Full torus

Full torus

In mathematics , a full torus is a 3-dimensional structure with exactly one handle . It is bordered by a torus .

Full torus as a solid of revolution

The set of points that are the distance from a circular line with a radius for a fixed one is a full torus. It is obtained by rotating the circular area from the radius about an axis of rotation lying in the circular plane and not intersecting the circle, the distance from the center of the circle being greater than the radius of the circular area. ${\displaystyle R}$ ${\displaystyle a\leq r}$ ${\displaystyle r<R}$ ${\displaystyle r}$ ${\displaystyle R}$

Parameterization

A parameterization of the full torus is

${\displaystyle {\vec {X}}(a,t,p)={\begin{pmatrix}x\\y\\z\end{pmatrix}}=R\cdot {\begin{pmatrix}\cos(t)\\\sin(t)\\0\end{pmatrix}}+a\cdot {\begin{pmatrix}\cos(t)\cdot \cos(p)\\\sin(t)\cdot \cos(p)\\\sin(p)\end{pmatrix}}={\begin{pmatrix}(R+a\cdot \cos(p))\cos(t)\\(R+a\cdot \cos(p))\sin(t)\\a\cdot \sin(p)\end{pmatrix}}}$

with . ${\displaystyle 0\leq a\leq r,0\leq t\leq 2\pi ,0\leq p\leq 2\pi }$

Volume of the full torus

The volume of the full torus can be calculated as a triple integral using the Jacobi determinant (the determinant of the functional matrix). The Jacobi matrix for the parameterization of the full torus can be given as follows:

${\displaystyle J_{f}={\frac {\partial \left(x,y,z\right)}{\partial \left(r,t,p\right)}}={\begin{pmatrix}\partial _{r}x&\partial _{t}x&\partial _{p}x\\\partial _{r}y&\partial _{t}y&\partial _{p}y\\\partial _{r}z&\partial _{t}z&\partial _{p}z\\\end{pmatrix}}={\begin{pmatrix}\cos(t)\cos(p)&-R\sin(t)-r\sin(t)\cos(p)&-r\cos(t)\sin(p)\\\sin(t)\cos(p)&R\cos(t)+r\cos(t)\cos(p)&-r\sin(t)\sin(p)\\\sin(p)&0&r\cos(p)\end{pmatrix}}}$

It follows:

${\displaystyle \det(J_{f})=r\cdot \left(r\cos(p)+R\right)}$

The functional determinant here is therefore equal to the norm of the surface normal vector.

${\displaystyle V=\int _{V}\mathrm {d} V=\int _{\Gamma }\det(J_{f})\ \mathrm {d} \Gamma =\int _{0}^{2\pi }\int _{0}^{2\pi }\int _{0}^{r}\left(Rr+r^{2}\cos(p)\right)\ \mathrm {d} r\mathrm {d} p\mathrm {d} t=2\pi ^{2}r^{2}R\;{\color {OliveGreen}=\int A_{O}\mathrm {d} r}}$

So we get for the volume of the full torus . ${\displaystyle V=2\pi ^{2}r^{2}R}$

The formula for the volume can be interpreted in such a way that the area of ​​the circle is multiplied by the circumference (see Guldin's second rule ). To understand this, this can be put in analogy to the cylinder volume. The same applies to the area of ​​the surface, here the perimeters and are multiplied with each other (see Guldin's first rule ). This is also analogous to the cylinder surface . ${\displaystyle A_{r}=\pi r^{2}}$ ${\displaystyle U_{R}=2\pi R}$ ${\displaystyle V_{\text{zyl}}=\pi r^{2}l}$ ${\displaystyle U_{r}=2\pi r}$ ${\displaystyle U_{R}=2\pi R}$ ${\displaystyle O_{\text{zyl}}=2\pi rl}$

Moment of inertia of a full torus

The moment of inertia of a full torus with the density in relation to the -axis (axis of symmetry) can be through ${\displaystyle \rho }$ ${\displaystyle z}$

${\displaystyle I=\rho \int _{T}(x^{2}+y^{2})\,\,\mathrm {d} ^{3}x}$

be calculated. The transformation can now be carried out on torus coordinates. The Jacobi determinant is also included in the integral.

${\displaystyle I=\rho \int _{t=0}^{2\pi }\int _{p=0}^{2\pi }\int _{r'=0}^{r}|\det J_{\text{Torus}}|\cdot (R+r'\cdot \cos(p))^{2}\,\,\,\mathrm {d} r'\mathrm {d} p\,\mathrm {d} t=\rho \int _{t=0}^{2\pi }\int _{p=0}^{2\pi }\int _{r'=0}^{r}r'\cdot (R+r'\cdot \cos(p))^{3}\,\,\,\mathrm {d} r'\mathrm {d} p\,\mathrm {d} t}$

With partial integration and the toroidal mass one obtains: ${\displaystyle M}$

${\displaystyle I=2\pi ^{2}\cdot \rho \cdot R\cdot r^{2}\left({\frac {3}{4}}\cdot r^{2}+R^{2}\right)}$

${\displaystyle I=M\cdot \left({\frac {3}{4}}\cdot r^{2}+R^{2}\right)}$

Full torus in the topology

A full torus is a handle body of the gender . The edge of the full torus is a torus. ${\displaystyle g=1}$

Topologically, a full torus is homeomorphic to the product of the circular disk and the circular line. You can embed the full torus as a rotationally symmetrical full torus in the . ${\displaystyle D^{2}\times S^{1}}$ ${\displaystyle \mathbb {R} ^{3}}$

Its topological invariants are calculated as follows:

${\displaystyle \pi _{1}(S^{1}\times D^{2})\cong \pi _{1}(S^{1})\cong \mathbb {Z} ,}$

${\displaystyle H_{k}(S^{1}\times D^{2})\cong H_{k}(S^{1})\cong {\begin{cases}\mathbb {Z} &{\mbox{ falls }}k=0,1\\0&{\mbox{ sonst }}\end{cases}}.}$

The 3-sphere , i.e. the three-dimensional space together with an infinitely distant point, can be represented as the union of two full gates that only overlap in their surface. They are obtained, for example, from the Hopf fibers , in which the base space is understood as the union of the northern and southern hemispheres; The grain is trivial over both halves. The division of the 3-Sphere into two full gates is used, for example, in the construction of the Reeb foliage . ${\displaystyle S^{2}}$