Laminae With Holes
annulus | disk with square hole | disk with equilateral triangular hole | elliptical annulus | equilateral triangle with circular hole | rectangle with rectangular hole | square with circular hole | square with square hole | Acura symbol-like lamina | Honda symbol-like lamina | Yamaha symbol-like lamina | bunny lamina | first Easter egg | second Easter egg | third Easter egg | fourth Easter egg | jack-o'-lantern face lamina | pirate face lamina | smiley face lamina (total: 19)
b^2<=x^2 + y^2<=a^2
x^2 + y^2<=a^2 and abs(x)>=b/2 or abs(y)>=b/2
sqrt(3) b<=3 (sqrt(3) x + y) or sqrt(3) (b + 3 x)<=3 y or sqrt(3) b + 6 y<=0 and x^2 + y^2<=a^2
r^2<=x^2/a^2 + y^2/b^2<=1
sqrt(3) (a + 3 x)>=3 y and sqrt(3) a + 6 y>=0 and sqrt(3) a>=3 (sqrt(3) x + y) and x^2 + y^2>=b^2
abs(x)<=a/2 and abs(y)<=b/2 and abs(x)>=c/2 or abs(y)>=d/2
abs(x)<=a/2 and abs(y)<=a/2 and x^2 + y^2>=b^2
abs(x)<=a/2 and abs(y)<=a/2 and abs(x)>=b/2 or abs(y)>=b/2
max(min(-7500 (x^2/a^2)^(5/4) - 8323 (y^2/a^2)^(5/4) + 1305, 9614 (x^2/a^2)^(5/4) + 8568 (y^2/a^2)^(5/4) - 1173), min(36677 - 864 ((224 x)/a + (31 y)/a), 1311 ((74 x)/a + (35 y)/a) - 4582, -(33176 x)/a - (12376 y)/a + 6523, (38880 x)/a + (513 y)/a - 1234), min(9498 - 767 ((360 x)/a + (31 y)/a), (38880 x)/a - (3348 y)/a + 1339, 6 - (59 y)/a, (27 y)/a - 1), min((1144 x)/a - (427 y)/a + 194569/864, (224 x)/a - (31 y)/a + 55240/1311, -(74 x)/a + (35 y)/a - 1559/432, -(449280 x)/a + (5928 y)/a - 14237))>=0
max(min(-(456 abs(x/a)^(17/4))/(25 (1/13 (y/a - 17/42) + 1)^(17/4)) - 1969/38 abs(y/a)^(17/4) + 1, (99 abs(x/a)^(17/4))/(4 (1/13 (y/a - 17/42) + 1)^(17/4)) + 2651/37 abs(y/a)^(17/4) - 1), min(-(99 abs(x/a)^(17/4))/(4 (1/13 (y/a - 17/42) + 1)^(17/4)) - 2651/37 abs(y/a)^(17/4) + 1, -(x/a - 185/17)^2 - (y/a - 46/55)^2 + 3136/25, -(x/a + 185/17)^2 - (y/a - 46/55)^2 + 3136/25, 349/175 abs(x/a)^(11/2) + 5269/23 abs(x/a)^(17/4) - y/a - 1/12, (3 abs(x/a)^6)/19 + (5269 abs(x/a)^4)/23 + y/a + 1/6))>=0
rectangle with rectangular hole | (-a/2, -b/2) | (a/2, -b/2) | (a/2, b/2) | (-a/2, b/2) | (-c/2, -d/2) | (c/2, -d/2) | (c/2, d/2) | (-c/2, d/2) square with square hole | (-a/2, -a/2) | (a/2, -a/2) | (a/2, a/2) | (-a/2, a/2) | (-b/2, -b/2) | (b/2, -b/2) | (b/2, b/2) | (-b/2, b/2)
rectangle with rectangular hole | 8 square with square hole | 8
annulus | 0
annulus | A = π (a^2 - b^2) disk with square hole | A = π a^2 - b^2 disk with equilateral triangular hole | A = π a^2 - (sqrt(3) b^2)/4 elliptical annulus | A = π a b (1 - r^2) equilateral triangle with circular hole | A = (sqrt(3) a^2)/4 - π b^2 rectangle with rectangular hole | A = a b - c d square with circular hole | A = a^2 - π b^2 square with square hole | A = a^2 - b^2 Yamaha symbol-like lamina | A = a^2 (271817/1427150 + (1691 sqrt(7/10510))/16854 - (33 sqrt(3310123/44713))/40000 - (14 sqrt(48525107/44713))/65025 + (61 sqrt(22817731/1051))/3468000 + (2831 sqrt(186707671/1051))/67416000 - sqrt(1159/5282731) π - sqrt(1159/5282731) cos^(-1)((3 sqrt(174217/61))/200) + 3/40 sqrt(89/1051) cos^(-1)((2 sqrt(6230))/159) - sqrt(1159/5282731) sec^(-1)(255 sqrt(61/2553953)) + 3/40 sqrt(89/1051) sec^(-1)(510/sqrt(256379)) + 3/20 sqrt(89/1051) sec^(-1)(3180/sqrt(2097839))) jack-o'-lantern face lamina | A = (16 π - 103/16) a^2 smiley face lamina | A = π (a^2 + 2 r^2 - 2 R^2) + r^2 (-sec^(-1)(-(2 r (d_l - d_u))/((d_l - d_u)^2 + r^2 - R^2))) + R^2 sec^(-1)((2 R (d_l - d_u))/((d_l - d_u)^2 - r^2 + R^2)) - 1/2 sqrt((d_l - d_u + r - R) (d_l - d_u - r + R) (d_l - d_u + r + R) (-d_l + d_u + r + R))
annulus | x^_ = (0, 0) disk with square hole | x^_ = (0, 0) disk with equilateral triangular hole | x^_ = (0, 0) elliptical annulus | x^_ = (0, 0) equilateral triangle with circular hole | x^_ = (0, 0) rectangle with rectangular hole | x^_ = (0, 0) square with circular hole | x^_ = (0, 0) square with square hole | x^_ = (0, 0) jack-o'-lantern face lamina | x^_ = (0, -(367 a)/(824 - 2048 π))
annulus | J_x invisible comma x = 1/4 π (a^4 - b^4) disk with square hole | J_x invisible comma x = 1/12 (3 π a^4 - b^4) disk with equilateral triangular hole | J_x invisible comma x = 1/96 (24 π a^4 - sqrt(3) b^4) elliptical annulus | J_x invisible comma x = 1/4 π a b^3 (1 - r^4) equilateral triangle with circular hole | J_x invisible comma x = 1/96 (sqrt(3) a^4 - 24 π b^4) rectangle with rectangular hole | J_x invisible comma x = 1/12 (a b^3 - c d^3) square with circular hole | J_x invisible comma x = 1/12 (a^4 - 3 π b^4) square with square hole | J_x invisible comma x = 1/12 (a^4 - b^4) jack-o'-lantern face lamina | J_x invisible comma x = ((236653 - 1179648 π) a^2)/(118656 - 294912 π)
annulus | J_y invisible comma y = 1/4 π (a^4 - b^4) disk with square hole | J_y invisible comma y = 1/12 (3 π a^4 - b^4) disk with equilateral triangular hole | J_y invisible comma y = 1/96 (24 π a^4 - sqrt(3) b^4) elliptical annulus | J_y invisible comma y = 1/4 π a^3 b (1 - r^4) equilateral triangle with circular hole | J_y invisible comma y = 1/96 (sqrt(3) a^4 - 24 π b^4) rectangle with rectangular hole | J_y invisible comma y = 1/12 (a^3 b - c^3 d) square with circular hole | J_y invisible comma y = 1/12 (a^4 - 3 π b^4) square with square hole | J_y invisible comma y = 1/12 (a^4 - b^4) jack-o'-lantern face lamina | J_y invisible comma y = ((63025 - 393216 π) a^2)/(39552 - 98304 π)
