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MODERN GEOMETRY

by: Cassidy Grimes

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MODERN GEOMETRY MATH 532

Cassidy Grimes

GPA 3.51

Ma Filaseta

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Ma Filaseta
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This 3 page Class Notes was uploaded by Cassidy Grimes on Monday October 26, 2015. The Class Notes belongs to MATH 532 at University of South Carolina - Columbia taught by Ma Filaseta in Fall. Since its upload, it has received 25 views. For similar materials see /class/229525/math-532-university-of-south-carolina-columbia in Mathematics (M) at University of South Carolina - Columbia.

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Date Created: 10/26/15
MATH 5327361 LECTURE NOTES 10 Notes on Translations and Rotations at We associate with each point Ly the column y which we will sometimes write as x y 1T A translatton of the Euclidean plane is a function f which maps each point z y to z a y b for some real numbers a and b To make matters more precise we shall refer to f as a translation by a b We may view such a translation as mapping z y 1T into z a y b 1T Since xa 1 0 a z yb 01 b y 1 0 01 1 we may therefore think of f as simply being multiplication by the matrix above We shall refer to the above matrix as T0117 If P represents the point a b we will sometimes write Tp Thus 1 0 a Tp 0 1 b 0 0 1 represents a translation of the Euclidean plane by P If P 0 0 observe that Tp maps each point to itself In this case we will call Tp the identity transformation Now consider a point A 1111 and a real number 15 A rotatton of the Euclidean plane about A by an angle t is a function f which maps each point B Ly to O z y where O is the same distance as B from A and where the angle measured counterclockwise from the vector B to the vector E is t It will be convenient to also nd a matrix representation of such a rotation Suppose for the moment that A 0 0 We can write B in polar coordinates as r 6 Then 0 has the polar coordinate representation r 6 15 Hence x r cos0 b 71030 cos 7 rsin6 sin acos 7 y sin and 1 rsin0 b 71030 sin rsin6 cos xsin y cos In matrix notation we may combine these as a 7 cos isin y sin COS y 39 In general with A 1111 we may obtain z y by translating the Euclidean plane rst by izl 711 and then performing the above rotation about the origin and then translating the Euclidean plane by 1111 Thus 1 zit 2222 132 1 ma 7 ysma no 7 coslt gtgt 111 ma z s1n y COS 7 961 81H y11 COS We may rewrite this as x 003 7 3in 1l 7 003 yl 3in z y 3in 003 7z1 3in y1l 7 003 y 1 0 0 1 1 Thus a rotation f can also be viewed in terms of matrix multiplication We call the above 3 gtlt 3 matrix R A With the above information we may now view a combination of translations and rotations in terms of matrix multiplication For example if we wish to translate the Euclidean plane by A 2 3 and then rotate about the point B 11 by 7r 6 and then translate by O 75 7 each point z y in the Euclidean plane will be moved to z y where z z y TORw6BTA y 1 1 This is a good place to do some examples and to make up some related homework Our main goal here is to establish and apply the following result Theorem Let Di and 6 be real numbers not necessarily distinct and let A and B be points not necessarily distinct If Oi 6 is not an integer multiple of 27139 then there is point C such that RagRa Range fa 6 is an integer multiple 0f27r then R BRaA is a translation Before demonstrating the theorem it would be a good idea to discuss the analogous result for a composition of 2 translations the rst by a b and the second by c d Geometrically it should be clear that the result of such a composition is a translation by a c b d Alternatively one can show by taking the product of matrices that TabTcd Tltacbd To see why the theorem holds write A zhyl and B 2112 Then 0035 Sir15 9621 COSWD 12 9115 R BRaA 3in6 0036 7z2 3in6 y2l 7 0036 0 0 1 003Oi 7 3ina 1l 7 003a yl 3ina gtlt 3ina 003Oi 7z1 3inOi y11 7 003a 0 0 1 003 3in a6 73ina6 u am mmm v 0 0 1 where u 1 0036l 7 003a yl 3ina 0036 1 3ina 3in6 7mamml7cmo mu7cmw mamm 1l 7 003Oi 6 yl 3in04 6 m7muwmmw7wmm and 1 1 sin l 7 cosa yl sina sin 7 1 cosa sin yl cos l 7 cosa 7 2 sin y2l 7 0036 7z1 sina B y1l 7 cosa 6 i 952 i 951 31115 12 7 1100 0035 Observe that if a B is an integer multiple of 27139 then the above matrix represents a translation by u 1 so that the second part of the theorem follows Suppose now that a B is not an integer multiple of 27139 We will have that there is a C such that R BRaA is a rotation at O by the angle a B if we can nd a pair 3113 such that 9631 7 00w 6 ya sina 6 7 962 7 9601 7 0086 12 7 in 31MB and 7 smlta B y3lt17 coslta 6 7 7ltz2 7 z1gtsinlt gtlty2 7 y1gtlt17 cow We have two equations in the 2 unknowns 3 and y3 There is a solution provided that l 7 cosa B 311104 5 detlt 7sinoz B 17 00304 6 7g 039 Observe that one does not need to use anything fancy here simply solve for 3 and y3 above and the equivalent of the determinant being non zero above follows We get that 0 exists provided that 272cosa 7amp0 Since we are now only considering a B which are not integer multiples of 27139 the theorem is established

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