# Common volume of three cylinders with unequal radii

I would like to solve it please for the case where the radii can be a similar size - so the case where this statement is NOT true: $$\mathbf r_1^2 \mathbf \geq \mathbf r_2^2 \mathbf + \mathbf r_3^2$$

How do you solve for the common volume of 3 cylinders with unequal radii? (If you could please include the integral needed - I think it might need a cartesian equation system? Or if you have any good idea of what direction or things I could read up on to please learn to solve this. Thank you so much for your help! $$x^2 + y^2 = r_1^2$$ $$x^2 + z^2 = r_2^2$$ $$y^2 + z^2 = r_3^2$$ where $$r_1 \neq r_2 \neq r_3$$

I can find that the common volume for equal radii using triple integration with circular co-ordinates to get the answer below (following this) $$V_c = 8\cdot(2-\sqrt 2)\cdot r^3$$

Essentially I want to get to a place where I can find the equation for the common volume of three cylinders with different radii and at different angles. So if anyone has an idea of how to calculate for cylinders at different angles other than 90 that would be great too! Thank you!

• I suppose it's going to be at least as difficult as this problem: math.stackexchange.com/questions/1167660/… – David K Jun 18 at 13:00
• Hi. Yes unfortunately :'( I have looked through the 2 cylinder case and I can get the answer after looking for the elliptic integrals using the formula below. But I don't know how to set up the integral for a cartesian system with 3 cylinders, it seems that everyone only uses the circular co-ordinate system. $$V_c = \frac {8 \cdot r_1 \cdot r_2^2 \cdot k^2} {3} \cdot (2 \cdot R_f (0,1-k^2,1) – \frac {1+k^2}{3} R_d (0,1-k^2,1)$$ – 657933 Jun 18 at 13:12

According to this [first] picture made in GeoGebra, this enclosed volume [seems to be] strictly inside the intersection of the 2 smaller cylinders. Am I right? [No! I was not!] This second picture shows that the third one must join the diagonal point where the other two meet, viewed along the z axis or be larger than that if it is to not interfere with the answer for the two first cylinders.

The intersection of the solid parts of [two] cylinders are on a surface called hyperbolic cylinder (Could be something else if they are not perpendicular, like part of another quadric). We find it's equation by eliminating the common variable in two equations (see picture). You will have to decide where you position your cylinders (smaller radius along x axis, the other along y). Then you need change the angle of only one of them, and find the new equation (with a slope parameter).

• Basically my real goal is to find the common volume for different radii and at different angles, but I am working it one step at a time! Thank you so much though. If you know anything for both that would be great! – 657933 Jun 18 at 14:16
• But the 3 center lines (axis) do intersect, don't they? – ChristianBC Jun 18 at 14:21
• Oh sorry I was being a bit slow! So I essentially don't need to calculate for three cylinders. Thats so great. I think I need to redo my question without my poor maths involved. Thank you so much for all your help! – 657933 Jun 18 at 14:32
• The statement is false unless the largest radius is much larger than the rest. For example, when $r_1 \ge r_2 \ge r_3$, your statement fails unless $r_1^2 \ge r_2^2 + r_3^2$. – achille hui Jun 18 at 16:01
• Using this statement then I just use the formula for the common volume between two cylinders essentially? And if that statement were to fail would I need to find the integral in the cartesian co-ordinates to solve the case for three cylinders? Thank you for everyones help I really appreciate it! – 657933 Jun 19 at 8:09

Let $$a with $$c<\sqrt{a^2+b^2}$$ be the radii of the three cylinders $$x^2+y^2\leq a^2,\qquad x^2+z^2\leq b^2,\qquad y^2+z^2\leq c^2\ ,$$ and denote by $$S$$ the part of the intersection lying in the positive octant. The set $$S$$ is standing on the quarter disc $$S':\qquad x\geq0,\quad y\geq 0, \quad x^2+y^2\leq a^2$$ in the $$(x,y)$$-plane. The upper boundary of $$S$$ is determined by the two other cylinders. Therefore the height $$z(x,y)$$ of $$S$$ at the point $$(x,y)\in S'$$ satisfies $$z^2(x,y)=\min\{b^2-x^2, \ c^2-y^2\}\ .$$ The two entries in the $$\min$$ are equal when $$y^2=c^2-b^2+x^2$$. It follows that z(x,y)=\left\{\eqalign{\sqrt{b^2-x^2}\qquad&(y\leq\sqrt{c^2-b^2+x^2})\cr \sqrt{c^2-y^2}\qquad&(y\geq\sqrt{c^2-b^2+x^2})\cr}\right.\ . We now have to calculate $${\rm vol}(S)=\int_{S'}z(x,y)\>{\rm d}(x,y)\ .\tag{1}$$ The hyperbolic arc $$y=\sqrt{c^2-b^2+x^2}$$ intersects the quarter circle $$x^2+y^2=a^2$$ at the point $$P=\left(\sqrt{a^2+b^2-c^2\over2}, \ \sqrt{a^2+c^2-b^2\over2}\right)=:(u,v)\ .$$ The hyperbolic arc and the lines $$x=u$$, $$y=v$$ through $$P$$ partition the quarter disc $$S'$$ into four sections. From $$(1)$$ we then obtain \eqalign{{\rm vol}(S)&=\int_0^u \sqrt{c^2-b^2+x^2}\>\sqrt{b^2-x^2}\>dx+\int_u^a\sqrt{a^2-x^2}\>\sqrt{b^2-x^2}\>dx \cr&\qquad + \int_{\sqrt{c^2-b^2}}^v \sqrt{y^2-(c^2-b^2)}\>\sqrt{c^2-y^2}\>dy+\int_v^a\sqrt{a^2-y^2}\>\sqrt{c^2-y^2}\>dy\ .\cr}

• Thank you so much for your help Christian! I was wondering if it was possible to reduce the integrals below to Carlson's form in order to solve them? I can only find the form for $\int \sqrt {a^2 - t^2} \sqrt {b^2 - t^2}$ $$\int \sqrt {c^2 - b^2 + x^2} \sqrt {b^2 - x^2}$$ $$\int \sqrt {y^2 - (c^2 - b^2)} \sqrt {c^2 - y^2}$$ – 657933 Jun 20 at 13:39
• I have no expertise with elliptic integrals. – Christian Blatter Jun 20 at 13:53
• Thank you for your detailed answer. Last question sorry but do you know any good books that I could read so I can better understand this topic? – 657933 Jun 20 at 18:10
• I've tried drawing the graph of the hyperbolic arc and the circle equation but it seems that it only works if $a^2 > b^2 + c^2$ if $a^2$ is smaller, the curves do not intersect. – 657933 Jun 20 at 18:44
• Read my first line: $a<b<c$. – Christian Blatter Jun 20 at 18:51