Cococubed.com Volume from a plane slicing a cube
Home Astronomy Research    2024 Radiative Opacity    2024 Neutrino Emission from Stars    2023 White Dwarfs & 12C(α,γ)16O    2023 MESA VI    2022 Earendel, A Highly Magnified Star    2022 Black Hole Mass Spectrum    2021 Skye Equation of State    2021 White Dwarf Pulsations & 22Ne    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      Phyllotaxis      MESA      MESA-Web      FLASH      Zingale's software      Brown's dStar      GR1D code      Iliadis' STARLIB database      Herwig's NuGRID      Meyer's NetNuc AAS Journals    2024 AAS YouTube    2024 AAS Peer Review Workshops 2024 ASU Energy in Everyday Life 2024 MESA Classroom Outreach and Education Materials Other Stuff:    Bicycle Adventures    Illustrations    Presentations 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.
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