10. ROTATIONS (I) I Main Topics

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1 I Main Topics A Uses of rota=on in geolog (and engineering) B Concepts behind rota=on C Appendi (Geometr on a Sphere for Plate Tectonics) 10/5/15 GG303 1 II Uses of rota=on in geolog (and engineering) I A Representa=on of quan==es in an easier to understand form B Plate tectonics C Evalua=ng stresses and strains D Eamina=on of features in =lted bedding 1 Pre- =l=ng orienta=on of sedimentar structures (e.g. ripples) 2 Pre- =l=ng orienta=on of beds below angular unconformi=es 3 Pre- =l=ng orienta=on of paleomagne=c orienta=ons E To determine in- situ orienta=ons of features from drill cores F Need to consider whether object is rotated and coordinate aes are fied or whether the object is fied and coordinate aes are rotated. This affects the sign(s) and sequence of the angle(s) of rota=on. 10/5/15 GG

2 III Concepts behind rota=on A An orthogonal coordinate sstem with aes,, can be rotated to coincide with another orthogonal coordinate sstem,, b using the direc=on cosines rela=ng the aes of the two sstems. 10/5/15 GG303 3 A direc=on cosines (cont.) Dir. cosines ais ais ais ais a ' = a a ' = a a ' = a ais a ' = a a ' = a a ' = a ais a ' = a a ' = a a ' = a Note: a ' is not equal to a because θ ' is not equal to θ 10/5/15 GG

3 A An orthogonal coordinate sstem with aes,, can be rotated to coincide with another orthogonal coordinate sstem,, b using the direc=on cosines rela=ng the aes of the two sstems. = a a a a a a a a a and = a a a a a a a a a Note: a ' a because θ ' θ ;, and 10/5/15 GG303 5 B An orthogonal coordinate sstem can be rotated to coincide with another orthogonal coordinate sstem b three consecu=ve rota=ons (Cale): Right- hand rule: If right thumb is along rota=on ais, then fingers curl in posi=ve θ direc=on 10/5/15 GG

4 B three consecu=ve rota=ons (Cale) (cont.) 1 Rotate the,, sstem about the - ais b angle θ 1 = a a a a a a a a a = 0 cosθ 1 sinθ 1 0 sinθ 1 cosθ 1 2 Rotate the,,' sstem about the - ais b angle θ 2 = a a a a a a a a a = cosθ 2 0 sinθ sinθ 2 0 cosθ 2 3 Rotate the ","," sstem about the - ais b angle θ 3 = a a a a a a a a a cosθ 3 sinθ 3 0 = sinθ 3 cosθ /5/15 GG303 7 C An orthogonal coordinate sstem can be rotated to coincide with another orthogonal coordinate sstem b one rota=on about a speciall chosen ais (Euler's theorem). 10/5/15 GG

5 C rota=on about a speciall chosen ais (cont.) = a a a a a a a a a a a a a a a a a a a a a a a a a a a Third Rotation Second Rotation First Rotation a a a = a a a a a a Single Rotation 10/5/15 GG303 9 C rota=on about a speciall chosen ais (cont.) Direc=on cosines in terms of three rota=ons (θ 1, θ 2, θ 3 ) about aes,, and, respec=vel a = cosθ 3 cosθ 2 a = sinθ 3 cosθ 2 a = - sinθ 2 a = - sinθ 3 cosθ 1 3 sinθ 2 sinθ 1 a = sinθ 3 sinθ 1 3 sinθ 2 sinθ 1 a = cosθ 3 cosθ 1 + sinθ 3 sinθ 2 sinθ 1 a = - cosθ 3 sinθ 1 + sinθ 3 sinθ 2 cosθ 1 a = cosθ 2 sinθ 1 a = cosθ 2 cosθ 1 (b mul=pling matrices of slide 7) 10/5/15 GG

6 C rota=on about a speciall chosen ais (cont.) Direc=on cosines in terms of one rota=on angle (θ) about ais with direc=on cosines η 1, η 2, η 3 a = η 1 η 1 (1- cosθ) a = η 2 η 1 (1- cosθ) - η 3 sinθ a = η 3 η 1 (1- cosθ) + η 2 sinθ a = η 1 η 2 (1- cosθ) + η 3 sinθ a = η 2 η 2 (1- cosθ) a = η 3 η 2 (1- cosθ) - η 1 sinθ a = η 1 η 3 (1- cosθ) a = η 2 η 3 (1- cosθ) + η 1 sinθ a = η 3 η 3 (1- cosθ) 10/5/15 GG III Concepts behind rota=on (cont.) D Angles between lines Measure the acute angle between lines D Angles between ras, aes, or vectors ( directed half- lines ) Angles can be as great as 180 Angle between lines 1 and 2 Is 32 10/5/15 GG

7 IV Appendi (Geometr on a Sphere for Plate Tectonics) 10/5/15 GG