Mathematical Pointillism

• In mathematics field, relative to spots, dots science : included spots constraints
  Normal metric : ,
  $$d(A,B)= || \overrightarrow{AB}||_2 = || \overrightarrow{B}-\overrightarrow{A}||_2 =\sqrt{(x_B-x_A)^2+(y_B-y_A)^2+(z_B-z_A)^2}$$
  Proposition(
  R : radius of sphere
  Exhaustive for comprehension
  $$x = d(U,V)\cdot cos(U)$$,$$y = d(U,V)\cdot sin(U)$$, $$z = R\cdot sin(\frac{2\cdot\pi\cdot d(U,V)}{2\cdot \pi\cdot R}\cdot 2\cdot \pi) = R\cdot sin(\frac{2\cdot\pi\cdot d(U,V)}{R})$$
  )
  Surface metrics distances ( breadcrumbs : surface integrals, minimal path (Dijkstra's algorithm when discrete), geodesics : Constraints on the surface : sphere, donut(tore) + iso-distance OR-tools, Scilab Scicos
  Example 1 : 3 dots iso-distance on sphere gives a triangle
  Example 2 : known a cone equation, find equilateral triangle whose $$(max za - min zc) > 10$$ for instance or .........................................................whose $$a=(xa,ya,za)$$
  Example 3 : known 5 locations on Earth, find 3 36000 altitude satellites, so that average distance(sum(sum, by satellites of distances)) is minimal
  Example 4 : $$1 < d(x,y) < 2$$ $$\equiv$$ $$d(x,y)>1$$ AND $$d(x,y)<2$$ $$\equiv$$
  
  $$ \left\{ \begin{array}{c} d(x,y)>1 \\ d(x,y)<2 \\ \end{array} \right. $$
  ... Relative to CP i.e. Constraint Programming(Python Google ORtools, swi-prolog), graphs, GPS(Earth locations, satellites best spread), CSP and COP
  Example 5 : density(-> continuum : discontinuous to continuous link through dot of highest density ~ shortest path (Dijkstra,...)), chemistry Pauli orbitals distributions (->stochastics)
  Example 6 : Penrose related : with France frontiers(), find gravity point, 100km x 100km squares (or less by frontiers) so that number of squares is minimal, or average real polygone surface is max, and so on.
  Example 7 : 5 iso distance dots on a donut torus, with first one not given on inside circumference
• Subfamily of solving
  • discrete equations
  • implicit equations
  -> Boolean graphs (when not easyly solvable, use brute force pixel loops -> inequalities -> dots are coarse summed up to lines ( $$x^2 + y^2=1$$) or area patterns)
• Astronomy where stars can be related to dots
• Fractals

By Peter MOUEZA
Index des ressources