III. Vector data. First, create a unit circle which presents the margin of the stereonet. tan. sin. r=1. cos

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1 EDV in der Geologie, SS001 Vector dt Aims In session one ou will crete our own stereonet in Ecel. This cn plot poles to plnes nd linetions. Session two requires some bsic knowledge of liner lgebr, especill cross nd dot product. Using these techniques, ou will clculte the ngle between two different tpes of linetions. These re the mullion neck es mesured in the Devonin beds between Monschu nd Lmmersdorf, nd the d-linetion (being-clevge intersection linetion). Bsed on frequenc digrms ou now cn speculte bout the origin of mullion, nd their reltion to clevge (if n). Built of our own stereonet progrmme using Ecel You do lot of vector dt nlsis during our stud! All our field mesurements like being, clevge, fold es, or linetions re vectors. All of these dt cn be presented s lines, either poles or linetions. And becuse ou re n educted person, ou not onl mesure such dt, but interpret them in vrious ws. Finll, ou will come up with nice stor, how the structures evolved. All ou ppl is some simple trigometr. II I r1 III sin cos IV tn segment sin cos tn I II III IV Fig.1. Remember the definition of sine nd cosine. First, crete unit circle which presents the mrgin of the stereonet. Geologie-Endogene Dnmik, RWTH Achen 1

2 EDV in der Geologie, SS001 plot of unit circle: use series of ngles form 0-360, e.g. step size then plot, coordintes 0 cos 0 sin 0 cos sin etc. Now trnsform the geologicl dt to plot them in stereonet: polr cooridntes 0 R z 0 crtesin (rectngulr) cooridntes Fig.. Representtion of of point in spce in polr (stereogrphic projections) nd crtesin coordintes (to clculte with). Simple trigonemtric construction using figure llows the trnsformtion to crtesin coordintes (dip direction /, dip or plunge / ): 0cos( ) 0sin( ) z Rsin( ) with O Rcos( ): È ÈRcos( )cos( ) Rcos( )sin( ) Î z Î Rsin( ) Geologie-Endogene Dnmik, RWTH Achen

3 EDV in der Geologie, SS001 φ ' Fig.3. Cross-section cross sphere. Green lines re the projection lines. Lower horizontl touching the circle in one point is the projection of the Schmidt net (lower hemisphere equl re projection). The distnce towrds nd is: ( ) ( ) r sin 90 - f towrds ' r sin 90 - f towrds '' Now we cn clculte the crtesin coordintes in Ecel: '' 1 p sin sin - Ë cos sin 1 p - Ë Input dt re dip direction nd dip (poles to plnes) or dip direction nd plunge (linetions), herefter nmed nd, respectivel. Geologie-Endogene Dnmik, RWTH Achen 3

4 EDV in der Geologie, SS001 È cos cos È cos sin Î z sin Î Some liner lgebr: Cross product / vector product (Kreuzprodukt) A B Note tht the result of the cross product is vector! All ou need is two lines intersecting ech other. The cross product is the norml t the intersection of these two lines. Do ou know n geologicl ppliction of the cross product? Imgine fold with its limbs represented s poles to being. Using the cross product ou cn esil clculte the fold is from two poles to S 0. vector product A B n A B sinq A B re of prllelogrm Remember sme bsic mths - determinnts determinnt formul: b d -bc c d È1 Èb1 Èb b b Î Î b 3 3 Î b b - b - b Dot product / sclr product (Sklrprodukt) The dot product of two vectors is sclr. This is just number nd contins no other informtion, such s temperture. Remember tht vectors contin two informtion, which re the mgnitude nd the orienttion in spce. Grphicll the dot product mirrors the length of the one vector on the other. You cn lso Geologie-Endogene Dnmik, RWTH Achen 4

5 EDV in der Geologie, SS001 clculte the ngle between two vectors. Geologicl speking, ou cn clculte the ngle between two poles to being on either limb of fold, giving the interlimb ngle of the fold. In this eercise ou will clculte the ngle between two linetions, tht is the mullion neck es nd the delt-linetion. It is the bse of model developed for the formtion of mullions. You might remember tht mullions re generll ttributed to compressionl sstem. But how re the formed? Some bsic mths sclr product AB A B cosq Tking two vectors, the dot product cn be clculted s follows: È Î 1 3 È1 + + Î A A A Tke the outer circle of our Schmidt net s 1. Using the cos -1 ou now cn simpl clculte the ngle between the two vectors. Literture Hobbs, B.E., Mens, W.D., Willims,.F An outline of structurl geolog. Wile, 571pp.. esp. p Qude, H Die Lgekugelprojektion in der Tektonik. Ds Schmidtsche Netz und seine Anwendung. Clusthler Tektonische Hefte 0, 196pp.. Uri, J.L., Speth, G., vn der Zee, W., Hilgers, C Evolution of mullion (formerl boundin) structures in the Vriscn of the Ardennes nd Eifel. Journl of the Virtul Eplorer, 3, Geologie-Endogene Dnmik, RWTH Achen 5