anchor ring | horn torus | napkin ring | ring | ring torus | spindle torus

x(u, v) = cos(u) (a cos(v) + c) y(u, v) = sin(u) (a cos(v) + c) z(u, v) = a sin(v)

(c - sqrt(x^2 + y^2))^2 + z^2 = a^2

4

g = 1

S = 4 π^2 a c

ds^2 = (a cos(v) + c)^2 du^2 + a^2 dv^2

dA = a abs(c + a cos(v)) du dv

x^_ = (0, 0, 0)

V = 2 π^2 a^2 c

I = ((5 a^2)/8 + c^2/2 | 0 | 0 0 | (5 a^2)/8 + c^2/2 | 0 0 | 0 | (3 a^2)/4 + c^2)

K(u, v) = cos(v)/(a (a cos(v) + c))

(for a torus with center at the origin, rotational axis of symmetry about the z-axis, radius c, from the center of the hole to the center of the torus tube, and radius a of the tube; c>a corresponds to a ring torus, c = a to a horn torus (which is tangent to itself at the point at the origin), and c

g_(uu) = (a cos(v) + c)^2 g_(vv) = a^2

Γ | u | | | uv = -(a sin(v))/(a cos(v) + c) Γ | u | | | vu = -(a sin(v))/(a cos(v) + c) Γ | v | | | uu = (sin(v) (a cos(v) + c))/a

E(u, v) = (a cos(v) + c)^2 F(u, v) = 0 G(u, v) = a^2

e(u, v) = cos(v) (a (-cos(v)) - c) f(u, v) = 0 g(u, v) = -a

left double bracketing bar x(u, v) right double bracketing bar = sqrt(a^2 + 2 a c cos(v) + c^2)

N^^(u, v) = (-cos(u) cos(v) sgn(cos(v) a + c), -cos(v) sgn(cos(v) a + c) sin(u), -sgn(cos(v) a + c) sin(v))

algebraic surfaces | closed surfaces | quartic surfaces | surfaces of revolution

rhythmic gymnastics hoop