USING TABLET/SMART-PHONE SPREADSHEETS FOR SOLVING COMMON

STRUCTURAL GEOLOGY LAB/FIELD PROBLEMS BY CROSS-PRODUCT OF 3D

VECTORS

David T. AllisonDepartment of Earth SciencesUniversity of South Alabamadallison@southalabama.edu

http://www.usouthal.edu/geography/allison/research/VectorMethods.pptx

Presentation Outline

• Mathematical and Geometrical Basis of 3D Vector Manipulation

• Implementation of Spreadsheets with Examples

• Special Considerations for Spreadsheets Running on Tablet and/or Smart Phones

3D Coordinate system for orientation data

Orthogonal coordinate system using directional angles alpha, beta, and gamma.

Directional components of the (x, y, z) axes are equal to cos(α), cos(β), and cos(γ) respectively

+X (East)

+Y (North)

+Z

Projection of data vector upon Z axis (cos )

Data Vector=unit length

-X (West)

-Y (South)

Projection of data vector upon 3D axes = directionalcomponets (x,y,z)

-Z

Key mathematical concepts for manipulating 3D vectors

• Data Conversion: standard azimuth and plunge of a linear orientation can be converted to directional components (x,y,z) or directional angles (α,β,γ)

• Dot Product: calculates the angle between two non-parallel vectors

• 3D Vector addition: operates in the same fashion as 2D “head-to-tail” method but with the additional z component

• Cross Product: calculates the vector that is perpendicular to the plane containing 2 non-parallel vectors

• Rotation: the rotation of a 3D vector about a 3D rotation axis uses a combination of the above calculations

Converting Orientation Data to 3D Vectors

• Planar orientations must be converted to poles

• Azimuth and plunge of a linear orientation can be converted to directional components with below equations:• x = sin (azimuth) * sin (90-plunge)• y = cos (azimuth) * sin (90-plunge)• z = cos (90-plunge)

• Note that the directional angles , , and are related to the directional components by:• = cos-1 (x)• = cos-1 (y)• = cos-1 (z)

co s( ) co s( ) co s( ) co s( ) co s( ) co s( ) co s( ) 1 2 1 2 1 2

+X (East)

+Y (North)

+Z

Data vector 1

Data vector 2

Planecontainingdata vectors 1& 2

Dot Product of 2 Non-Parallel Vectors• For 2 non-parallel data vectors with directional angles

(α1,β1,γ1) and (α2,β2,γ2) respectively:

• Cross Product Method: given two non-parallel vectors calculates the orientation of the pole (perpendicular) to the plane that contains the two given vectors.

• Orientation data must be converted to directional components.

• The dot-product is used to calculate the angle θ between the given non-parallel vectors.

• The answer is calculated by 3 separate equations: one for each axis component.

• The magnitude of the cross-product vector is not important for orientation calculations, but is = (vector 1)(vector 2)(sin θ)

co s( )[co s( ) co s( ) co s( ) co s( )]

s in ( )

1 2 1 2

co s( )[co s( ) co s( ) co s( ) co s( )]

s in ( )

1 2 1 2

co s( )[co s( ) co s( ) co s( ) co s( )]

s in ( )

1 2 1 2

Cross Product

Geometry of the Cross Product Vector

Data vector 2

Data vector 1

Perpendicular vector fromcross product withmagnitude = (vector 1)(vector)(sin θ)

Plane defined by data vectors 1 and 2

Equal-Area Lower Hemisphere

start

R

-45-90

-135

-180

-225-270

-315

-180-360

Rotational path generated by a horizontal rotation axis

• Rotation of a vector (030, 0=“start”) about an axis (000, 0=“R”) through 360 degrees clockwise as viewed from the center of the net toward the trend of the rotation axis (R)

• Note: rotation angles are “mathematical” therefore clockwise angles are negative

Equal Area ProjectionN

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R1

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Horizontal versus non-horizontal rotational axes

• Rotation about a horizontal (plunge=0) axis generates a stereonet small circle path

• Rotation about a plunging axis generally creates an elliptical path that does not match either a small circle or great circle on the stereonet

θP

+S (x,y,z)

A(a,b,c) P

+S= original data attitude

+Q

V= rotated dataattitude

-Q

-S

r

Rotationalaxis

-S

Lowerhemispheresurface

Circle ofrotationplane

Circle ofrotation

X

YO

Datavector

Geometry of the Rotational 3D Vector Method

OP OA

OA

OS

OP is the rotational axis multiplied by the dot product of the rotation axis and data vector. This yields the vector with head at the center of the circle of rotation (OP).

PQ OA

OS

PQ is the vector perpendicular to the cross product of OA and OS. The magnitude of the cross product is equal to(OA)(OS)(sin θ) where θ is the angle between OA and OS. Since OA and OS are unity, PQ is exactly the magnitude to "touch" the circle of rotation. PS is then calculated by taking the cross product of PQ and OA.

