Projections and tricks tools to test petrogenetic ipotheses Pietro Armienti (Università di Pisa)...
25
Projections and tricks tools to test petrogenetic ipotheses Pietro Armienti (Università di Pisa) based on the papers Do We Really Need Mantle Components to Define Mantle Composition? Armienti and Gasperini - J. Petrology 48: 693 and Three-dimensional representation of geochemical data from a multidimensional compositional space. Armienti and Longo - Int, Jour. of Geosciences. In press
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Transcript of Projections and tricks tools to test petrogenetic ipotheses Pietro Armienti (Università di Pisa)...
Projections and trickstools to test petrogenetic ipotheses
Pietro Armienti (Università di Pisa)
based on the papersDo We Really Need Mantle Components to Define Mantle Composition?
Armienti and Gasperini - J. Petrology 48: 693 and
Three-dimensional representation of geochemical data from a multidimensional compositional space.
Armienti and Longo - Int, Jour. of Geosciences. In press
This analysis of data is based on an algorithm
which projects data from a multi-dimensional space in
three-dimension
y
x
AB
E
E*
B A E* = B + 1 v
1 v1
v1 = A - B
The projection scheme from Rn to R3 may be envisaged by looking at the analogue for the projection from a plane onto a segment AB.
In a 2D space two points B, A
visualize the 1D space onto which to map E (x, y)
onto E* ().
In a similar way four points “materialize” the 3D space onto which to project data from Rn
EM I
HIMU
EM II
DMM
EMI-EMII
OIB-EMII
DMM-EM1
OIB-EM1
OIB-DMM
DMM-EMII
linea
Canary
Hawaii
St. Helena
SAmoa
Canary
Azzorre
Australi
EM I
HIMU
EM II
DMM
EMI-EMII
OIB-EMII
DMM-EM1
OIB-EM1
OIB-DMM
DMM-EMII
linea
St. Helena
SAmoa
Azzorre
Australi
For simplicity’s sake, the four points (“end-members”) are mapped onto the vertices of a regular tetrahedron
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitiveSp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitiveSp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitiveSp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive SpCpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive SpCpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive SpCpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive SpCpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitiveSp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
Sp
Cpx
OVP18 106
Ol
linea
ovp
BRP
BR
WR
Da FusioneAmphKaersutite
pargasite
Clinopirosseni
Ortopirosseni
Olivina
Spinello
EMI-EMII
OIB-DMM
DMM-EMII
OIB-EM1
OIB-EMII
DMM-EM1
Lave primitive
By choosing a suitable set of four end-
members, each defined by its components (e.g.
minerals and major elements), the
tetrahedron vertices attract the projection of the given analyses. The
method allows to project points with negative
values of the end-members to obey mass
balance constraints
(b)Di
(d)Q
(c)Ol
(a)S
Sp
Ol
Di
CaTs
active values C M A Sselect
projectiona CaTs 18,693 ,000 33,987 20,030 0
b Di 14,020 10,008 ,000 30,045 1
c OL ,000 26,687 ,000 20,030 0
d Q ,000 ,000 ,000 60,090 0y/n 1,0 1,0 1,0 1,0 none
(b)Di
(d)Q
(a)S
(c)Ol
Sp
CaTs
Di
Ol
… allowing also to project data onto a face of the tetrahedron from the opposite vertex.
The same scheme allows also to project relevant phase relations in petrologic diagrams
In systems with many components, compositions are often re-calculated in terms of end members, like in normative calculations that allow to recast the analysis of a rock in terms of fictive anhydrous minerals.
