

Home 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 FermiDirac 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 twobody problem Circular and elliptical 3 body The pendulum Phyllotaxis MESA MESAWeb FLASH Zingale's software Brown's dStar GR1D code Iliadis' STARLIB database Herwig's NuGRID Meyer's NetNuc Presentations Illustrations 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 & ^{12}C(α,γ)^{16}O 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.



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 coauthorship as appropriate. 
