Bour minimal surface | Catalan surface | catenoid | Costa minimal surface | double sphere | Enneper minimal surface | helicoid | Henneberg minimal surface | Lichtenfels surface | Scherk's first minimal surface | Scherk's second minimal surface | trinoid (total: 12)

doubled sphere

| parametric equations Bour minimal surface | x(r, ϕ) = r cos(ϕ) - 1/2 r^2 cos(2 ϕ) y(r, ϕ) = -r sin(ϕ) (r cos(ϕ) + 1) z(r, ϕ) = 4/3 r^(3/2) cos((3 ϕ)/2) Catalan surface | x(u, v) = a (u - sin(u) cosh(v)) y(u, v) = a (1 - cos(u) cosh(v)) z(u, v) = 4 a sin(u/2) sinh(v/2) catenoid | x(u, v) = a cos(u) cosh(v/a) y(u, v) = a sin(u) cosh(v/a) z(u, v) = v Costa minimal surface | x(u, v) = 1/2 ((2 π^3)/Γ(1/4)^4 + Re((4 π^2 (ζ(-u - i v + i/2;Γ(1/4)^8/(16 π^2), 0) - ζ(-u - i v + 1/2;Γ(1/4)^8/(16 π^2), 0)))/Γ(1/4)^4 - ζ(u + i v;Γ(1/4)^8/(16 π^2), 0) + π u)) y(u, v) = 1/2 ((4 π^2 Im(ζ(-u - i v + i/2;Γ(1/4)^8/(16 π^2), 0) - ζ(-u - i v + 1/2;Γ(1/4)^8/(16 π^2), 0)))/Γ(1/4)^4 + Im(ζ(u + i v;Γ(1/4)^8/(16 π^2), 0)) + π Re(v) + (2 π^3)/Γ(1/4)^4) z(u, v) = 1/2 sqrt(π/2) log(abs((℘(u + i v;Γ(1/4)^8/(16 π^2), 0) - Γ(1/4)^4/(8 π))/(℘(u + i v;Γ(1/4)^8/(16 π^2), 0) + Γ(1/4)^4/(8 π)))) double sphere | x(u, v) = a cos(u) sin(v) y(u, v) = a sin(u) sin(v) z(u, v) = a cos(v) Enneper minimal surface | x(r, ϕ) = r cos(ϕ) - 1/3 r^3 cos(3 ϕ) y(r, ϕ) = -1/3 r (r^2 sin(3 ϕ) + 3 sin(ϕ)) z(r, ϕ) = r^2 cos(2 ϕ) helicoid | x(u, v) = u cos(v) y(u, v) = u sin(v) z(u, v) = c v Henneberg minimal surface | x(u, v) = 2 sinh(u) cos(v) - 2/3 sinh(3 u) cos(3 v) y(u, v) = 2 sinh(u) sin(v) + 2/3 sinh(3 u) sin(3 v) z(u, v) = 2 cosh(2 u) cos(2 v) Lichtenfels surface | x(u, v) = sqrt(2) Re(cos(1/3 (u + i v)) sqrt(cos(2/3 (u + i v)))) y(u, v) = -sqrt(2) Re(sin(1/3 (u + i v)) sqrt(cos(2/3 (u + i v)))) z(u, v) = sqrt(2) Im(F(1/3 (u + i v)|2)) Scherk's first minimal surface | x(u, v) = a u y(u, v) = a v z(u, v) = a log(cos(u) sec(v)) Scherk's second minimal surface | x(r, v) = a log((r^2 + 2 r cos(v) + 1)/(r^2 - 2 r cos(v) + 1)) y(r, v) = a log((r^2 - 2 r sin(v) + 1)/(r^2 + 2 r sin(v) + 1)) z(r, v) = 2 a tan^(-1)((2 r^2 sin(2 v))/(r^4 - 1)) trinoid | x(r, ϕ) = 1/18 (-4 log(r^2 - 2 r cos(ϕ) + 1) + (6 r ((r^2 + 1) cos(ϕ) + r))/(r^4 + 2 (r^3 + r) cos(ϕ) + 2 r^2 cos(2 ϕ) + r^2 + 1) + 2 log(r^4 + 2 (r^3 + r) cos(ϕ) + 2 r^2 cos(2 ϕ) + r^2 + 1)) y(r, ϕ) = 1/9 (-2 sqrt(3) tanh^(-1)((sqrt(3) r sin(ϕ))/(r^2 + r cos(ϕ) + 1)) - (3 r sin(ϕ) (r^4 + 2 (r^3 + r) cos(ϕ) + 