PS PQ

OA

PX cos r( ) PS

PY sin r( ) PQ

PX is the projection of the rotated data vector (PV) upon the PS vector. The rotation amount is “r”

PY is the projection of the rotated data vector (PV) upon the PQ vector.

OV OP

PX

PY

By adding OP, PX, and PY "head-to-tail", the rotated data vector is calculated in terms of the orthogonal coordinatesystem defined above.

Method of 3D Vector Addition Utilized to Process Rotations

MathCAD © Worksheet link:http://www.usouthal.edu/geography/allison/GY403/RotationByComponents.mcdhttp://www.usouthal.edu/geography/allison/GY403/RotationByComponents.pdf

Programming example of rotational calculations

• Given a data vector (x1,y1,z1) and a rotation axis vector (x2,y2,z2) and a rotation angle r, the following equations calculate the new rotated orientation:

• tp = (x1*x2+y1*y2+z1*z2) * (1-cos(r))

• rot_x = cos(r)*x1+tp*x2+[sin(r)*(y2*z1-z2*y1)]

• rot_y = cos(r)*y1+tp*y2-[sin(r)*(x2*z1-z2*x1)]

• rot_z = cos(r)*z1+tp*z2+[sin(r)*(x2*y1-y2*x1)]

• Note that the rotated position may result in a negative z component that would plot in the upper hemisphere of a stereonet (i.e. a negative plunge). In that case the (x, y, z) components should be multiplied by -1 to “reflect” it back to the lower hemisphere projection.

Implementation of 3D Vector Analysis as Excel 2010 SpreadsheetsProblem Mathematical Method Spreadsheet ApplicationAngle between 2 Linear orientations Dot Product N/AOrientation of intersection of 2 non-parallel planes Cross Product IntersectingPlanes.xlsmOrientation of Plane containing to 2 non-parallel lines Cross Product CommonPlane.xlsmRotate a line around a rotational axis by a specified angle Cross Product, Dot Product Rotation.xlsm

• Quickoffice spreadsheets are simplified versions of Excel 2010

• Quickoffice runs on Android, iPad, iPhone OS

• Formatting:• Blue cells: data entered• Magenta cells: labels or formulae• Green cells: calculation results

Spreadsheet Implementation: Intersecting Planes

(IntersectingPlanes.xlsm)- 2 Fold LimbsProjection: Equal Area

Data I.D.: Ex 1A, Prob. 1

Strike Azimuth Dip Dip Quad. Pole Az. Pole Pl.Plane 1: 310.00 70.00 E 220.000 20.000Plane 2: 40.00 20.00 E 310.000 70.000

Pole 1 Pole 2Cos(alpha) Cos(beta) Cos(gamma) Cos(alpha) Cos(beta) Cos(gamma) Theta(rad.s) Theta(deg.s)

-0.604 -0.720 0.342 -0.262 0.220 0.940 1.244 71.253

IntersectionCos(alpha) Cos(beta) Cos(gamma)

3D -0.794 0.505 -0.339Lower hemi.: 0.794 -0.505 0.339

Intersection Azimuth Plunge

122.454 19.840

Fold HingeN

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122.5, 19.8

310, 70E040, 20E

• In this case the intersecting planes were 2 planar fold limbs, therefore, the intersection is the hinge orientation (122.5, 19.8)

Limb 2Limb 1

Hinge

NETPROG diagram

Application of Cross-Product and Dot-Product Example: yields attitude of fold hinge given the two limb attitudes

Data vector 2=Pole to limb 2

Data vector 1=Pole to limb 1

angle betweenpoles 1 & 2

ChevronFold hinge

Fold limb 2Fold limb 1

Plane containingpole 1 & 2

Intersecting Planes: Apparent Dip ExampleProjection: Equal Area

Data I.D.: App. Dip

Strike Azimuth Dip Dip Quad. Pole Az. Pole Pl.Plane 1: 50.00 40.00 E 320.000 50.000Plane 2: 290.00 90.00 E 200.000 0.000

Pole 1 Pole 2Cos(alpha) Cos(beta) Cos(gamma) Cos(alpha) Cos(beta) Cos(gamma) Theta(rad.s) Theta(deg.s)

-0.413 0.492 0.766 -0.342 -0.940 0.000 1.898 108.747

IntersectionCos(alpha) Cos(beta) Cos(gamma)

3D 0.760 -0.277 0.588Lower hemi.: 0.760 -0.277 0.588

Intersection Azimuth Plunge

110.000 36.005

Apparent Dip ExampleN

S

W E

110, 36

050, 40E

290, 90

App. Dip

Plane 1

Plane 2

Intersection

• Given strike & dip of 050, 40E (Plane 1), calculate apparent dip along vertical plane trending 110

