torus
Assuming "torus" is a mathematical surface | Use as or or instead

Input interpretation

torus (surface)

Example plots

Example plots

Alternate names

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

Equations

Parametric equations

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

Cartesian equation

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

Surface properties

Algebraic degree

4

Genus

g = 1

Surface area

S = 4 π^2 a c

Squared line element

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

Area element

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

Centroid of solid

x^_ = (0, 0, 0)

Volume of solid

V = 2 π^2 a^2 c

Moment of inertia tensor of solid

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)

Gaussian curvature

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<a to a self-intersecting spindle torus (for which some of the properties given here are not well-defined or do not hold))

Metric properties

Metric tensor

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

Christoffel symbol of the second kind

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

Coefficients of the first fundamental form

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

Coefficients of the second fundamental form

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

Vector properties

Vector length

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

Normal vector

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))

Properties

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

Objects with this shape

Sport objects

rhythmic gymnastics hoop

Related entities

Filled region

torus

Cross sections

circle

Related Wolfram Language symbols

Torus
POWERED BY THE WOLFRAM LANGUAGE

Related Queries:

Have a question about using Wolfram|Alpha?Contact Pro Premium Expert Support »

Give us your feedback »