1))/(r^6 - 2 r^3 cos(3 ϕ) + 1)) z(r, ϕ) = 1/((3 - 3 r^6)/(2 r^3 cos(3 ϕ) - 2) + 3) | Cartesian equation Bour minimal surface | 43046721 z^16 - 1836660096 z^14 + 544195584 x^3 z^12 - 15509574144 x^2 z^12 - 1632586752 x y^2 z^12 - 15509574144 y^2 z^12 + 26121388032 z^12 - 29023764480 x^4 z^10 - 29023764480 y^4 z^10 + 80299081728 x^3 z^10 + 232190115840 x^2 z^10 - 58047528960 x^2 y^2 z^10 - 240897245184 x y^2 z^10 + 232190115840 y^2 z^10 - 123834728448 z^10 - 12039487488 x^6 z^8 - 14619377664 y^6 z^8 + 232190115840 x^5 z^8 + 602404356096 x^4 z^8 - 20639121408 x^2 y^4 z^8 - 696570347520 x y^4 z^8 + 602404356096 y^4 z^8 + 233910042624 x^3 z^8 - 165112971264 x^2 z^8 - 59337474048 x^4 y^2 z^8 - 464380231680 x^3 y^2 z^8 + 1204808712192 x^2 y^2 z^8 - 701730127872 x y^2 z^8 - 165112971264 y^2 z^8 + 183458856960 x^7 z^6 + 501454209024 x^6 z^6 - 550376570880 x y^6 z^6 + 623760113664 y^6 z^6 + 302707113984 x^5 z^6 - 55037657088 x^4 z^6 - 917294284800 x^3 y^4 z^6 + 770527199232 x^2 y^4 z^6 - 908121341952 x y^4 z^6 - 55037657088 y^4 z^6 - 183458856960 x^5 y^2 z^6 + 2605115768832 x^4 y^2 z^6 - 605414227968 x^3 y^2 z^6 - 110075314176 x^2 y^2 z^6 + 43486543872 x^9 z^4 + 130459631616 x^8 z^4 - 114152177664 x y^8 z^4 + 285380444160 y^8 z^4 + 97844723712 x^7 z^4 - 451172892672 x^3 y^6 z^4 - 252765536256 x^2 y^6 z^4 - 293534171136 x y^6 z^4 - 5435817984 y^6 z^4 - 81537269760 x^5 y^4 z^4 + 1247520227328 x^4 y^4 z^4 - 489223618560 x^3 y^4 z^4 + 32614907904 x^2 y^4 z^4 - 48922361856 x^7 y^2 z^4 + 1916125839360 x^6 y^2 z^4 - 97844723712 x^5 y^2 z^4 - 48922361856 x^4 y^2 z^4 + 57982058496 y^10 z^2 - 231928233984 x^2 y^8 z^2 - 28991029248 x y^8 z^2 - 115964116992 x^4 y^6 z^2 + 183609851904 x^3 y^6 z^2 + 695784701952 x^6 y^4 z^2 - 318901321728 x^5 y^4 z^2 + 521838526464 x^8 y^2 z^2 + 86973087744 x^7 y^2 z^2 + 4294967296 y^12 - 51539607552 x^2 y^10 + 231928233984 x^4 y^8 - 463856467968 x^6 y^6 + 347892350976 x^8 y^4 = 0 double sphere | (-a^2 + x^2 + y^2 + z^2)^2 = 0 Enneper minimal surface | -729 x^6 + 2187 x^4 y^2 - 1215 x^4 z^3 - 4374 x^4 z^2 + 729 x^4 z - 2187 x^2 y^4 - 6318 x^2 y^2 z^3 - 1458 x^2 y^2 z - 432 x^2 z^6 - 3888 x^2 z^5 - 6480 x^2 z^4 + 3888 x^2 z^3 + 729 y^6 - 1215 y^4 z^3 + 4374 y^4 z^2 + 729 y^4 z + 432 y^2 z^6 - 3888 y^2 z^5 + 6480 y^2 z^4 + 3888 y^2 z^3 + 64 z^9 - 1152 z^7 + 5184 z^5 = 0 helicoid | y = x tan(z/c) Henneberg minimal surface | -729 x^12 - 1215 z^3 x^10 - 4374 z^2 x^10 + 