Print( "Optimisation time: " str( time. fmin_l_bfgs_b( ObjFun, z, bounds = b, m = 12, maxfun = 500000, approx_grad = True) # Gradient based fmin_l_bfgs_b from scipy.optimize # Randomly generate a sensible starting guess Show, in fact, that the area of that rectangle is r2. write( svgHeader( cols * w *( celldim margin) margin * celldim, rows * h *( celldim margin) margin * celldim))ī = * N Find the dimensions of the rectangle with the most area that can be inscribed in a semi-circle of radius r. The straight line distance he must paddle is a little harder. In this case, the lake has radius 5 mi., so the distance he must walk around the lake is mi. A window consisting of a rectangle topped by a semicircle is to have an outer. Outf = open( "BubblesOptArray100.svg", "w") # Prepare to draw circles in. A geometry theorem: An angle of measure /itex\theta /itex with vertex on a circle of radius r cuts of arc with angular measure and so length. Recall that the arc length of a circle with radius r and angle 0 is re. Line = 0.1 # Line thickness for graphics (native scale) Show that of all the rectangles with a given perimeter, the one with the. Margin = 1 # Gap between plots (native scale). of the largest isosceles triangle that can be inscribed in a circle of radius 4. Rows, cols = 12, 12 # Graphical array sizeĬelldim = 10 # Final graphics scaling factor W, h = 10, 10 # Native rectangle dimensions '''Write valid svg header specifying bounding box w wide and h high''' '''Draw a rectangle with top left corner x,y, width w, height h, line thickness s, fill colour fillcol, line colour strokecol, exploded slightly so inner edge is rectangle specified by 1st 4 params.''' Return( ''' \n''' %( x, y, r - 4 * s / 10, s, fillcol, strokecol))ĭef drawRect( x, y, w, h, s = 2, fillcol = "none", strokecol = "black"): ![]() '''Draw a circle with centre x,y radius r, line thickness s, fill colour fillcol and line colour strokecol.''' float)Ĭlist, clist, clist = x, y, rĭef drawCircle( x, y, r, s = 2, fillcol = "none", strokecol = "black"): '''Convert z vector to an array of circle centres and radii''' 2.1 Optimization models The problem is at one hand a geometrical problem and on the other hand a continuous global optimization problem. Problem 4 Determine the smallest square of side n that contains n points with mutual distance of at least 1. # Actual objective function (fraction of area covered) and non-overlapping circles where the radius of circles is 1. # Some linear inequality constraints to be satisfiedĬonstraints = numpy. # Split z into x,y,r triplets: z,z,z -> x1,y1,r1 Circle coords and dimensions are represented by a single list z''' '''Calculate packing density of N circles in W*H rectangle (N,W,H defined on function initialisation). Circle coords and dimensions are represented by a single list z'''Ī, b, c = ind, ind, ind correct expression for cost of two circles in terms of r (seen anywhere) A1. '''Function closure which initialises and creates a function for calculating the packing density of N circles in an W*H rectangle. A closed cylindrical can with radius r centimetres and height h centimetres. '''Generate all (2-way) unique pairings of N objects''' Also, a typographical error in calculating the ratio of a sphere’s volume to a cube’s has been corrected.'''Generate sensible initial guess vector (random circle coordinates and radii)''' The expression for A ( s) can be simplified slightly to s 2 ( 1 / ) ( 5 2 s) 2. ![]() If we allow 0 radius or side, as we should here, we have the bounds 0 s 2.5. ![]() This column has been revised to reflect that the arrangement of spheres in Exercise 2 is a "simple cubic" packing, not a "body-centered cubic" packing. Substituting in the formula for area, we see that we want to maximize/minimize A ( s), where. It is interesting to note that this is not the densest possible packing of octagons in the plane. How does that compare to the area of the hexagon? A hexagon of side length s is really six equilateral triangles of side length s, each with area $latex \frac$ ≈ 0.8284
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