Volume of pyramid intersection

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It's just the volume of the first one, because this is entirely below the second.

This is "obvious" if you visualise climbing the "ridgeline" from the origin to the second apex: this ridgeline passes through the first apex, because that is in the same direction from the origin but at half the distance. After you pass the first apex, the second pyramid keeps increasing but the first starts decreasing. So the faces of the two pyramids which lie along the $x$ axis coincide, as do those which lie along the $y$ axis; but the other faces of the first pyramid lie below the corresponding faces of the second.

If this is difficult to visualise you can confirm it algebraically, though it is a fair bit of work. To give one example: consider $(x,y)$ in the triangle with vertices at $(30,0)$, $(20,20)$ and $(10,10)$. One side of this triangle is the line with equation $x+2y=30$, and in the region we have $x+2y\ge30$.

The face of the first pyramid which lies above the region is part of the plane containing $(30,0,0)$, $(30,30,0)$ and $(10,10,10)$: it has equation $$x+2z=30\ .$$ The face of the second pyramid which lies above the region is part of the plane containing $(0,0,0)$, $(30,0,0)$ and $(20,20,20)$, with equation $$y-z=0\ .$$ Therefore we have $$x+2y\ge30\quad\Rightarrow\quad \frac{30-x}{2}\le y\quad\Rightarrow\quad z_1\le z_2\ .$$

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Updated on August 01, 2022

Comments

  • user191316
    user191316 over 1 year

    Suppose that there are two square pyramids on the $xyz$-plane.

    Both have base coordinates of $(0,0,0)$, $(30,0,0)$, $(0,30,0)$, and $(30,30,0)$.

    One pyramid has its apex at $(10,10,30)$, while the other has its apex at $(20,20,30)$.

    What is the volume of their intersection?after change in coordinates

  • user191316
    user191316 almost 9 years
    Could you offer some more explanation on the last part? I'm struggling to see how those equations yield the conclusion that the first pyramid must be the intersection of the two pyramids.
  • David
    David almost 9 years
    Draw a plan of the situation, indicating the apices of the pyramids and the edges connecting them to the base. You will see that this divides the square into six regions. The calculations I have given show that the height $z_1$ of the first pyramid is less than the height $z_2$ of the second in one of these regions. You can do similar calculations for the other five regions.
  • user191316
    user191316 almost 9 years
    How does proving that the heights are less prove that the first pyramid is within the second pyramid? Also, I'm assuming that the "square" mentioned is the base? Also, out of curiosity; could vectors be used to prove this idea? Proving that the four lines connecting the apices for one pyramid lie on the same vector of the four lines connecting the apices for another pyramid, would that prove this idea? Finally, I just noticed the edit made by the community. Does it change this solution?
  • robjohn
    robjohn almost 9 years
    This has been deleted temporarily. We will undelete when the question is unlocked. Ping one of us if it has not.