Agoston M.K. Computer Graphics and Geometric Modelling: Mathematics

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and

The theorem now follows from Lemma 2.2.4.3 applied to M

2.2.4.5. Corollary. Motions preserve angles.

See Figure 2.7. There is a converse to the results proved above.

2.2.4.6. Theorem. A map that preserves the length of vectors and the angles

between them also preserves distance, that is, it is a motion.

Proof. Exercise 2.2.4.1.

2.2.5 Some Existence and Uniqueness Results

Let P

, P

,...,P

and P

¢, P

¢,...,P

¢ , k ≥ 1, be two sequences of points in the plane.

We would like to determine when there is a motion M that sends P

to P

¢. Since

motions always preserve distances, a minimal requirement is that |P

| = |P

¢P

¢| for

all i and j. Is this enough though?

Case 1. k = 1.

There is no problem in this case. For example, the translation T(Q) = Q + P

would do the job. In fact, so would M = RT, where R is any rotation about P

¢. In other

words, there are an inﬁnite number of distinct motions that send P

to P

¢.

Case 2. k = 2.

Assume, without loss of generality, that P

π P

. Consider the translation T deﬁned

in Case 1 that sends P

to P

¢. By hypothesis, |P

¢T(P

)| = |P

¢P

¢|. Let a be the angle

between the vectors P

¢T(P

) and P

¢P

¢ and let R be the rotation about the point P

through the angle a. See Figure 2.8. It is easy to show that the motion M = RT does

what we want, as does the motion M¢ = SRT, where S is the reﬂection about the line

through P

¢ and P

¢. M and M¢ are clearly distinct.

Case 3. k = 3.

MM M MAC A C

() ()

2.2 Motions 79

M(C) M(B)

M(A)

Figure 2.7. Motions preserve angles.

Let M and M¢ be the motions deﬁned in Case 2 that send P

and P

to P

¢ and

¢, respectively. By hypothesis, |P

¢M(P

)| = |P

¢P

¢| for i = 1,2. The next lemma

shows that either M or M¢ does what we want, namely, either M(P

) = P

¢ or

M¢(P

) = P

¢.

2.2.5.1. Lemma. Let A, B, and C be three noncollinear points in the plane. The

only vectors X in the plane that satisfy the two equations |AX| = |AC| and |BX| = |BC|

are X = C and X = R(C), where R is the reﬂection about the line L determined by A

and B.

Proof. Assume that X π C.

Claim. The midpoint D = (C+X) of the segment [C,X] lies on the line L.

See Figure 2.9. Once the claim is proved we are done because the deﬁnition of D

implies that X = C + 2CD, which is where the reﬂection sends the point D. Consider

the following identities:

CD • AD C X C C X A

AX AC AC AX

AX AC

()

•

80 2 Afﬁne Geometry

T(P

)

Figure 2.8. Moving two points to another two

points.

X = R(C)

Figure 2.9. Proving Lemma 2.2.5.1.

Similarly, one can show that CD • BD = 0. It follows easily from this that the

vectors AD and BD are parallel and that D lies on L. This proves the claim and the

lemma.

Case 4. k > 3.

We claim that if the ﬁrst three points P

, P

, and P

are linearly independent, then

the map deﬁned in Case 3 that sends them to P

¢, P

¢, and P

¢, respectively, will already

send all the other points P

, i > 3, to P

¢. Figure 2.10 shows how the argument pro-

ceeds. In Figure 2.10(a) we show three circles with centers P

, P

, and P

and radius

, P

, and P

, respectively. The point P

lies on the intersection of these circles.

Figure 2.10(b) shows the corresponding circles around the image points. One has to

show that P

will get sent to the intersection of those circles and that this is the same

as the point P

¢.

We have just given a constructive proof of the following theorem.

2.2.5.2. Theorem. (The Existence Theorem for Motions) Given points P

, P

,...,

and P

¢, P

¢,...,P

¢ with the property that |P

| = |P

¢P

¢| for all i and j, then there

is motion M, so that M(P

) = P

¢.

