Figure 17.1Shape defined in relation to the circular form.Shape factors of four non-circular shapes are shown schematically.(a) Circle = reference shape: isometric, fully convex, continuously curved;(b) square: isometric (a / b = 1.00), fully convex, angular;(c) ellipse: elongated (a / b > 1.00), fully convex, continuously curved;(d) lobate shape; isometric, convex-concave, continuously curved;(e) angular fragment: isometric, convex-concave, angular;shape factors: SF1 = perimeter ratio P / Pequ; SF2 = area ratio 4πA / P2;A = measured area of shape; P = measured length of outline; Pequ = perimeter of area equivalent circle.

circle = standardSF1 = P / Pequ = 1SF2 = 4πA / P2 = 1

SF1 = 1.12SF2 = 0.79

SF1 = 1.29SF2 = 0.60

SF1 = 1.30SF2 = 0.59

SF1 = 1.31SF2 = 0.59

a

c

e

b

d

Figure 17.2Defining long and short axes.See chapter 14 for PAROR method. Long and short axes derived from projections are shown for an ellipse (a), a rectangle (b) and a parallelogram (c); stippled ellipses are best-fit ellipses; (d) table of values; stippled rectangles are bounding boxes for different long and short axes.a = long axis; b = short axis of best-fit ellipse;xS = shortest projection, xL = longest projection;xP1 = projection at 90° to xL ('perpendicular to long'); xP2 = projection at 90° to xS ('perpendicular to short');α1 = orientation of xL; α2 = orientation of xP2.

xLxSxP1

xP2a

b

xS

xLa

b

xP1

xP2

xL

xS

xP1

xP2

α1α2

a

b(a) (b) (c)

a/b 2.00 2.00 2.00

xL/xS 2.00 2.25 2.44

xL/xP1 2.00 1.26 1.56

xP2/xS 2.00 2.00 2.25

a

c

b

d

Figure 17.3Shape factors of 4 test shapes.SF1 = P / Pequ; SF2 = 4π · A / P2 = 1 / (SF1)2, where P = measured perimeter, Pequ = perimeter of area equivalent circle, A = measured area;a / b = aspect ratio derived from best-fit ellipse;L / S = aspect ratio derived from longest and shortest projection;four rectangles and four ellipses with same shape factors as test shapes are shown for comparison.

# SF1 SF2 a / bshape

a / brectangle

a / bellipse

L / Sshape

L / Srectangle

L / Sellipse

1 1.48 0.46 1.5 4.6 4.9 1.6 4.7 4.9

2 1.48 0.46 1.7 4.6 4.9 1.6 4.7 4.9

3 1.41 0.50 1.6 4.0 4.4 1.6 4.1 4.4

4 1.18 0.72 1.4 1.9 2.7 1.6 2.2 2.7

1 2 3 4

1 2 3 4

1 2 3 4

Figure 17.4Shape factors as a function of elongation.(a) Shape factor SF1 = perimeter ratio P / Pequ is shown for ellipses (red curve) and rectangles (black curve) with increasing aspect ratio a / b; dots indicate the shape factors of shapes in Figure 17.3;(b) detail of plot (a).

a / b

SF1 = P / Pequ

a / b

SF1 = P / Pequ

#4

#1 #2#3

a

b

Figure 17.5Projection of convex and concave outlines.(a) Particle projection, B, of a fully convex shape; (compare PAROR, chapter 14);(b) surface projection, A, of the same shape as (a) (compare SURFOR, chapter 15);(c) particle projection, B, of convex-concave shape;(d) surface projection, A, of same shape as (c).

B A

B A

a

c

b

d

Figure 17.6Shape factors derived from the convex hull.(a) Convex-concave shape with area, A, and perimeter, P;(b) convex hull around shape (a) with area, AE, and perimeter, PE;(c) heavy black line = excess perimeter, ΔP = P - PE; dark green = defect area, ΔA = AE - A.

