6. Face and hand proportions

M5 ¼ M13 þ M12 (19)

M12 ¼ M23 þ M22 (22)

M15 ¼ M27 þ M26 (25)

M14 ¼ M24 þ M25 (28)

<sup>φ</sup> (20)

<sup>φ</sup> <sup>þ</sup> <sup>1</sup> (21)

<sup>φ</sup> (23)

<sup>φ</sup> <sup>þ</sup> <sup>1</sup> (24)

<sup>φ</sup> (26)

<sup>φ</sup> <sup>þ</sup> <sup>1</sup> (27)

<sup>φ</sup> (29)

<sup>φ</sup> <sup>þ</sup> <sup>1</sup> (30)

<sup>M</sup><sup>12</sup> <sup>¼</sup> <sup>M</sup><sup>5</sup>

<sup>M</sup><sup>13</sup> <sup>¼</sup> <sup>M</sup><sup>5</sup>

The M12 is from the navel to rib cage, the upper limit of M12 is same level as tangent to of circle 4. Since M12 is the total of M23 and M22 according to (22). Hence, M22, and M23 can be

<sup>M</sup><sup>22</sup> <sup>¼</sup> <sup>M</sup><sup>12</sup>

<sup>M</sup><sup>23</sup> <sup>¼</sup> <sup>M</sup><sup>12</sup>

The M15 is from eyebrows to top of the head, the upper limit of M12 is same level as tangent to of circle 6. Since M12 is the total of M27 and M26 according to (25). Hence, M26, and M27 can

<sup>M</sup><sup>26</sup> <sup>¼</sup> <sup>M</sup><sup>15</sup>

<sup>M</sup><sup>27</sup> <sup>¼</sup> <sup>M</sup><sup>15</sup>

The M14 is from the throat to eyebrows. M14 is tangent to circles 3 and 5 while the upper limit is tangent to circle 6. Note that, M14 is the addition of M24 and M25, and is derived according

<sup>M</sup><sup>24</sup> <sup>¼</sup> <sup>M</sup><sup>14</sup>

<sup>M</sup><sup>25</sup> <sup>¼</sup> <sup>M</sup><sup>14</sup>

The M24 upper limit will be tangent to circle 10 and the lower limit is tangent to circle 5. M24 is tangent to circles 3 and 5 while the upper limit is tangent to circle 6. Note that, M24 is the

addition of M29 and M28, and is derived according (32) and (33):

driven according to (23) and (24):

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be driven according to (26) and (27):

(29) and (30):

The golden proportion is repeated in the human face and is used by plastic surgeons to follow as guidelines. Human face from the bottom of the chin to the hairline in length to the edge of the eyebrow is 1:1.618 or 1: φ as illustrated in Figure 12, the red rectangle. Likewise, proportion is illustrated in the same figure using the green rectangle; the green rectangle is edges of the eyebrows and the height is from the tip of the nose to the top of the eyebrows. The third golden proportion is seen in the blue rectangle. The blue rectangle extends from the eyebrow to the bottom lip, and from the space between the eyes to the end of the eyebrow. Based on the previous one can calculate the width of the face to equal 1: length of the face. The length of the face = M26 + M25 + M29 and the length of the green box equals M25 and the length of the blue rectangle is 2 � M25. Once the length is calculated, then the width can be easily calculated (Figure 8).

Figure 8. Face and hand golden proportions.

