/sci/ - Science & Math » Thread #11547740

3MiB, 3840x2160, doughnuts.png

View Same Google iqdb SauceNAO

Anonymous Thu 09 Apr 13:27:18 2020 No.11547740 View Reply Original Report

Quoted By: >>11547747 >>11547940 >>11548302 >>11548381 >>11548427 >>11548497 >>11548514 >>11548548 >>11548831

Without looking it up, derive a formula for the volume of a torus.

Anonymous

Anonymous Thu 09 Apr 2020 13:29:27 No.11547747 Report

Quoted By: >>11547753

>>11547740

without attempting to act like you are not a faggot, describe the taste of cum

Anonymous

Anonymous Thu 09 Apr 2020 13:31:59 No.11547753 Report

Quoted By:

>>11547747
A bit bitter, a bit salty, like a less-strong snot taste

Anonymous

Anonymous Thu 09 Apr 2020 14:59:38 No.11547940 Report

Quoted By: >>11547963

>>11547740
Recall the parametrisation

$\vec{r}(\theta, \phi)=((a \cos \theta+b) \cos \phi,(a \cos \theta+b) \sin \phi, a \sin \theta)$

Use this to calculate the metric tensor using coordinate vector fields $\frac{\partial}{\partial \theta}, \frac{\partial}{\partial \phi}$ and then immerse into $\mathbb{R}^3$ to calculate the first fundamental form $\mathbb{I}$

Then use the Hodge-star operator calculate the Riemannian volume form and integrate the constant function $1dV_g$

The calculation of the integral is left as an exercise

Anonymous

Anonymous Thu 09 Apr 2020 15:01:18 No.11547942 Report

Quoted By:

The flat torus has 2-volume 1.

Anonymous

Anonymous Thu 09 Apr 2020 15:02:46 No.11547948 Report

Quoted By: >>11548159

just cut and straighten it and it turns into a cylinder with volume 2*Pi^2*r^2*R

Anonymous

Anonymous Thu 09 Apr 2020 15:06:45 No.11547963 Report

Quoted By:

>>11547940
>left as an exercise
Kek

Anonymous

Anonymous Thu 09 Apr 2020 15:59:16 No.11548159 Report

Quoted By: >>11548317

>>11547948
thats what I was thinking. though I wonder if because the interior has a smaller circumference than the outer circumference that it wouldnt be able to be "straightened" without changing its form thus is volume.

Anonymous

Anonymous Thu 09 Apr 2020 16:41:21 No.11548302 Report

Quoted By:

>>11547740
V1 - V2 = VT
Where V1 is the volume of water before dunking the torus in the bucket, V2 is the volume after doing so, and VT is the volume of the torus. Error includes water being spilled due to jostling, water sticking to the torus, and limited precision tools.

Anonymous

Anonymous Thu 09 Apr 2020 16:43:55 No.11548317 Report

Quoted By:

>>11548159
Cut it into wedges, then I think you just have to check that you can approximate those wedges as cylinders.

Vitruvian666 !!666fz4vFOJ1

Vitruvian666 !!666fz4vFOJ1 Thu 09 Apr 2020 16:59:55 No.11548381 Report

Quoted By:

>>11547740
Not a math or geometry fag, but wouldn't this require that you know whatever the constant value between a Torus of X size and a perfect sphere of X size.

Since you'd probably deduce it, as a sphere and subtract what's missing...

Or, you cut the torus on one side, to make it a cylinder, then compute volume with the cylinder formula.

Again, just need to know the radius or diameter of the ring, and the diameter of the full torus.

Wouldn't be too bad.

I would actually choose the cylinder method.

Anonymous

Anonymous Thu 09 Apr 2020 17:10:52 No.11548427 Report

Quoted By:

>>11547740
Whats a torus?

Anonymous

Anonymous Thu 09 Apr 2020 17:34:46 No.11548497 Report

Quoted By: >>11548504 >>11548804

>>11547740

This follows from Pappus's centroid theorem: the volume of the torus is equal to the surface of the plane figure - a circle - times the traveled distance of said circle's centroid C.

Surface of the plane figure - a circle - is (?r2) where r is the circle's radius i.e. from C out to the torus's edge.
Traveled distance is the circumference (2?R) where R is the radius from C to the center of the torus.

Final formula: (?r2)(2?R)

(And seriously fuck off if you expect someone to reproduce the proof for the theorem from memory.)

Anonymous

Anonymous Thu 09 Apr 2020 17:36:45 No.11548504 Report

Quoted By: >>11548573

>>11548497
> (?r2)

(?r^2) obviously.

Fuck the crappy board software folding the Unicode super-script 2 into the normal one.

Anonymous

Anonymous Thu 09 Apr 2020 17:39:34 No.11548514 Report

Quoted By:

>>11547740
Area of the cross section multiplied to get the rotation.

Anonymous

Anonymous Thu 09 Apr 2020 17:51:06 No.11548548 Report

Quoted By:

>>11547740
Some gay shit involving solids in revolution.

Anonymous

Anonymous Thu 09 Apr 2020 18:06:43 No.11548573 Report

Quoted By:

>>11548504
Just use LaTeX, faggot.

Anonymous

Anonymous Thu 09 Apr 2020 19:30:03 No.11548804 Report

Quoted By:

>>11548497
>Pappus's centroid theorem
That's neat

Anonymous

Anonymous Thu 09 Apr 2020 19:40:52 No.11548831 Report

Quoted By:

>>11547740
a = radius of inner circle cross-section where it is largest
b = radius of entire torus cross section where it is largest
(((b-a)/2)^2 * pi) * ((a+((b-a)/2))*2)*pi

Capcode	All Only User Posts Only Moderator Posts Only Admin Posts Only Developer Posts
Show Posts	All Only With Images Only Without Images
Deleted Posts	All Only Deleted Posts Only Non-Deleted Posts
Ghost Posts	All Only Ghost Posts Only Non-Ghost Posts
Post Type	All Only Sticky Threads Only Opening Posts Only Reply Posts
Results	All Grouped By Threads
Order	Latest Posts First Oldest Posts First

Your latest searches