P

A

PE

AE

ΔA

ΔP

a

b c

radiu

sΔ

Figure 17.7Combining two shape factors.Plot of deltA (defect area) versus deltP (excess perimeter);radiusΔ (radius length) = √ (deltP2 + deltA2).

deltP(%)

deltA

(%)

Figure 17.8Comparison of shape descriptors for convex-concave forms.SF1 = P / Pequ; SF2 = 4π · A / P2 = 1 / (SF1)2;deltP = excess perimeter = (P-PE) / P;deltA = defect area = (AE-A) / A;PARIS = Percentile Average Relative Indented Surface = 2 · (P-PE) / PE · (P-PE) / PE;radiusΔ = √ (deltP2 + deltA2);P = measured perimeter, Pequ = perimeter of area equivalent circle, A = measured area; PE = perimeter of convex hull; AE = area of convex hull.

4 53 71 2 6

# SF1 SF2 PARIS (%) deltP (%) deltA (%) radiusΔ (%)

1 1.00 1.000 0 0 0 0.02 1.15 0.761 0 0 0 0.03 1.12 0.791 0 0 0 0.04 1.07 0.871 1.1 0.5 1.2 1.35 1.20 0.695 14.6 6.8 9.3 11.56 1.25 0.645 10.4 4.9 11 12.07 2.12 0.223 192.4 49 6.6 49.5

Figure 17.9Ranking of shapes by different shape descriptors.SF1: increasing deviation from circular form ('non-circularity');PARIS factor: increasing fraction of concave outline;deltP: increasing fraction of concave outline;deltA: increasing fraction of concave area;radiusΔ: increasing fraction of concave outline and area;light gray shapes indicate that circle, square and ellipse cannot be ranked, PARIS, deltP, deltA and radiusΔ are identical = 0%.

rank 1. 2. 3. 4. 5. 6. 7.

weakest strongestweakest strongestweakest strongestweakest strongestweakest strongestweakest strongestweakest strongest

SF1

PARIS

deltP

deltA

radiusΔ

4 53 71 2 6

4 5 71

6

4 5 72

6

4 5

37 6

4 5 76

Figure 17.10Vertex angles of polygon.Polygonal chain (black outline) is shown with a counterclockwise loop (arrows); vertex angles, αi, > 0° (blue) if rotation is in the sense of closing (convex corner), αi, < 0°(red) if rotation is in the sense of opening (concave corner); for any polygon ∑ αi = 360°.

85°

128°

-114°

98°

98°

65° αi

Figure 17.11Vertex angles of test shapes.Angles, αi, are plotted against relative length of outlines (0 ≤ Lr ≤ 1.00); blue = positive (convex), red = negative (concave).

1 2 3

4 5

6 7

1 2 3

4 5

6 7

α α α

relative length, Lr

relative length, Lrrelative length, Lr

vert

ex a

ngle

vert

ex a

ngle

vert

ex a

ngle

Figure 17.12Histograms of vertex angles.Frequencies are shown up to 100% for shapes 1 to 3, and up to 50% for shapes 4 to 7, positive values of vertex angles, αi, (convex corners) are highlighted in blue, negative vales (concave corners) are highlighted in red;Ω = sum of frequencies from -180° to 0°;Ω' = sum of frequencies from -180° to -10°;αext = value of maximum absolute vertex angle;Δα = αmax - αmin.

1 2 3

4 5

6 7

1 2 3

4 5

6 7

vertex angle

(%) (%) (%)

(%) (%)

(%) (%)

-180° 0° +180° -180° 0° +180° -180° 0° +180°

-180° 0° +180° -180° 0° +180°

Ω = 0% Ω' = 0%

Ω = 0% Ω' = 0%

Ω = 0% Ω' = 0%

Ω = 15% Ω' = 15%

Ω = 23% Ω' = 19%

Ω = 33% Ω' = 33%

Ω = 25% Ω' = 25%

αext = 174° αext = 100°

αext = 56° αext = 45°

αext = 24° αext = 10° αext = 90°

Δα = 327° Δα = 192°

Δα = 110° Δα = 78°

Δα = 20° Δα = 0° Δα = 0°

-180° 0° +180° -180° 0° +180°

-180° 0° +180° -180° 0° +180°vertex anglevertex angle

Figure 17.13Comparison of shape descriptors for angular forms.P = measured perimeter, Pequ = perimeter of area equivalent circle, A = measured area; PE = perimeter of convex hull; AE = area of convex hull;Ω = sum of frequencies form -180° to 0°;Ω' = sum of frequencies form -180° to -10°;αext = value of maximum absolute vertex angle;Δα = αmax - αmin;σα = standard deviation of vertex angles.