Gary Meisner [19] lists seven vertical golden proportions in the face and eight horizontal golden proportions. Next, the 15 proportions will be described based on Meisner's guide to beauty. The seven vertical golden proportions are: center of pupils to bottom of chin the golden ratio point is at center of lips, center of pupils to bottom of chin the golden ratio point is at nose at nostrils, center of pupils to nose base the golden ratio point is at nose flair top, top arc of eyebrows to bottom of eyes the golden ratio point is at top of eyes, center of pupils to center of lips the golden ratio point is at nose at nostrils, top of lips to bottom of lips the golden ratio point is at center of lips, nose at nostrils to center of lips the golden ratio point is at top of lips. Eight horizontal golden proportions are: side of face to opposite side of face, the golden ratio point is at inside of near eye; side of face to inside of opposite eye, the golden ratio point is at inside of near eye; center of face to side of face, the golden ratio point is at outside edge of eye; side of face to inside edge of eye, the golden ratio point is at outside edge of eye; side of face to outside edge of eye, the golden ratio point is at outside of eyebrow; center of face to width of mouth, the golden ratio point is at width of nose; side of mouth to opposite side of mouth, the golden ratio point is at cupid's bow. Meisner designated 33 points on human face as seen in [19], such points can be driven from the derived rules in Figure 9, namely the length of the face = M25 + M26 + M29 and the width of the face = length of face/1.618. Since the width and length of face are known then Meisner's points (1, 33, 19, 20) can be calculated: Meisner's points (1, 33) are the upper and lower limit of the face height, respectively. Meisner's points (19, 20) are the left and right limits of the width of the face respectively. Meisner's point (15) = width of face/2.618 and Meisner's point (16) = width of face/2.618 both according to H1 and H2 in Meisner's work [19]. Meisner's points (13, 14) can be calculated as follows: Meisner's point (13) = width of face/(2 � 2.618) same for Meisner's point (14) = width of face/ (2 � 2.618). Meisner's point (7) = Meisner's point (13)/1.618 and Meisner's point (8) = Meisner's point (14)/ 1.618). Meisner's point (29) = width of face / (2 � 1.618) and Meisner's point (30) = width of face/ (2 � 1.618). Meisner's Point (24) = half width of face to Meisner's point (30)/1.618 and Meisner's point (23) = half width of face to Meisner's point (30)/1.618. Meisner's points (27,28) are cupid bow hence to derive them divide the distance from Meisner's point (29–30) over golden ration, Meisner's point (29, 30) = distance Meisner's point (29, 30) � 1.618. According to the previous 12 golden ratio rules were derived to the face vertically. Next, the same will be conducted to derive more rules horizontally. Again, since the height of the face is known, one can derive seven golden ratio rules according to Meisner [19].

First Meisner's points (19, 20) heights: Meisner's point (19) height = length of face/ (1 + 1.618) and Meisner's point (20) height = face length/ (1 + 1.618). Meisner's points (21, 22) are calculated from bottom as follows: Meisner's point (21) = face length/1.618 and the same for Meisner's point (22) = face length/1.618. Meisner's points (3, 4) measuring from bottom are face length/1.618, hence Meisner's point (3) = face length/1.618 and m Meisner's point (4) = face length/1.618. Meisner's points (11, 12) the center of the pupils are calculated as follows: Meisner's point (12) = difference of height between Meisner's point (20) and Meisner's point (4)/1.618 from top, and Meisner's point (11) = difference of height between Meisner's point (19) and Meisner's point (3) /1.618 from top. Meisner's point (31) is calculated as follows Meisner's point (31) = from Meisner's point (11, 12) to Meisner's point (33)/ (1 + 1.618). Meisner's points (23, 24) are calculated as follows: Meisner's point (23) = from Meisner's point (11, 12) to

Meisner's point (33)/ (1 + 1.618) and Meisner's point (24) = from Meisner's point (11, 12) to Meisner's point (33)/ (1 + 1.618). Meisner's point (2) is calculated as follows: Meisner's point (2) = from Meisner's point (11, 12) to Meisner's point (33)/ (1 + 1.618), measuring from the top. To calculate Meisner's points (9, 10), the following must be done: Meisner's point (9) = difference from Meisner's point (3) to Meisner's point (19)/1.618 and Meisner's point (10) = difference from Meisner's point (4) to Meisner's point (20)/1.618. Meisner's point (17) = difference from Meisner's point (9) to Meisner's point (19)/1.618 and Meisner's point (18) = difference from Meisner's point (10) to Meisner's point (20)/1.618. Meisner's point (5) = difference from Meisner's point (3) to Meisner's point (7)/1.618 and Meisner's point (6) = difference from

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Figure 9. Human body measurements algorithm.

