# Common volume of three finite cylinders on a 2D plane

Its quite a big question sorry... but I really am just looking for the starting point for three cylinders (not two sorry) on a 2D plane. Hopefully after that I can find the common volume for either unequal radii or at different angles like the photo below. Thank you for your help!

A shot in the dark of what I have so far.. (not too sure if it is of use) Cylinder equations $$x^2 + y^2 = r_1^2$$ $$x^2 + z^2 = r_2^2$$ $$y^2 + z^2 = r_3^2$$ where $$r_1 \neq r_2 \neq r_3$$

The common volume for infinite cylinders with equal radii using triple integration with circular co-ordinates is: $$V_c = 8\cdot(2-\sqrt 2)\cdot r^3$$

(following this great video)

• The picture suggests that you want the volume where $x,y,z\geq0$. Is this the case? – saulspatz Jun 18 at 15:16
• Even for two cylindres this problem seems to be a nightmare. – Yves Daoust Jun 18 at 15:19
• Yes please or the picture shows best of what I want to do. Sorry if my maths is quite poor! Thank you! – 657933 Jun 18 at 15:23
• For two cylinders in the finite case, I can divide the infinite cylinder common volume case by 4 to get the finite case I think. I followed this paper reference to get the two cylinder case: ElMaraghy, W., Valluri, S., Skubnik, B. and Surry, P. (1994). Intersection volumes and surface areas of cylinders for geometrical modelling and tolerancing. Computer-Aided Design, 26(1), pp.29-45. – 657933 Jun 18 at 15:29
• What is the radius of the smallest cylinder? What are the equations of the axes of the cylinders? Or do you want solve the problem in general? – zoli Jun 25 at 15:43