Ne
Ol
Di
En
Ab
SiO2 TiO2 Al2O3 Fe2O3 FeO MnO MgO CaO Na2O K2O p2O5
p.M. 60,0855 79,88 101,96 159,694 71,847 70,938 40,305 56,08 61,9795 94,1966 141,95
Analisi wt% 47,72 1,71 17,91 2,42 7,61 0,00 5,60 9,95 3,80 1,84 0,45
47,72 1,71 17,91 2,42 7,61 0,00 5,60 9,95 3,80 1,84 0,45
47,72 1,71 17,91 2,42 7,61 0,00 5,60 9,95 3,80 1,84 0,45
B
D
A
C
X
Y
Z
r
r
O
E F
"canonical" orientation of the tetrahedron whose centre is in the origin of the axes and the XA,YA,ZA coordinates of vertex A are (0,0,r) while the edge CB forms an angle f with the Y axis (It follows for the XB,YB, ZB coordinates of B
13) XB = r*sin ()*sin() 14) YB = r*sin ()*cos()15) ZB = r*cos()
being the angle AOB ;
B
D
A
C
X
Y
Z
r
r
O
E F
coordinates of vertices C e D are obtained substituting with (+120) and (+240) respectively. Varying causes the tetrahedron to rotate around Z.
Rotating around the X axis of an angle , the new coordinates
Xi', Yi', Zi' are related to the old ones by the equations :
16) Xi’ = Xi*cos()+z*cos()17) Yi’ = Yi18) Zi’ = -Xi*sin()+Zi*cos()
To plot a point P(Ap,Bp,Cp,Dp) within the tetrahedron and get P(Xp,Yp,Zp) we can start projecting E on the edge AB on the basis of the ratio Bp/Ap, calculating the coordinates of this point P'(Xp’,Yp’,Zp’);19a) Xp ’= (XA + Bp/Ap * XB)/(1+ Bp/Ap)19b) Yp’ = (YA + Bp/Ap * YB)/(1+ Bp/Ap)19c) Zp’ = (ZA + Bp/Ap * ZB)/(1+ Bp/Ap)
A
B C
D
P’
then P is projected onto the face ABC, as a function of the Cp/(Ap+Bp) and the coordinates of this point P”(X p”, Y p”, Z p”);
20a) Xp” = (Xp’+ (Cp/(Ap+Bp)* Xc)/(1+ Cp/(Ap+Bp)) 20b) Yp” = (Yp’+ (Cp/(Ap+Bp)* Yc)/(1+ Cp/(Ap+Bp))20c) Zp” = (Yp’+ (Cp/(Ap+Bp)* Zc)/(1+ Cp/(Ap+Bp))
A
B C
D
P’
P’’
At last, the coordinates of P within the tetrahedron are found on the segment P”D tracing on P”D a segment P”P as required by the lever rule : P”P/DP= Dp/(Ap+Bp+Cp). 21a) Xp = (Xp” +(Dp/(Ap+Bp+Cp).)*XD)/(1+(Dp/(Ap+Bp+Cp)) 21b) Yp = (Yp” +(Dp/(Ap+Bp+Cp).)*YD)/(1+(Dp/(Ap+Bp+Cp))21c) Zp = (Zp” +(Dp/(Ap+Bp+Cp).)*ZD)/(1+(Dp/(Ap+Bp+Cp))
A
B C
D
P’
P’’P
Let us assume that a system E, whose composition is described in terms of n components (e.g. major elements), can be referred to a set of four end members A,B,C,D and that each of them can be represented in terms of the same set of n components. E, A, B, C and D can be represented as points in n-dimensional space (Rn ).
E(E1, …En)
Considering A, B, C, D as the vertices of a tetrahedron in Rn, we need relations that are able to represent the point E(E1, …En) in n-dimensional space in terms of four end members : A(A1,...,An), B (B1,...,Bn), C (C1,...,Cn), D (D1,...,Dn).