4 53 71 2 6

# Ω (%) Ω' (%) αext (°) Δα σα1 0 0 10 0 02 0 0 24 20 63 0 0 90 0 04 15 15 45 78 205 23 19 56 110 246 33 33 100 192 597 25 25 174 327 105

Figure 17.14Ranking of shapes by different descriptors for angularity.Ω = sum of frequencies from -180° to 0°;Ω' = sum of frequencies from -180° to -10°;αext = value of maximum absolute vertex angle.Δα = αmax - αmin;σα = standard deviation of vertex angles;light gray shapes indicate that shapes cannot be ranked: Ω and Ω' of circle, square and ellipse are identical = 0%; Δα and σα of the circle and the square are identical = 0°.

rank 1. 2. 3. 4. 5. 6. 7.

weakest strongestweakest strongestweakest strongestweakest strongestweakest strongestweakest strongestweakest strongest

Ω

Ω'

αext

Δα

σα

4 5 3 71 2 6

1 4 5 72

6

4 53 7 6

4 5 76

1 3

2

4 5 762

Software Box 17.1Dialog with program iSHAPES; answers are numbered and highlighted, see text for explanation.

----------------------------------------------------------- *** ishapes *** 2012-02-28, rh ----------------------------------------------------------- calculates shape factors and angles of polygons (open outlines are closed) deltP=(PE-P)/P (≠PARIS) deltA=(AE-A)/A prints results on screen and in files minimum number of points per particles = 3 maximum number of points per particles = 4000 number of particles is unlimited... ----------------------------------------------------------- input file: line 1: bti title (must have) line 2: n total number of points for each particle: x,y floating x-y coordinates | ... ...etc. | Xend,Yend end coordinates ----------------------------------------------------------name of input file:gouge.scmend coordinate of input file (0, 9999, ... one number): 9999want results on screen (0), in files (1), or both (2):2name of output file (angles) ? [gouge.scm.ang] (=default) > name of output file (shapes) ? [gouge.scm.shp] (=default) > name of output file (envelopes) ? [gouge.scm.elp](=default) >

1

2

3456

--------results:-------- # n(in) n(out) S/L L/p phiL paris deltP deltA

1 18 18 0.607 1.625 170. 1.05 0.53 2.45 2 13 13 0.776 1.167 140. 2.17 1.07 2.69 3 9 9 0.825 1.077 3. 0.00 0.00 0.00... 13 10 10 0.548 1.651 35. 1.27 0.64 2.63 14 8 8 0.833 1.074 54. 0.00 0.00 0.00

file with 14 grains ( 177 points) evaluated-------------------------------------------------------output files: - gouge.txt.scm.ang - gouge.txt.scm.shp - gouge.txt.scm.elp------------------------------------------------------- (screen output is not saved)

Figure 17.15Manual and automatic digitization of test shapes.Previously used test shapes are digitized and represented by polygonal chains:(a) manually digitized outlines (polygonal chains);(b) automatically digitized outlines, minimal distance between points = 20 pixels, smoothing error = 1 pixel;(c) automatically digitized outlines, minimal distance between points = 10 pixels, smoothing error = 0.5 pixel;circles mark locations of digitized vertices.

E

F

H

G

A

B

D

C

a b c d

Figure 17.16Shape descriptors of manually and automatically digitized shapes.Outlines are digitized as in Figure 17.15.a (black), 17.15.b (green), 17.15.c (red);L / S = aspect ratio, where L = longest projection, S = shortest projection;SF1 = P / Pequ = shape factor 1;PARIS = Percentile Average Relative Indented Surface;deltP = excess perimeter;deltA = defect area.