• Apparent dip plane is equivalent to 290, 90 (Plane 2) strike & dip

Spreadsheet Implementation: Common Plane (CommonPlane.xlsm) to 2 Non-parallel Linear Data- Find Strike & Dip

from 2 Apparent DipsProjection: Equal AreaLinear 1 Linear 2

Data ID Azimuth plunge Azimuth plungeStrike & Dip Ex. 310.000 15.000 210.000 35.000

Linear 1 Linear 2Cos(alpha) Cos(beta) Cos(gamma) Cos(alpha) Cos(beta) Cos(gamma)

-0.740 0.621 0.259 -0.410 -0.709 0.574Theta ThetaAngle(rad.s) Angle(deg.s)

1.560 89.367Cross-product Cos(alpha) Cos(beta) Cos(gamma) Pole Pole3D 0.540 0.318 0.779 Azimuth PlungeLower hemisphere 0.540 0.318 0.779 59.462 51.194

Plane True Dip Vector True Dip Vector True Dip VectorStrike azimuth Dip Quadrant Azimuth Plunge Cos(gamma)

329.462 38.806 W 239.462 38.806 0.627

Strike & Dip ExampleN

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App. Dip 1

App. Dip 2

Strike & Dip: 329.5,38.8W

Pole to Plane

• Note that Cross-Product calculates pole to plane that contains the 2 apparent dip linear vectors

• The true dip trend is always 180 degrees from the pole trend, and the dip angle is always = 90 – pole plunge

Lineation vector 2

Lineationvector 1

anglebetween poles1 & 2

Cross-product vector oflineations 1 & 2 =perpendicular (pole) toplane containinglineations 1 & 2

Plane containinglineations 1 & 2

Application of Common Plane Spreadsheet to Strike & Dip

Calculation from 2 Apparent Dips

Rotational Problem Scenarios• Rotational fault• Retro-deforming a fold limb• Rotating cross-bedding to original

attitude

Spreadsheet Implementation: Rotation of a line about a rotational

axis (Rotation.xlsm) - Rotational Fault

Given a rotational fault axis (300,30) and that bedding (090,40S) was rotated 120 degrees calculate the new bedding attitude = 39.5, 72.3W

Rotation Path

N 39.5 E 72.3 WRotated Bedding

120

Projection Equal Area

Data set Az. data Pl. data Az. axis Pl. axisBedding 0.000 50.000 300.000 30.000

Data vector Axis vectorCos(alpha) Cos(beta) Cos(gamma) Cos(alpha) Cos(beta) Cos(gamma)

0.000 0.643 0.766 -0.750 0.433 0.500T_dot Axis/Data Axis/DataFactor Theta(rad.s) Theta(deg.s)

0.992 0.848 48.597Rotated

Cos(alpha) Cos(beta) Cos(gamma)3D -0.735 0.606 -0.305Lower hemisphere 0.735 -0.606 0.305Rotated Rotated

Azimuth Plunge129.489 17.729

Rotated planar Rotated planar Rotated planar dipStrike Dip quadrant

39.489 72.271 W

current incrementrotation angle 120.000 5.000

increment macro: <ctrl>+idecrement macro: <ctrl>+d

Rotational Fault ExampleN

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P

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48.6

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Fault Plane

Special Considerations for Tablet/Smart Phone Spreadsheets

• Spreadsheet layout should be compact for limited screen area

• Currently “named” cells are not supported• Graphics are generally not practical or are not

supported• VB macros are not supported• Downloadable spreadsheets have been

tested with Quickoffice on the Android OS

Compact Layout of “CommonPlane.xlsx” in Quickoffice

Excel “Named” Cell Constraints

• “Named cells” uses symbolic names to represent cell addresses to clarify formulae

• Named cells cannot be used with current Tablet/Smart Phone Excel compatible spreadsheets (example from “CommonPlane.xlsx”

• =SIN(RADIANS(Az_1))*SIN(RADIANS(90-Pl_1))

• =SIN(RADIANS(B4))*SIN(RADIANS(90-C4))

Excel Graphics and VB Macros• VB macros are not supported in current Tablet/Smart Phone

applications• Graphics are not practical with smart phones but may be possible

on tablets

Web Site Resources• Excel 2010 Spreadsheets with graphics and dynamic VB macros:

• http://www.usouthal.edu/geography/allison/GY403/CommonPlane.xlsm

• http://www.usouthal.edu/geography/allison/GY403/IntersectingPlanes.xlsm

• http://www.usouthal.edu/geography/allison/GY403/Rotation.xlsm

• Smart Phone/ Tablet compatible spreadsheet versions:• http://

www.usouthal.edu/geography/allison/GY403/CommonPlane.xlsx

• http://www.usouthal.edu/geography/allison/GY403/IntersectingPlanes.xlsx

• http://www.usouthal.edu/geography/allison/GY403/Rotation.xlsx

• NETPROG stereonet application:• http://www.usouthal.edu/geography/allison/w-netprg.htm

• QuickOffice web site:• http://www.quickoffice.com/

Concluding Scenario…