Proof. See [Gans69] for missing details in the discussion above.

Now that we have answered the question of the existence of certain motions, let

us look at the issue of uniqueness more closely?

2.2.5.3. Theorem. A motion that has two distinct ﬁxed points ﬁxes every point on

the line determined by those points.

Proof. Let M be a motion and assume that M(A,B) = (A,B) for two distinct points

A and B. Let L be the line determined by A and B and let C be any point of L. If C

= A + tAB, then Lemma 2.2.2 implies that M(C) = M(A) + tM(A)M(B). In other words,

M(C) = C and the theorem is proved.

2.2.5.4. Theorem. Any motion of the plane that leaves ﬁxed three noncollinear

points must be the identity.

2.2 Motions 81

(a) (b)

Figure 2.10. Case 4 of the existence theorem.

82 2 Afﬁne Geometry

Proof. Let A, B, and C be three noncollinear points and let M be a motion with

M(A,B,C) = (A,B,C). Let P be any other point in the plane. We would like to show that

M(P) = P. By Theorem 2.2.5.3, M is the identity on the three lines determined by the

points A, B, and C. If P lies on these lines we are done; otherwise, Lemma 2.2.3 implies

that P lies on a line through two distinct points that lie on two of these lines. Using

Theorem 2.2.5.3 we can again conclude that M ﬁxes P.

2.2.5.5. Corollary. Two motions of the plane that agree on three noncollinear points

must be identical.

Proof. Let M and M¢ be motions and assume that M(A,B,C) = M¢(A,B,C) for three

noncollinear points A, B, and C. Consider the motion T = M

-1

M¢. Since T(A,B,C) =

(A,B,C), Theorem 2.2.5.4 implies that T is the identity, that is, M = M¢.

2.2.5.6. Corollary. Every motion of the plane is a composite of a translation, a rota-

tion, and/or possibly a reﬂection.

Proof. This follows from the construction in Case 3 above and Corollary 2.2.5.5.

Theorem 2.2.5.3 raises the question whether a motion of the plane that ﬁxes two

distinct points is actually the identity map. That is not the case. Reﬂections, such

as the map T(x,y) = (x,-y), can leave all the points of a line ﬁxed but still not be the

identity.

2.2.5.7. Theorem. A motion M of the plane that ﬁxes two distinct points A

and B is either the identity map or the reﬂection about the line L determined by A

and B.

Proof. By Theorem 2.2.5.3, M ﬁxes all the points on the line L. Let C be any point

not on L. Lemma 2.2.5.1 shows that C gets mapped by M either to itself or to its reﬂec-

tion C¢ about the line L. The theorem now follows from the Corollary 2.2.5.5 since we

know what M does on three points.

2.2.6 Rigid Motions in the Plane

2.2.6.1. Lemma. Every rotation R of the plane can be expressed in the form R =

= T

, where R

is a rotation about the origin and T

and T

are translations.

Conversely, if R

is any rotation about the origin through a nonzero angle and if T is

a translation of the plane, then both R

T and TR

are rotations.

Proof. Suppose that R = TR

-1

, where R

is a rotation about the origin and T is a

translation. By Theorem 2.2.4.2 we can move the translations to either side of R

which proves the ﬁrst part of the lemma. The other part can be proved by showing

that certain equations have unique solutions. For example, to show that TR

is a rota-

tion, one assumes that it is a rotation about some point (a,b) and tries to solve the

equations

xa yb y c-

()

-+cos sin +a = x cos sinqqqq

for a and b. The details are left as an exercise.

2.2.6.2. Theorem. The set of all translations and rotations of the plane is a sub-

group of the group of all motions. The set of rotations by itself is not a group.

Proof. To prove the theorem one uses Lemma 2.2.6.1 to show that the composites

of translations and rotations about an arbitrary point are again either a translation

or a rotation.

Deﬁnition. A motion of the plane that is a composition of translations and/or

rotations is called a rigid motion or displacement.