A B C D E F G H

A B C D E F G H

L / S

SF1

(= P/Pequ)

PARIS (%)

deltP (%)

deltA (%)

manual auto d=20auto d=10

E F HGA B DC

shapes:

mode of digitization:

-180° 0° +180°

perc

enta

ge (

%)

Ω = 14% Ω' = 14%αext = -169°

A

Ω = 15% Ω' = 15%αext = +45°

E

Ω = 14% Ω' = 14%αext = -116°

B

Ω = 23% Ω' = 19%αext = +56°

F

Ω = 15% Ω' = 11%αext = -61°

C

Ω = 33% Ω' = 33%αext = +100°

G

Ω = 0% Ω' = 0%αext = +91°

D

Ω = 25% Ω' = 25%αext = -174°

Hmanual

auto d = 20

Ω = 26% Ω' = 3%αext = -164°

A

Ω = 0% Ω' = 0%αext = +23°

E

Ω = 25% Ω' = 11%αext = ±73°

B

Ω = 29% Ω' = 23%αext = -67°

F

Ω = 13% Ω' = 10%αext = -59°

C

Ω = 33% Ω' = 17%αext = +58°

G

Ω = 12% Ω' = 0%αext = +45°

D

Ω = 28% Ω' = 16%αext = -171°

H

Ω = 42% Ω' = 3%αext = -129°

A

Ω = 10% Ω' = 0%αext = +21°

E

Ω = 44% Ω' = 5%αext = -89°

B

Ω = 23% Ω' = 18%αext = +40°

F

Ω = 16% Ω' = 7%αext = -50°

C

Ω = 41% Ω' = 8%αext = ±62°

G

Ω = 39% Ω' = 0%αext = +43°

D

Ω = 35% Ω' = 11%αext = -160°

H

auto d = 10

-180° 0° +180°

-180° 0° +180°

-180° 0° +180°-180° 0° +180°

-180° 0° +180°

-180° 0° +180°-180° 0° +180°

-180° 0° +180°

perc

enta

ge (

%)

perc

enta

ge (

%)

perc

enta

ge (

%)

perc

enta

ge (

%)

perc

enta

ge (

%)

perc

enta

ge (

%)

perc

enta

ge (

%)

perc

enta

ge (

%)

Figure 17.17Descriptors of angularity of manually and automatically digitized shapes.Histograms of vertex angles are shown for test shapes of Figure 17.15:(a) for manually digitized outlines;(b) for automatically digitized outlines (wide spacing of vertices);(c) for automatically digitized outlines (close spacing of vertices);Ω = sum of frequencies form -180° to 0°;Ω' = sum of frequencies form -180° to -10°;αext = value of maximum absolute vertex angle.

Figure 17.18Comparison of descriptors of angularity for manual and automatic digitization.Outlines are digitized as in Figure 17.15.a (black), 17.15.b (green), 17.15.c (red);Ω = sum of frequencies form -180° to 0°;Ω' = sum of frequencies form -180° to -10°;αext = extreme angle = value of maximum absolute vertex angle;Δα = range of angles = αmax - αmin;σα = standard deviation of vertex angles.

A B C D E F G H

A B C D E F G H

Ω

Ω'

A B C D E F G H

αext

A B C D E F G H

Δα

A B C D E F G H

σα

manual auto d=20auto d=10

E F HGA B DC

shapes:

mode of digitization:

Figure 17.19Shape analysis of fault rocks.SEM micrographs (BSE contrast) and digitized outlines(a) of fragments with angular shapes ('cracked');(b) of gouge with rounded shapes ('gouge');(c) plot of 'defect area', deltA, versus 'excess perimeter', deltP;(d) plot of PARIS factor versus aspect ratio, L/S;(e) list of average values for (a) and (b).

a/b PARIS (%)average

deltP(%)average

deltA (%)average

cracked 2.3 9.1 4.3 19.6

gouge 1.4 0.5 0.2 1.2

PARIS (%)

