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Cococubed.com


Volume from a plane slicing a cube

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Contact: F.X.Timmes
my one page vitae,
full vitae,
research statement, and
teaching statement.

The intersection of a plane and a cube is a geometric computation with applications in computer graphics, solid modeling, and computational astrophysics (e.g., fraction of cell that may be partial ionized or covered by a burning front).

The tool cube_and_plane.tbz computes the volume of a plane slicing through a cube. This slicing results in an inscribed irregular polygon with 3 to 6 intersection points, as shown below. This tool extends the idea in Salama & Kolb 2005 by constructing an ordered list of the face vertices, and hence the volume of the irregular polyhedron.

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On the unit cube with vertices and faces enumerated and ordered as shown by the figure above, there are only three independent paths from the front vertex ${\bf V}_1$ to the back vertex ${\bf V}_8$ as marked by the black, dark gray, and light gray lines. The intersecting plane is defined by the length of the normal ray dl (aka the radius vector), and the spherical angles $0 \le \theta \le 2\pi$ and $0 \le \phi \le \pi$. The Cartesian coordinates of the normal ray are \begin{equation} {\bf n}= [dl \cos(\theta), \ dl \sin(\theta)\sin(\phi), \ dl \cos(\phi) ] \ . \label{eq1} \tag{1} \end{equation} Take the equation of the plane, $ax + by + cd + d = 0)$, to be \begin{equation} dl = {\bf \hat{n}} \cdot {\bf x} \label{eq2} \tag{2} \end{equation} where ${\bf \hat{n}}$ is the unit normal vector. Edge $E_{i \rightarrow j}$ between vertices ${\bf V}_i$ and ${\bf V}_j$ is described by the straight line \begin{equation} E_{i \rightarrow j} = {\bf V}_i + \lambda ( {\bf V}_j - {\bf V}_i ) \label{eq3} \tag{3} \end{equation} and thus the intersection point between the plane and the straight line spanned by $E_{i \rightarrow j}$ is \begin{equation} \lambda = \frac{ dl - {\bf \hat{n}} \cdot {\bf V}_i }{{\bf \hat{n}} \cdot ({\bf V}_j - {\bf V}_i ) } \label{eq4} \tag{4} \end{equation} There is an intersection only if $0 \le \lambda \le 1$, otherwise the plane does not intersect the edge within the cube.

The gif animation on the left shows the number of vertices of the intersecting polygon in the $\theta - \phi$ plane as a function of the length of the normal ray $dl$ and the gif animation on the right shows the volume of a unit cube. Whil the changes in the volume are smooth. the number of vertices of the intersecting polygon show wild patterns.

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Number of vertices of the intersecting polygon.
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Volume of the unit cube.

 



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