annulus | J_zz = 1/2 π (a^4 - b^4) disk with square hole | J_zz = 1/6 (3 π a^4 - b^4) disk with equilateral triangular hole | J_zz = 1/48 (24 π a^4 - sqrt(3) b^4) elliptical annulus | J_zz = -1/4 π a b (r^4 - 1) (a^2 + b^2) equilateral triangle with circular hole | J_zz = 1/48 (sqrt(3) a^4 - 24 π b^4) rectangle with rectangular hole | J_zz = 1/12 (a^3 b + a b^3 - c d (c^2 + d^2)) square with circular hole | J_zz = 1/6 (a^4 - 3 π b^4) square with square hole | J_zz = 1/6 (a^4 - b^4) jack-o'-lantern face lamina | J_zz = (2 (9216 π - 1663) a^2)/(2304 π - 927)
annulus | J_x invisible comma y = 0 disk with square hole | J_x invisible comma y = 0 disk with equilateral triangular hole | J_x invisible comma y = 0 elliptical annulus | J_x invisible comma y = 0 equilateral triangle with circular hole | J_x invisible comma y = 0 rectangle with rectangular hole | J_x invisible comma y = 0 square with circular hole | J_x invisible comma y = 0 square with square hole | J_x invisible comma y = 0 jack-o'-lantern face lamina | J_x invisible comma y = 0
annulus | r_x = 1/2 sqrt(a^2 + b^2) r_y = 1/2 sqrt(a^2 + b^2) disk with square hole | r_x = sqrt((b^4 - 3 π a^4)/(b^2 - π a^2))/(2 sqrt(3)) r_y = sqrt((b^4 - 3 π a^4)/(b^2 - π a^2))/(2 sqrt(3)) elliptical annulus | r_x = 3/2 b sqrt(r^2 + 1) r_y = 3/2 a sqrt(r^2 + 1) equilateral triangle with circular hole | r_x = sqrt((sqrt(3) a^4 - 24 π b^4)/(sqrt(3) a^2 - 4 π b^2))/(2 sqrt(6)) r_y = sqrt((sqrt(3) a^4 - 24 π b^4)/(sqrt(3) a^2 - 4 π b^2))/(2 sqrt(6)) rectangle with rectangular hole | r_x = 1/2 sqrt((a b^3 - c d^3)/(3 a b - 3 c d)) r_y = 1/2 sqrt((a^3 b - c^3 d)/(3 a b - 3 c d)) square with circular hole | r_x = sqrt((a^4 - 3 π b^4)/(a^2 - π b^2))/(2 sqrt(3)) r_y = sqrt((a^4 - 3 π b^4)/(a^2 - π b^2))/(2 sqrt(3)) square with square hole | r_x = sqrt(a^2 + b^2)/sqrt(2) r_y = sqrt(a^2 + b^2)/sqrt(2)
annulus | K = 1/2 π (a^4 - b^4)
rectangle with rectangular hole | a | b | a | b | c | d | c | d square with square hole | a | a | a | a | b | b | b | b
annulus | p = 2 π (a + b) disk with square hole | p = 2 π a + 4 b disk with equilateral triangular hole | p = 2 π a + 3 b elliptical annulus | p = 4 a (r^2 + 1) E(1 - b^2/a^2) equilateral triangle with circular hole | p = 3 a + 2 π b rectangle with rectangular hole | p = 2 (a + b + c + d) square with circular hole | p = 4 a + 2 π b square with square hole | p = 4 (a + b)
annulus | 2 a disk with square hole | 2 a disk with equilateral triangular hole | 2 a elliptical annulus | max(2 a, 2 b) equilateral triangle with circular hole | a rectangle with rectangular hole | sqrt(a^2 + b^2) square with circular hole | sqrt(2) a square with square hole | sqrt(2) a Yamaha symbol-like lamina | a first Easter egg | (8 a)/3 second Easter egg | (8 a)/3 third Easter egg | (8 a)/3 fourth Easter egg | (8 a)/3
annulus | χ = (2 (a^2 cos^(-1)(b/a) - b sqrt(a^2 - b^2)))/(π (a^2 - b^2))
concave laminae | laminae with holes