Spindle torus

The three different types of tori: spindle torus (-r <R <r), horn torus (r = R) and ring torus (r <R). At R = 0 the spindle torus degenerates into a sphere.

In differential geometry , a branch of mathematics , a spindle torus is a certain self-penetrating surface in three-dimensional space. It is created by rotating a circle around an axis of rotation that lies in the plane of the circle and whose distance from the center of the circle is smaller than the radius of the circle.

Spindle torus as a surface of revolution

A circle with a radius and a center point has the equation ${\displaystyle r}$ ${\displaystyle (R,0)}$

${\displaystyle (x-R)^{2}+y^{2}=r^{2}}$

and shows different arcs depending on the size of in the right half plane of the Cartesian coordinate system. If these arcs are rotated around the y-axis, the result is spindle torques. At ${\displaystyle -r<R\leq r}$

${\displaystyle -r<R<0}$

shows a spindle torus with two points, at

${\displaystyle R=0}$

the degeneration to the sphere and at

${\displaystyle 0<R\leq r}$

the indentations (apple shape) that open from the torus hole. ${\displaystyle r<R}$

Parameterization of the spindle torus

The spindle torus with can through ${\displaystyle -r<R<r}$

${\displaystyle x=(R+r\cos v)\cos u}$

${\displaystyle y=(R+r\cos v)\sin u}$

${\displaystyle z=r\sin v}$

with be parameterized. ${\displaystyle u,v\in \left[0,2\pi \right]}$

Volume of the spindle torus

The volume element is where the distance from the axis of rotation, h denotes the height and the angle of rotation. Due to the existing cylinder symmetry, the external volume can be found in the area with as ${\displaystyle \mathrm {d} V=\mathrm {d} h\cdot \mathrm {d} \rho \cdot \rho \cdot \mathrm {d} \phi ,}$ ${\displaystyle \rho }$ ${\displaystyle \phi }$ ${\displaystyle -r<R\leq r}$ ${\displaystyle h=f(\rho )={\sqrt {r^{2}-(\rho -R)^{2}}}}$

${\displaystyle V=2\int _{\text{halber Torus}}\mathrm {d} V=4\pi \int _{0}^{R+r}\rho f(\rho )\mathrm {d} \rho ={\frac {2\pi }{3}}(2r^{2}+R^{2}){\sqrt {r^{2}-R^{2}}}+\pi r^{2}R\left(\pi +2\arctan \left({\frac {R}{\sqrt {r^{2}-R^{2}}}}\right)\right).}$

Ab is then the volume (the lower limit in the integral is now instead of 0) . The surface is also apparent here, the derivative of the volume to the radius : . ${\displaystyle R\geq r}$ ${\displaystyle R-r}$ ${\displaystyle V=2\pi ^{2}r^{2}R}$ ${\displaystyle r}$ ${\displaystyle O=\mathrm {d} V/\mathrm {d} r}$

Flat peach

Trivia

Numerous types of fruit are similar to the spindle torus or the horn torus.

Web links

• Spindle Torus (MathWorld)