Gary Meisner [19] lists seven vertical golden proportions in the face and eight horizontal golden proportions. Next, the 15 proportions will be described based on Meisner's guide to beauty. The seven vertical golden proportions are: center of pupils to bottom of chin the golden ratio point is at center of lips, center of pupils to bottom of chin the golden ratio point is at nose at nostrils, center of pupils to nose base the golden ratio point is at nose flair top, top arc of eyebrows to bottom of eyes the golden ratio point is at top of eyes, center of pupils to center of lips the golden ratio point is at nose at nostrils, top of lips to bottom of lips the golden ratio point is at center of lips, nose at nostrils to center of lips the golden ratio point is at top of lips. Eight horizontal golden proportions are: side of face to opposite side of face, the golden ratio point is at inside of near eye; side of face to inside of opposite eye, the golden ratio point is at inside of near eye; center of face to side of face, the golden ratio point is at outside edge of eye; side of face to inside edge of eye, the golden ratio point is at outside edge of eye; side of face to outside edge of eye, the golden ratio point is at outside of eyebrow; center of face to width of mouth, the golden ratio point is at width of nose; side of mouth to opposite side of mouth, the golden ratio point is at cupid's bow. Meisner designated 33 points on human face as seen in [19], such points can be driven from the derived rules in Figure 9, namely the length of the face = M25 + M26 + M29 and the width of the face = length of face/1.618. Since the width and length of face are known then Meisner's points (1, 33, 19, 20) can be calculated: Meisner's points (1, 33) are the upper and lower limit of the face height, respectively. Meisner's points (19, 20) are the left and right limits of the width of the face respectively. Meisner's point (15) = width of face/2.618 and Meisner's point (16) = width of face/2.618 both according to H1 and H2 in Meisner's work [19]. Meisner's points (13, 14) can be calculated as follows: Meisner's point (13) = width of face/(2 � 2.618) same for Meisner's point (14) = width of face/ (2 � 2.618). Meisner's point (7) = Meisner's point (13)/1.618 and Meisner's point (8) = Meisner's point (14)/ 1.618). Meisner's point (29) = width of face / (2 � 1.618) and Meisner's point (30) = width of face/ (2 � 1.618). Meisner's Point (24) = half width of face to Meisner's point (30)/1.618 and Meisner's point (23) = half width of face to Meisner's point (30)/1.618. Meisner's points (27,28) are cupid bow hence to derive them divide the distance from Meisner's point (29–30) over golden ration, Meisner's point (29, 30) = distance Meisner's point (29, 30) � 1.618. According to the previous 12 golden ratio rules were derived to the face vertically. Next, the same will be conducted to derive more rules horizontally. Again, since the height of the face is known, one

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can derive seven golden ratio rules according to Meisner [19].

First Meisner's points (19, 20) heights: Meisner's point (19) height = length of face/ (1 + 1.618) and Meisner's point (20) height = face length/ (1 + 1.618). Meisner's points (21, 22) are calculated from bottom as follows: Meisner's point (21) = face length/1.618 and the same for Meisner's point (22) = face length/1.618. Meisner's points (3, 4) measuring from bottom are face length/1.618, hence Meisner's point (3) = face length/1.618 and m Meisner's point (4) = face length/1.618. Meisner's points (11, 12) the center of the pupils are calculated as follows: Meisner's point (12) = difference of height between Meisner's point (20) and Meisner's point (4)/1.618 from top, and Meisner's point (11) = difference of height between Meisner's point (19) and Meisner's point (3) /1.618 from top. Meisner's point (31) is calculated as follows Meisner's point (31) = from Meisner's point (11, 12) to Meisner's point (33)/ (1 + 1.618). Meisner's points (23, 24) are calculated as follows: Meisner's point (23) = from Meisner's point (11, 12) to Figure 9. Human body measurements algorithm.

Meisner's point (33)/ (1 + 1.618) and Meisner's point (24) = from Meisner's point (11, 12) to Meisner's point (33)/ (1 + 1.618). Meisner's point (2) is calculated as follows: Meisner's point (2) = from Meisner's point (11, 12) to Meisner's point (33)/ (1 + 1.618), measuring from the top. To calculate Meisner's points (9, 10), the following must be done: Meisner's point (9) = difference from Meisner's point (3) to Meisner's point (19)/1.618 and Meisner's point (10) = difference from Meisner's point (4) to Meisner's point (20)/1.618. Meisner's point (17) = difference from Meisner's point (9) to Meisner's point (19)/1.618 and Meisner's point (18) = difference from Meisner's point (10) to Meisner's point (20)/1.618. Meisner's point (5) = difference from Meisner's point (3) to Meisner's point (7)/1.618 and Meisner's point (6) = difference from Meisner's point (4) to Meisner's point (8)/1.618 from the top. Meisner's point (32) = the difference between Meisner's point (33) to Meisner's point (31)/1.618 from bottom. The Meisner's points (25, 26) can be calculated as follows: difference between (19,20) to (31)/1.618, hence calculate Meisner's point (25) = difference between (Meisner's point (18) and Meisner's point (31)/1.618 then Meisner's point (26) = difference between Meisner's point (20) to Meisner's point (31)/1.618.