Rigid motions are closely related to orientation-preserving maps. We deﬁned that

concept in Section 1.6 for linear transformations and we would now like to extend

the deﬁnition to motions. Motions are not linear transformations, but by Theorem

2.2.4.2 they differ from one by a translation. Intuitively, we would like to say that a

motion M of the plane is “orientation preserving” if for every three noncollinear points

A, B, and C the ordered pairs of basis vectors (AB,AC) and (M(A)M(B),M(A)M(C))

determine the same orientation of R

. See Figure 2.11. This deﬁnition would be messy

to work with and so we take a different approach.

Let M be a motion in R

. By Theorem 2.2.4.2 we can write M uniquely in the form

M = TM

, where T is a translation and M

is a motion that ﬁxes the origin. Theorem

2.2.4.1 implies that M

is a linear transformation.

Deﬁnition. The motion M is said to be orientation preserving if M

is. Otherwise, M

is said to be orientation reversing.

2.2.6.3. Theorem.

(1) A motion M is orientation preserving if and only if M

-1

is orientation

preserving.

(2) The composition MM¢ of two motions M and M¢ is orientation preserving if and

only if either both are orientation preserving or both are orientation reversing.

(3) The composition M

...M

of motions M

is orientation preserving if and

only if the number of orientation-reversing motions M

is even.

Proof. The proof is left as an exercise. It makes heavy use of Theorem 2.2.4.2 to

switch translations from one side of a motion that ﬁxes the origin to the other.

x a in y b os in y os d-

()

++sc+b=xscqqqq

2.2 Motions 83

M(B)

M(C)

M(A)

Figure 2.11. An orientation-preserving motion.

2.2.6.4. Theorem.

(1) Translations and rotations of the plane are orientation-preserving motions.

(2) Reﬂections are orientation-reversing motions.

Proof. The fact that translations are orientation-preserving motions follows imme-

diately from the deﬁnition since the identity map is certainly orientation preserving.

To prove that rotations are orientation preserving, it sufﬁces to show, by Theorem

2.2.6.3, that any rotation R about the origin is orientation preserving since an arbi-

trary rotation is a composition of translations and a rotation about the origin. The

fact that such an R is orientation preserving follows from Theorems 1.6.6 and 2.2.2.1

and the fact that the matrix for the linear transformation R has determinant +1. This

proves (1).

To prove (2) note that the reﬂection S

about the x-axis is a linear transformation

with equation (2.12) that clearly has determinant -1 and hence is orientation revers-

ing. Next, Theorem 2.2.3.6 showed that an arbitrary reﬂection can be written in the

form T

-1

RT, where T is a translation and R is a rotation about the origin. Prop-

erty (2) now follows from (1) and Theorem 2.2.6.3.

We can also justify Theorem 2.2.6.4(2) geometrically based on the intuitive idea

mentioned earlier that a motion of the plane is orientation reversing if for some three

noncollinear points A, B, and C the ordered pairs of basis vectors (AB,AC) and

(M(A)M(B),M(A)M(C)) determine opposite orientations for R

. To see this we shall

use the same notation as in the deﬁnition of a reﬂection in Section 2.2.3. If P is a

point not on L, then clearly AQ and QP form a basis for R

and

The determinant of the matrix of coefﬁcients that relates the original basis to the

transformed one is -1. This means that the two bases are in opposite orientation

classes.

2.2.6.5. Theorem. A motion of the plane is orientation preserving if and only if it

is a rigid motion.

Proof. Exercise.

Although it takes three points to specify a general motion of the plane, two points

sufﬁce in the special case of rigid motions.

2.2.6.6. Theorem. If M is a rigid motion of the plane and if M ﬁxes two distinct

points, then M is the identity.

Proof. This theorem is an immediate consequence of Theorems 2.2.5.7 and 2.2.6.4.

2.2.6.7. Corollary. Two rigid motions of the plane that agree on two distinct points

must be identical.

TTQ P QP AQ QP

()()

=¢=◊+-

()

◊01.