L/S

cracked gouge

incre

asing

matu

rity

deltA (%)

deltP (%)

cracked gouge

incre

asing

matu

rity

cracked

gouge15 µm

30 µm

a

b

c d

e

Figure 17.20Angularity of fault rocks.The fault rock microstructures shown in Figure 17.19.a and 17.19.b are analyzed.(a) Vertex angles, α, along relative length of outline of one particle of 'cracked' microstructure, (0 ≤ Lr ≤ 1.00), shape of particle is shown next to graph;(b) same as (a) for one 'gouge' particle;(c) same as (a) for all particles of 'cracked' microstructure;(d) same as (a) for all 'gouge' particles;(e) histogram of vertex angles of 'cracked';(f) histogram of vertex angles of 'gouge';Ω = fraction of histogram with α ≤ 0°; Ω' = fraction of histogram with α ≤ -10°.

cracked gouge

Lr

particle perimeter

α

Lr

Lr Lr

particle perimeter

particle perimeter particle perimeter

α

α αΔα

= 2

52°

Δα =

151

°

Ω = 9.7 %Ω' = 7.6 %

(%)

gouge

-90 0 +90 α(°)vertex angles (°)

Ω = 31.8 %Ω' = 26.8 %

(%)

cracked

-90 0 +90 α(°)vertex angles (°)

a

c

e

b

d

f

Figure 17.21Shape analysis of dynamically recrystallized marble.Samples are from Schmid et al. (1987).From left to right, micrographs (ultrathin section, crossed polarizers), digitized outlines and histograms of vertex angles are shown for:(a) CT1, deformed at 600°C;(b) CT2, deformed at 800°C to low strain (γ < 1);(c) CT7, deformed at 800°C to moderate strain (γ ~ 2.5).Ω' = fraction of histogram with α ≤ -10°.

CT1

CT2

CT7

Ω' = 13.0 %

Ω' = 16.1 %

Ω' = 18.4 %

-90 0 +90 α(°)

a

b

c

Figure 17.22Effect of grain boundary migration on grain shape.Evaluation of microstructures shown in Figure 17.21.(a) Plot of 'defect area', deltA, versus 'excess perimeter', deltP;(b) plot of PARIS factor versus aspect ratio, L/S;(c) list of average values for CT1, CT2, and CT7.

deltA (%) PARIS (%)

L/Saverage

PARIS (%)average

deltP(%)average

deltA (%)average

CT1 2.65 4.9 2.3 10.7

CT2 2.00 15.0 6.5 15.8

CT7 2.20 20.9 8.6 15.4

CT1CT2CT7

CT1CT2CT7

deltP (%) L/S

increasing GBM

increasing GBM

a b

c

Figure 17.23Effect of annealing on angularity.Four stages of annealing of dynamically recrystallized Octochloropropane ('see-through' experiment by Youngdo Park, Jin-Han Ree, images courtesy Wynn Means).(a) Optical micrographs, crossed polarizers;(b) digitized outlines;(c) histograms of vertex angles; Ω = fraction of histogram with α < 0°; Ω' = fraction of histogram with α ≤ -10°.

-90 0 +90 α(°)

Ω' = 16.9 %

Ω' = 24.6 %

Ω' = 8.5 %

Ω' = 2.4 % Ω = 31.4 %

Ω = 36.5 %

Ω = 41.2 %

Ω = 33.8 %

OCP10

OCP7

OCP4

OCP1

a b c

deltA (%) PARIS (%)

deltP (%) L/S

a b

c

Figure 17.24Effect of annealing on grain shape.Evaluation of microstructures shown in Figure 17.23.(a) Plot of 'defect area', deltA, versus 'excess perimeter', deltP;(b) plot of PARIS factor versus aspect ratio, L/S;(c) list of average values for OCP10 (dynamically recrystallized) to OCP1 (annealed).

L/Saverage

PARIS (%)average

deltP(%)average

deltA (%)average

OCP10 1.81 18.10 7.75 20.50

OCP7 1.55 10.20 4.63 16.15

OCP4 1.65 2.69 1.30 7.31

OCP1 1.55 0.66 0.33 2.57

OCP10OCP7OCP4OCP1

OCP10OCP7OCP4OCP1

anne

alingan

neali

ng