8748 z x^10 + 9720 x^10 - 432 z^6 x^8 - 3888 z^5 x^8 + 2187 y^4 x^8 + 7776 z^4 x^8 - 9963 y^2 z^3 x^8 + 69120 z^3 x^8 - 79704 y^2 x^8 - 13122 y^2 z^2 x^8 + 31104 z^2 x^8 - 26244 y^2 z x^8 - 62208 z x^8 - 27648 x^8 + 64 z^9 x^6 + 4320 z^7 x^6 - 864 y^2 z^6 x^6 + 52032 z^6 x^6 - 15552 y^2 z^5 x^6 + 89856 z^5 x^6 + 190512 y^4 x^6 + 62208 y^2 z^4 x^6 - 179712 z^4 x^6 - 23814 y^4 z^3 x^6 - 416256 z^3 x^6 + 55296 y^2 x^6 - 8748 y^4 z^2 x^6 - 248832 y^2 z^2 x^6 - 138240 z^2 x^6 + 17496 y^4 z x^6 + 248832 y^2 z x^6 - 32768 x^6 - 768 z^10 x^4 + 192 y^2 z^9 x^4 - 768 z^9 x^4 - 2187 y^8 x^4 - 5376 z^8 x^4 - 7776 y^2 z^7 x^4 - 162816 z^7 x^4 - 190512 y^6 x^4 + 10560 y^2 z^6 x^4 - 595968 z^6 x^4 - 23328 y^4 z^5 x^4 - 573696 y^2 z^5 x^4 - 651264 z^5 x^4 - 1147392 y^2 z^4 x^4 - 86016 z^4 x^4 - 23814 y^6 z^3 x^4 - 511488 y^4 z^3 x^4 + 84480 y^2 z^3 x^4 - 49152 z^3 x^4 + 98304 y^2 x^4 + 8748 y^6 z^2 x^4 - 248832 y^2 z^2 x^4 - 196608 z^2 x^4 + 17496 y^6 z x^4 - 373248 y^4 z x^4 + 2304 z^11 x^2 + 4608 z^10 x^2 + 192 y^4 z^9 x^2 + 4608 y^2 z^9 x^2 - 27648 z^9 x^2 + 79704 y^8 x^2 - 55296 z^8 x^2 - 7776 y^4 z^7 x^2 - 276480 y^2 z^7 x^2 + 110592 z^7 x^2 - 55296 y^6 x^2 + 864 y^6 z^6 x^2 - 10560 y^4 z^6 x^2 + 221184 z^6 x^2 - 15552 y^6 z^5 x^2 - 573696 y^4 z^5 x^2 + 1105920 y^2 z^5 x^2 - 147456 z^5 x^2 - 98304 y^4 x^2 - 62208 y^6 z^4 x^2 + 1147392 y^4 z^4 x^2 - 294912 z^4 x^2 - 9963 y^8 z^3 x^2 + 84480 y^4 z^3 x^2 - 294912 y^2 z^3 x^2 + 13122 y^8 z^2 x^2 + 248832 y^6 z^2 x^2 + 248832 y^4 z^2 x^2 - 26244 y^8 z x^2 + 248832 y^6 z x^2 + 729 y^12 + 2304 y^2 z^11 - 9720 y^10 + 768 y^4 z^10 - 4608 y^2 z^10 + 64 y^6 z^9 - 768 y^4 z^9 - 27648 y^2 z^9 + 27648 y^8 + 5376 y^4 z^8 + 55296 y^2 z^8 + 4320 y^6 z^7 - 162816 y^4 z^7 + 110592 y^2 z^7 + 32768 y^6 + 432 y^8 z^6 - 52032 y^6 z^6 + 595968 y^4 z^6 - 221184 y^2 z^6 - 3888 y^8 z^5 + 89856 y^6 z^5 - 651264 y^4 z^5 - 147456 y^2 z^5 - 7776 y^8 z^4 + 179712 y^6 z^4 + 86016 y^4 z^4 + 294912 y^2 z^4 - 1215 y^10 z^3 + 69120 y^8 z^3 - 416256 y^6 z^3 - 49152 y^4 z^3 + 4374 y^10 z^2 - 31104 y^8 z^2 + 138240 y^6 z^2 + 196608 y^4 z^2 + 8748 y^10 z - 62208 y^8 z = 0 Scherk's first minimal surface | e^z cos(y) = cos(x)