TTA Q AQ AQ QP

()()

==◊+◊10

84 2 Afﬁne Geometry

2.2 Motions 85

Proof. The proof of this corollary is similar to the proof of Corollary 2.2.5.5.

2.2.6.8. Corollary. If two orientation-reversing motions of the plane agree on two

distinct points, then they must be identical.

Proof. If M and M¢ are the two orientation-reversing motions, then M¢ M

-1

is a rigid

motion that ﬁxes two distinct points and hence is the identity map. It follows that

M = M¢.

2.2.6.9. Theorem. A rigid motion of the plane that has a ﬁxed point p is a rotation

about p.

Proof. Exercise.

2.2.7 Summary for Motions in the Plane

We have deﬁned motions and have shown that a motion of the plane is completely

speciﬁed by what it does to three noncollinear points and that it can be described in

terms of three very simple motions, namely, translations, rotations, and reﬂections.

To understand such motions it sufﬁces to have a good understanding of these three

primitive types.

Planar motions are either orientation preserving or orientation reversing with

rigid motions being the orientation-preserving ones. Reﬂections are orientation

reversing. Another way to describe a planar motion is as a rigid motion or the com-

position of a rigid motion and a single reﬂection. In fact, we may assume that the

reﬂection, if it is needed, is just the reﬂection about the x-axis.

Combining various facts we know, it is now very easy to describe the equation of

an arbitrary motion of the plane.

2.2.7.1. Theorem. Every motion M of the plane is deﬁned by equations of the form

(2.19)

where a

+ b

= 1. Conversely, every such pair of equations deﬁnes a motion.

Proof. Let M(0) = (c,d) and deﬁne a translation T by T(P) = P + (c,d). Let M¢ =

-1

M. Then M = TM¢ and M¢ ﬁxes the origin.

Case 1. M is orientation preserving.

In this case M¢ is orientation preserving and must be a rotation about the origin

through some angle q (Theorem 2.2.6.9). Let a = cos q and b =-sin q. Clearly the equa-

tion for M has the desired form.

Case 2. M is orientation reversing.

¢=±- +

()

+ybxayd,

¢= + +xaxbyc

86 2 Afﬁne Geometry

Let S be the reﬂection about the x-axis, that is, S(x,y) = (x,-y). Since M¢ is orien-

tation reversing, it follows that the motion R = SM¢ is orientation preserving, but R

also ﬁxes the origin. Therefore, R must be a rotation about the origin through some

angle q. Note that SR = SSM¢ = M¢. Deﬁne a and b as in Case 1. It is again easy to see

that the equation for M = TM¢ = TSR has the desired form.

This proves the ﬁrst part of Theorem 2.2.7.1. The second part is Exercise 2.2.7.1.

See also the next example.

2.2.7.2. Example. Let us show directly, without using Theorem 2.2.7.1, that the

transformation M deﬁned by the equations

is a motion.

Solution. Deﬁne a translation T by T(x,y) = (x,y) + (5,7). Let R be the rotation about

the origin through the angle -p/6 and let S be the reﬂection about the x-axis. It is easy

to see that M = TSR and hence is a motion since it is a composite of motions.

Theorem 2.2.7.1 shows that motions can be represented by ﬁve real numbers (the

a, b, c, d, and ±1 depending on the sign). Rigid motions can be represented by four

real numbers. Chapter 20 in [AgoM05] describes a very compact way to represent

motions in terms of quaternions. The fact that a motion is deﬁned by ﬁve numbers

leads to another way to solve for a motion when it is given in terms of some points

and their images. One simply solves the equations in Theorem 2.2.7.1 for the unknown

coefﬁcients. Solving for ﬁve unknowns turns out to be not as complicated as it may

sound in this case.

Next, we would like to give a more complete geometric characterization of

motions than that given in Corollary 2.2.5.6.

2.2.7.3. Lemma. Every orientation-reversing motion M that ﬁxes the origin is a

reﬂection about a line through the origin.

