Volume from a plane slicing a cube


Astronomy research
Software instruments
   Stellar equation of states
   EOS with ionization
   EOS for supernovae
   Chemical potentials
   Stellar atmospheres

   Voigt Function
   Jeans escape
   Polytropic stars
   Cold white dwarfs
   Adiabatic white dwarfs

   Cold neutron stars
   Stellar opacities
   Neutrino energy loss rates
   Ephemeris routines
   Fermi-Dirac functions

   Polyhedra volume
   Plane - cube intersection
   Coating an ellipsoid

   Nuclear reaction networks
   Nuclear statistical equilibrium
   Laminar deflagrations
   CJ detonations
   ZND detonations

   Fitting to conic sections
   Unusual linear algebra
   Derivatives on uneven grids
   Pentadiagonal solver
   Quadratics, Cubics, Quartics

   Supernova light curves
   Exact Riemann solutions
   1D PPM hydrodynamics
   Hydrodynamic test cases
   Galactic chemical evolution

   Universal two-body problem
   Circular and elliptical 3 body
   The pendulum


   Zingale's software
   Brown's dStar
   GR1D code
   Iliadis' STARLIB database
   Herwig's NuGRID
   Meyer's NetNuc
cococubed YouTube
Bicycle adventures
Public Outreach
Education materials
2023 ASU Solar Systems Astronomy
2023 ASU Energy in Everyday Life

AAS Journals
2023 AAS YouTube
2023 AAS Peer Review Workshops
2023 MESA VI
2023 MESA Marketplace
2023 MESA Classroom
2023 Neutrino Emission from Stars
2023 White Dwarfs & 12C(α,γ)16O
2022 Earendel, A Highly Magnified Star
2022 Black Hole Mass Spectrum
2022 MESA in Don't Look Up
2021 Bill Paxton, Tinsley Prize

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.

* *

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.


Number of vertices of the intersecting polygon.

Volume of the unit cube.


Please cite the relevant references if you publish a piece of work that use these codes, pieces of these codes, or modified versions of them. Offer co-authorship as appropriate.