The previous two paragraphs enhance the two algorithms suggested in Figures 9 and 10 with more than 33 features that can be used in modeling a human face. The features are reflected as a continuum to the first algorithm in Figure 11.


The hand which is reflected as M20:M21 is 1:1.618, again M5 which is the elbow to tip of the shoulder, M5:M20 is 1:1.618. Hence when modeling a human figure, the following can be

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The fingers are proportional as follows: from tip of figure to distal phalange (C), from distal phalange to middle phalange is (B), and from middle phalange to proximal phalange is (A). The A:B: C of the whole arm is repeated in the figures proportions. The index finger from tip of

Length of hand name C: length of the arm (B) is proportion to 1:1.618.

Length of arm (B): length of elbow to tip of shoulder (A) is 1:1.618.

Figure 11. Human face segmentation a continuum from Figure 10.

derived:

Figure 10. Human body measurements algorithm.

Figure 11. Human face segmentation a continuum from Figure 10.

Meisner's point (4) to Meisner's point (8)/1.618 from the top. Meisner's point (32) = the difference between Meisner's point (33) to Meisner's point (31)/1.618 from bottom. The Meisner's points (25, 26) can be calculated as follows: difference between (19,20) to (31)/1.618, hence calculate Meisner's point (25) = difference between (Meisner's point (18) and Meisner's point (31)/1.618 then Meisner's point (26) = difference between Meisner's point (20) to Meisner's

The previous two paragraphs enhance the two algorithms suggested in Figures 9 and 10 with more than 33 features that can be used in modeling a human face. The features are reflected as

point (31)/1.618.

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a continuum to the first algorithm in Figure 11.

Figure 10. Human body measurements algorithm.

The hand which is reflected as M20:M21 is 1:1.618, again M5 which is the elbow to tip of the shoulder, M5:M20 is 1:1.618. Hence when modeling a human figure, the following can be derived:

Length of hand name C: length of the arm (B) is proportion to 1:1.618.

Length of arm (B): length of elbow to tip of shoulder (A) is 1:1.618.

The fingers are proportional as follows: from tip of figure to distal phalange (C), from distal phalange to middle phalange is (B), and from middle phalange to proximal phalange is (A). The A:B: C of the whole arm is repeated in the figures proportions. The index finger from tip of the figure to the first knuckles (distal phalange) is proportional to the second knuckle (middle phalange) 1: φ, which is equally similar to middle finger, ring finger, and little finger. The thumb from the tip to the (distal phalange) proportional to the proximal phalange is 1: φ and to the joint connecting the hand is 1: φ 2 as shown in Figure 10.

M1: feet bottom to navel 110.0086 M2: navel to top of head 67.98906 M3: feet bottom to knee bottom line 42.01955 M4: knee bottom line to navel 67.98906 M5: navel to pit of throat and elbow to tip of the shoulder 42.01955 M6: pit of throat to top of head 25.96951 M7: bottom of feet to begin of calves 16.05004 M8: from calves to bottom line of knee 25.96951 M9: bottom line of knee to mid thighs (beginning of finger tips) 25.96951 M11:mid thighs (beginning of finger tips) to navel 42.01955 M12: navel to middle of chest 25.96951 M13: middle of chest to pit of throat 16.05004 M14: pit of throat to point between the eyes 16.05004 M15: point between the eyes to top of head 9.919471 M16: bottom of foot to ankle 6.13057 M17: ankle to beginning of calves 9.919471 M18: beginning of knees to end of knees 9.919471 M19: top of knees to tip of hand fingers 16.05004 M20: hand tip of fingers to hand rest 16.05004 M21: hand rest to elbow and navel to man gentiles 25.96951 M22: navel to beginning of chest (abdomen) 16.05004 M23: beginning of chest to middle of chest 9.919471 M24: throat pit to below nose 9.919471 M25: below nose to top of eyebrows 6.13057 M26: top of eyebrows to hair line 3.788901 M27: hair line to top of head 6.13057 M28: from throat pit to Adam's apple 3.788901 M29: Adam's apple to below the nose 6.13057 The length of the face = M25 + M26 + M29 15.9 The width of the face = length of face/1.618 9.826 Left tip of eyebrow to the right tip of eyebrow = M25 � 1.618 9.869 From tip of eyebrow to between the eyes = 2 � M25/1.618 7.5779 Total height of a man = M1 + M2 177.989 Width of shoulders 1/4 of height of a man 44.49725 The foot is 1/6 of the height of a man 29.666

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Table 2. All M's calculated using Microsoft excel sheet.
