispef licence 1

[PDF] ispits maroc

[PDF] ispits maroc 2017

[PDF] ispits marrakech

[PDF] ispits tetouan 2017