Oval
Cartesian ovals | Cassini ovals | ellipse | limaçon | squircle (definition 1) | squircle (definition 2) | superellipse (total: 7)
ellipse | x(t) = a cos(t) y(t) = b sin(t) limaçon | x(t) = cos(t) (a cos(t) + b) y(t) = sin(t) (a cos(t) + b)
squircle (definition 2) | (s x^2 y^2)/k^4 - x^2/k^2 - y^2/k^2 + 1 = 0 and -k<=x<=k and -k<=y<=k
Cartesian ovals | (-(m^2 - n^2) (a^2 + x^2 + y^2) + 2 a x (m^2 + n^2) + k^2)^2 - 4 k^2 n^2 ((a + x)^2 + y^2) = 0 Cassini ovals | ((x - a)^2 + y^2) ((a + x)^2 + y^2) = b^4 ellipse | x^2/a^2 + y^2/b^2 = 1 limaçon | a^2 x^2 - 2 a x^3 - 2 a x y^2 - b^2 x^2 - b^2 y^2 + x^4 + 2 x^2 y^2 + y^4 = 0 squircle (definition 1) | x^4 + y^4 = r^4 squircle (definition 2) | -(s x^2 y^2)/k^4 + x^2/k^2 + y^2/k^2 - 1 = 0 superellipse | abs(x/a)^r + abs(y/b)^r = 1
ellipse | r(θ) = (a b)/sqrt((b^2 - a^2) cos^2(θ) + a^2) limaçon | r(θ) = a cos(θ) + b
closed | oval
ellipse | A = π a b limaçon | A = 3/2 b sqrt(a^2 - b^2) + π (a^2/2 + b^2) - (a^2/2 + b^2) cos^(-1)(b/a) squircle (definition 1) | A = (8 r^2 Γ(5/4)^2)/sqrt(π) squircle (definition 2) | A = (4 k^2 E(sin^(-1)(s)|1/s^2))/s superellipse | A = (sqrt(π) a b 4^(1 - 1/r) Γ(1 + 1/r))/Γ(1/2 + 1/r)
limaçon | A^* = 1/2 π (a^2 + 2 b^2)
ellipse | s = 4 a E(1 - b^2/a^2) limaçon | s = 4 (a + b) E((4 a b)/(a + b)^2) squircle (definition 1) | s = -(3^(1/4) r G_(5, 5)^(5, 5) (1|1/3, 2/3, 5/6, 1, 4/3 1/12, 5/12, 7/12, 3/4, 13/12))/(16 sqrt(2) π^(7/2) Γ(5/4))
Cartesian ovals | d = 4 Cassini ovals | d = 4 ellipse | d = 2 limaçon | d = 4 squircle (definition 1) | d = 4 squircle (definition 2) | d = 4
| eccentricity | focal parameter | semilatus rectum ellipse | e = sqrt(1 - b^2/a^2) | p = b^2/sqrt(a^2 - b^2) | L = b^2/a | foci | directrix ellipse | {(-sqrt(a^2 - b^2), 0), (sqrt(a^2 - b^2), 0)} | piecewise | {x = -a^2/sqrt(a^2 - b^2) ∨ x = a^2/sqrt(a^2 - b^2)} | ba | (otherwise)
| evolute | involute ellipse | ellipse evolute | ellipse involute limaçon | limaçon evolute |