Bour minimal surface | 43046721 z^16 - 1836660096 z^14 + 544195584 x^3 z^12 - 15509574144 x^2 z^12 - 1632586752 x y^2 z^12 - 15509574144 y^2 z^12 + 26121388032 z^12 - 29023764480 x^4 z^10 - 29023764480 y^4 z^10 + 80299081728 x^3 z^10 + 232190115840 x^2 z^10 - 58047528960 x^2 y^2 z^10 - 240897245184 x y^2 z^10 + 232190115840 y^2 z^10 - 123834728448 z^10 - 12039487488 x^6 z^8 - 14619377664 y^6 z^8 + 232190115840 x^5 z^8 + 602404356096 x^4 z^8 - 20639121408 x^2 y^4 z^8 - 696570347520 x y^4 z^8 + 602404356096 y^4 z^8 + 233910042624 x^3 z^8 - 165112971264 x^2 z^8 - 59337474048 x^4 y^2 z^8 - 464380231680 x^3 y^2 z^8 + 1204808712192 x^2 y^2 z^8 - 701730127872 x y^2 z^8 - 165112971264 y^2 z^8 + 183458856960 x^7 z^6 + 501454209024 x^6 z^6 - 550376570880 x y^6 z^6 + 623760113664 y^6 z^6 + 302707113984 x^5 z^6 - 55037657088 x^4 z^6 - 917294284800 x^3 y^4 z^6 + 770527199232 x^2 y^4 z^6 - 908121341952 x y^4 z^6 - 55037657088 y^4 z^6 - 183458856960 x^5 y^2 z^6 + 2605115768832 x^4 y^2 z^6 - 605414227968 x^3 y^2 z^6 - 110075314176 x^2 y^2 z^6 + 43486543872 x^9 z^4 + 130459631616 x^8 z^4 - 114152177664 x y^8 z^4 + 285380444160 y^8 z^4 + 97844723712 x^7 z^4 - 451172892672 x^3 y^6 z^4 - 252765536256 x^2 y^6 z^4 - 293534171136 x y^6 z^4 - 5435817984 y^6 z^4 - 81537269760 x^5 y^4 z^4 + 1247520227328 x^4 y^4 z^4 - 489223618560 x^3 y^4 z^4 + 32614907904 x^2 y^4 z^4 - 48922361856 x^7 y^2 z^4 + 1916125839360 x^6 y^2 z^4 - 97844723712 x^5 y^2 z^4 - 48922361856 x^4 y^2 z^4 + 57982058496 y^10 z^2 - 231928233984 x^2 y^8 z^2 - 28991029248 x y^8 z^2 - 115964116992 x^4 y^6 z^2 + 183609851904 x^3 y^6 z^2 + 695784701952 x^6 y^4 z^2 - 318901321728 x^5 y^4 z^2 + 521838526464 x^8 y^2 z^2 + 86973087744 x^7 y^2 z^2 + 4294967296 y^12 - 51539607552 x^2 y^10 + 231928233984 x^4 y^8 - 463856467968 x^6 y^6 + 347892350976 x^8 y^4 = 0 double sphere | (-a^2 + x^2 + y^2 + z^2)^2 = 0 Enneper minimal surface | -729 x^6 + 2187 x^4 y^2 - 1215 x^4 z^3 - 4374 x^4 z^2 + 729 x^4 z - 2187 x^2 y^4 - 6318 x^2 y^2 z^3 - 1458 x^2 y^2 z - 432 x^2 z^6 - 3888 x^2 z^5 - 6480 x^2 z^4 + 3888 x^2 z^3 + 729 y^6 - 1215 y^4 z^3 + 4374 y^4 z^2 + 729 y^4 z + 432 y^2 z^6 - 3888 y^2 z^5 + 6480 y^2 z^4 + 3888 y^2 z^3 + 64 z^9 - 1152 z^7 + 5184 z^5 = 0 helicoid | y = x tan(z/c) Henneberg minimal surface | -729 x^12 - 1215 z^3 x^10 - 4374 z^2 x^10 + 8748 z x^10 + 9720 x^10 - 432 z^6 x^8 - 3888 z^5 x^8 + 2187 y^4 x^8 + 7776 z^4 x^8 - 9963 y^2 z^3 x^8 + 69120 z^3 x^8 - 79704 y^2 x^8 - 13122 y^2 z^2 x^8 + 31104 z^2 x^8 - 26244 y^2 z x^8 - 62208 z x^8 - 27648 x^8 + 64 z^9 x^6 + 4320 z^7 x^6 - 864 y^2 z^6 x^6 + 52032 z^6 x^6 - 15552 y^2 z^5 x^6 + 89856 z^5 x^6 + 190512 y^4 x^6 + 62208 y^2 z^4 x^6 - 179712 z^4 