Proof. Let p be a nonzero point. If M ﬁxes p, then Theorem 2.2.5.7 implies that M

is the reﬂection about the line through the origin and p and we are done. Assume

therefore that p¢ = M(p) π p. Let q be the midpoint of the segment [p,p¢] and let S be

the reﬂection about the line through the origin and q. Clearly, S and M agree on 0

and p. By Corollary 2.2.6.8 they are the same map.

Deﬁnition. A glide reﬂection is the composite of a reﬂection about a line L followed

by a translation with nonzero translation vector parallel to L. (See Figure 2.12.)

2.2.7.4. Theorem.

Every orientation-reversing motion M is a reﬂection or a glide

reﬂection.

Proof. If M ﬁxes the origin, then the theorem is true by Lemma 2.2.7.3. Assume

therefore that M(0) = p is distinct from the origin and let T be the translation that

¢= - +yxy

¢= + +xxy

sends the origin to p. Let M¢ = T

-1

M. Then M¢ is an orientation-reversing motion that

ﬁxes the origin and hence a reﬂection by Lemma 2.2.7.3. Since M = TM¢, M is a glide

reﬂection and we are done.

2.2.7.5. Theorem. Every motion M is either a translation, rotation, reﬂection, or

glide reﬂection.

Proof. If M is a rigid motion, then M is a translation or rotation by Theorem 2.2.6.2.

If M is not a rigid motion, that is, if it is orientation reversing, then M is a reﬂection

or glide reﬂection by Theorem 2.2.7.4.

One ﬁnal word about why the term “congruent transformation” is sometimes used

instead of “motion.” The reader may recall the notion of “congruent ﬁgures” from

his/her Euclidean geometry course in high school, which most likely was never given

a really precise deﬁnition. Well, we can do so now.

Deﬁnition. Two ﬁgures are said to be congruent if there is a motion that carries one

into the other.

2.2.8 Frames in the Plane

Before leaving the subject of motions in the plane we want to discuss another

approach to deﬁning them – one that will be especially powerful in higher dimen-

sions.

Deﬁnition. A frame in R

is a tuple F = (u

,p), where p is a point and u

and u

deﬁne an orthonormal basis of R

. If the ordered basis (u

) induces the standard

orientation, then we shall call the frame an oriented frame. The lines determined by

p and the direction vectors u

and u

are called the x-, respectively, y-axis of the frame

F. The point p is called the origin of the frame F. (e

,0) is called the standard frame

of R

To simplify the notation, we sometimes use (u

) to denote the frame (u

,0).

Frames can be thought of as deﬁning a new coordinate system. See Figure 2.13.

They can also be associated to a transformation in a natural way. If F = (u

,p) is a

frame and if u

= (u

) and p = (m,n), then deﬁne a map T

by the equations

(2.20)

¢= + +y uxuyn

12 22

¢= + +x uxuym

11 21

2.2 Motions 87

A¢

B¢

Figure 2.12. A glide reﬂection.

In matrix form, T

is the map

(2.21)

Claim 1. T

is a motion.

Proof. Since (u

) is an orthonormal basis, we have that

If follows easily from this that the equations (2.20) have the form of the equations in

Theorem 2.2.7.1, proving the claim.

If we think of a frame as deﬁning a new coordinate system, then we can coordi-

natize the points in the plane with respect to it.

Deﬁnition. The coordinates of a point with respect to a frame are called the frame

coordinates. The frame coordinates with respect to the standard frame are called world

coordinates.

Since T

maps the origin (0,0) to p, (1,0) to p + u

, and (0,1) to p + u

, we can

think of T

as mapping frame coordinates to world coordinates.

There is a converse to Claim 1. Let M be a motion deﬁned by the equations

Let u

= (a,c), u

= (b,d), and p = (m,n).

Claim 2. (u

) is an orthonormal basis and (u

,p) is a frame.

¢= + +ycxdyn.

¢= + +xaxbym

u u u u and u u u u

11 21 12 22

10+==+ + =.

Txy xy

,, .

()

88 2 Afﬁne Geometry

Figure 2.13. Frames in the plane.