The algorithm has to assign to E four coordinates (AE,BE,CE,DE) that allow its projection in a 3D tetrahedron that can be easily plotted
B
D
A
C
X
Y
Z
O
E
G
The algorithm is derived extending in Rn the rules for the projection of a point E(e1, …en) in a R3 tetrahedron: the procedure can be summarized as below:
1. Assign in Rn the coordinates of the tetrahedron vertices: A(a1,...,an),
B(b1,...,bn), C(c1,...,cn), D(d1,...,dn) B
D
A(a1,a2,a3,a4)
C
G
E(e1,e2,e3,e4)
B
D
A
C
X
Y
Z
O
E
G
+
2 To assign the coordinate AE find in Rn the intersection E' (E'1,...,E'n ) between the line through AE and the hyperplane through BCD,
3 Compute the lengths of segments AE', EE' and AE
E’
4. following the lever rule, assign to AE the value : 100-100*(AE’-EE’)/AE’= 100*EE'/AE'.
Repeat the steps from 2 to 4 to compute BE,CE,DE changing the vertex and the plane onto which to project E.
B
D
A
C
X
Y
Z
O
E
G
+
Computing tetrahedral coordinates in this way ensures that for all the points falling inside the tetrahedron AE + BE + CE + DE = 100,
B
D
A
C
X
Y
Z
r
r
O
E
FG
E
E
E’
This sum is different from 100 for points that in Rn lay outside the tetrahedron.
B
D
A
C
X
Y
Z
r
r
O
E
FG
E
E
E’
To allow the representation of these points in R3 we have to allow for negative tetrahedral coordinates: this is easily accomplished by comparison of lengths of segments AE, EE’ and AE’, it is easy to realize that:
•For points that are inside the tetrahedron AE’=AE+EE’.
•For points that are below the face BCD, EE’ may be smaller or larger than AE’, but AE<AE’ and EE’ is to be taken in equation 12 with a negative sign.
•For points that are above vertex A, EE’ > AE’ and EE’ has to be taken in equation 12 with a positive sign,
B
D
A
C
X
Y
Z
r
r
O
E
FG
E
E
E’
To obtain the 3D projection E* of E:
1) assign tetrahedron vertices (end members) A,B,C,D Rn , and compute vectors
v1 =A-D, v2 = B-D, v3 =C-D
Set and ;
w1= v1 and
Lastly, set and
2 α
||v||
vv
1
12 2
3 ||v||
vv
1
1
122 wvw α
22
23 ||w||
wv 233 wwvw 1
2) For each point to project
compute the vector E’=E- D
and its orthogonal projection E” on <w1,w2,w3 > ,
that is E” = 1w1+2w2+3w3
where i = , i = 1, 2 , 3.
E” is the linear combination of w1,w2,w3 nearest to E’ in Rn. Since the distance is translation-invariant, it follows that
E*=D+1v1+2v2+3v3 is the point in the 3D-space passing through A,B,C,D nearest to D + E’=E .
•
2||w||
wE
i
i'
3) Finally, remark that
E*= D +1w1+2w2+3w3
= D+1(A–D)+2[ B–D - (A–D)] +
+3{ C–D-(A–D) -[ B–D- (A–D)]}=
=(1-2-3+3 A (2-3 B 3 C+
+ (1-2-3+2+3+3 -3 D
The sum of the four coefficients of the end members at the right hand of the above formula, is 1; therefore, they represent the tetrahedral coordinates i of E* with respect to the four vertices A,B,C,D:
Tetrahedral coordinates1= 1-2-3+3
2= 2-3
3= 3
4 = 1-2-3+2+3+3 -3 1 - 2 -
3The point E* coincides with E if and only if E itself is a linear combination in Rn of the end members with sum of coefficients equal to one. Otherwise, E* is distinct from E, while enjoying the property to be at the minimal Euclidean distance from it, among all points of the (affine) subspace passing through the end members.
Let now (T1,T2,T3,T4) be the vectors in R3 to which the tetrahedron vertices A,B,C,D are mapped on ;
Finally, E will be mapped in R3 onto E* by the relation
E*= 1 T1+2 T2+3 T3+4 T4.
For a regular tetrahedron a possible choice is :
T1=(0,0,1),
T2=(2*2/3,0,-1/3);
T3=(-2/3, 2/3, -1/3);
T4=(-2/3, -2/3, -1/3).