x^6 - 23814 y^4 z^3 x^6 - 416256 z^3 x^6 + 55296 y^2 x^6 - 8748 y^4 z^2 x^6 - 248832 y^2 z^2 x^6 - 138240 z^2 x^6 + 17496 y^4 z x^6 + 248832 y^2 z x^6 - 32768 x^6 - 768 z^10 x^4 + 192 y^2 z^9 x^4 - 768 z^9 x^4 - 2187 y^8 x^4 - 5376 z^8 x^4 - 7776 y^2 z^7 x^4 - 162816 z^7 x^4 - 190512 y^6 x^4 + 10560 y^2 z^6 x^4 - 595968 z^6 x^4 - 23328 y^4 z^5 x^4 - 573696 y^2 z^5 x^4 - 651264 z^5 x^4 - 1147392 y^2 z^4 x^4 - 86016 z^4 x^4 - 23814 y^6 z^3 x^4 - 511488 y^4 z^3 x^4 + 84480 y^2 z^3 x^4 - 49152 z^3 x^4 + 98304 y^2 x^4 + 8748 y^6 z^2 x^4 - 248832 y^2 z^2 x^4 - 196608 z^2 x^4 + 17496 y^6 z x^4 - 373248 y^4 z x^4 + 2304 z^11 x^2 + 4608 z^10 x^2 + 192 y^4 z^9 x^2 + 4608 y^2 z^9 x^2 - 27648 z^9 x^2 + 79704 y^8 x^2 - 55296 z^8 x^2 - 7776 y^4 z^7 x^2 - 276480 y^2 z^7 x^2 + 110592 z^7 x^2 - 55296 y^6 x^2 + 864 y^6 z^6 x^2 - 10560 y^4 z^6 x^2 + 221184 z^6 x^2 - 15552 y^6 z^5 x^2 - 573696 y^4 z^5 x^2 + 1105920 y^2 z^5 x^2 - 147456 z^5 x^2 - 98304 y^4 x^2 - 62208 y^6 z^4 x^2 + 1147392 y^4 z^4 x^2 - 294912 z^4 x^2 - 9963 y^8 z^3 x^2 + 84480 y^4 z^3 x^2 - 294912 y^2 z^3 x^2 + 13122 y^8 z^2 x^2 + 248832 y^6 z^2 x^2 + 248832 y^4 z^2 x^2 - 26244 y^8 z x^2 + 248832 y^6 z x^2 + 729 y^12 + 2304 y^2 z^11 - 9720 y^10 + 768 y^4 z^10 - 4608 y^2 z^10 + 64 y^6 z^9 - 768 y^4 z^9 - 27648 y^2 z^9 + 27648 y^8 + 5376 y^4 z^8 + 55296 y^2 z^8 + 4320 y^6 z^7 - 162816 y^4 z^7 + 110592 y^2 z^7 + 32768 y^6 + 432 y^8 z^6 - 52032 y^6 z^6 + 595968 y^4 z^6 - 221184 y^2 z^6 - 3888 y^8 z^5 + 89856 y^6 z^5 - 651264 y^4 z^5 - 147456 y^2 z^5 - 7776 y^8 z^4 + 179712 y^6 z^4 + 86016 y^4 z^4 + 294912 y^2 z^4 - 1215 y^10 z^3 + 69120 y^8 z^3 - 416256 y^6 z^3 - 49152 y^4 z^3 + 4374 y^10 z^2 - 31104 y^8 z^2 + 138240 y^6 z^2 + 196608 y^4 z^2 + 8748 y^10 z - 62208 y^8 z = 0 Scherk's first minimal surface | e^z cos(y) = cos(x)

| algebraic degree | genus Bour minimal surface | 16 | catenoid | (data not available) | g = 0 Costa minimal surface | | g = 1 double sphere | 4 | Enneper minimal surface | 9 | helicoid | (data not available) | g = 0 Henneberg minimal surface | 15 | | surface area | squared line element Bour minimal surface | | ds^2 = (r^4 + 1)^2/(r^6 - 2 r^3 cos(3 ϕ) + 1)^2 dr^2 + (r^5 + r)^2/(r^6 - 2 r^3 cos(3 ϕ) + 1)^2 dϕ^2 Catalan surface | | ds^2 = 2 a^2 cosh^2(v/2) (cosh(v) - cos(u)) du^2 + 2 a^2 cosh^2(v/2) (cosh(v) - cos(u)) dv^2 catenoid | | ds^2 = a^2 cosh^2(v/a) du^2 + cosh^2(v/a) dv^2 double sphere | S = 4 π a^2 | ds^2 = a^2 sin^2(v) du^2 + a^2 dv^2 Enneper minimal surface | | ds^2 = (r^2 + 1)^2 dr^2 + (r^3 + r)^2 dϕ^2 helicoid | | ds^2 = 1 du^2 + c^2 + u^2 dv^2 Henneberg minimal surface | | ds^2 = 8 cosh^2(u) (cosh(4 u) - cos(4 v)) du^2 + 8 cosh^2(u) (cosh(4 u) - cos(4 v)) dv^2 Scherk's first minimal surface | | ds^2 = a^2 sec^2(u) du^2 + -2 a^2 tan(u) tan(v) du dv + a^2 sec^2(v) dv^2 Scherk's second minimal surface | | ds^2 = (16 a^2 (r^2 + 1)^2)/(r^8 - 2 r^4 cos(4 v) + 1) dr^2 + (16 a^2 r^2 (r^2 + 1)^2)/(r^8 - 2 r^4 cos(4 v) + 1) dv^2 trinoid | | ds^2 = (r^4 + 1)^2/(r^6 - 2 r^3 cos(3 ϕ) + 1)^2 dr^2 + (r^5 + r)^2/(r^6 - 2 r^3 cos(3 ϕ) + 1)^2 dϕ^2 | area element | centroid of solid Bour minimal surface | dA = sqrt((r^2 (r^4 + 1)^4)/(r^6 - 2 r^3 cos(3 ϕ) + 1)^4) dr dϕ | Catalan surface | dA = 2 a^2 cosh^2(v/2) (cosh(v) - cos(u)) du dv | (data not available) catenoid | dA = a cosh^2(v/a) du dv | x^_ = (0, 0, 0) double sphere | dA = a^2 sin(v) du dv | x^_ = (0, 0, 0) Enneper minimal surface | dA = sqrt(r^2 (r^2 + 1)^4) dr dϕ | (data not available) helicoid | dA = sqrt(c^2 + u^2) du dv | Henneberg minimal surface | dA = 8 cosh^2(u) (cosh(4 u) - cos(4 v)) du dv | (data not available) Scherk's first minimal surface | dA = a^2 sqrt(tan^2(u) + sec^2(v)) du dv | (data not available) Scherk's second minimal surface | dA = 16 sqrt((a^4 r^2 (r^2 + 1)^4)/(r^8 - 2 r^4 cos(4 v) + 1)^2) dr dv | (data not available) trinoid | dA = sqrt((r^2 (r^4 + 1)^4)/(r^6 - 2 r^3 cos(3 ϕ) + 1)^4) dr dϕ | | volume of solid double sphere | V = (4 π a^3)/3 | moment of inertia tensor of solid | Gaussian curvature Bour minimal surface | | K(r, ϕ) = -(16 (r^7 - 2 r^4 cos(3 ϕ) + r)^2)/(r^4 + 1)^4 Catalan surface | (data not available) | K(u, v) = (sech^4(v/2))/(8 a^2 (cos(u) - cosh(v))) catenoid | | K(u, v) = -(sech^4(v/a))/a^2 double sphere | I = ((2 a^2)/5 | 0 | 0 0 | (2 a^2)/5 | 0 0 | 0 | (2 a^2)/5) | K(u, v) = 1/a^2 Enneper minimal surface | (data not available) | K(r, ϕ) = -4/(r^2 + 1)^4 helicoid | | K(u, v) = -c^2/(c^2 + u^2)^2 Henneberg minimal surface | (data not available) | K(u, v) = (sech^4(u))/(8 (cos(4 v) - cosh(4 u))) Scherk's first minimal surface | (data not available) | K(u, v) = -(sec^2(u) sec^2(v))/(a^2 (tan^2(u) + sec^2(v))^2) Scherk's second minimal surface | (data not available) | K(r, v) = -(r^8 - 2 r^4 cos(4 v) + 1)/(4 a^2 (r^2 + 1)^4) trinoid | | K(r, ϕ) = -(16 (r^7 - 2 r^4 cos(3 ϕ) + r)^2)/(r^4 + 1)^4