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Free UK Delivery on Eligible Order New upper bounds are given for the maximum number, τ n, of nonoverlapping unit spheres that can touch a unit sphere in n-dimensional Euclidean space, for n≤24. In particular it is shown that τ 8 = 240 and τ 24 = 196560 DIAMETER BOUNDS FOR EQUAL AREA PARTITIONS OF THE UNIT SPHERE PAUL LEOPARDI ∗ Abstract. The recursive zonal equal area (EQ) sphere partitioning algorithm is a practical algorithm for par-titioning higher dimensional spheres into regions of equal area and small diameter. Another such construction is due to Feige and Schechtman The unit sphere, centered at the origin in Rn, has a dense set of points with rational coordinates. We give an elementary proof of this fact that includes explicit bounds on the complexity of the coordinates: for every point v on the unit sphere in Rn; and every > 0; there is a point r = (r 1;r 2;:::;r n) such that: jjr vjj 1< : r is also a. However, [KL] does not study sphere packing di-rectly, but rather passes through the intermediate problem of spherical codes. In this paper, we develop linear programming bounds that apply directly to sphere packing, and study these bounds numerically to prove the best bounds known1 for sphere packing in dimensions 4 through 36. In dimensions 8.

In mathematics, a unit sphere is simply a sphere of radius one around a given center.More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of distance. A unit ball is the closed set of points of distance less than or equal to 1 from a fixed central point. Usually the center is at the origin of the space, so one. A unit sphere is a set of points that are distanced 1 unit away from a starting point (an origin ). The collection of points can exist in d -dimensions from one-dimensional to infinite-dimensional spaces The question proposes that the bounds of the integral over the interior of the unit sphere can be written as follows: ∫ − 1 1 ∫ − 1 − z 2 1 − z 2 ∫ − 1 − y 2 − z 2 1 − y 2 − z 2 x 2 + y 2 + z 2 d x d y d z. This is correct, although it is not to be recommended. (Integration over spherical coordinates, as shown in another. A globe showing the radial distance, polar angle and azimuthal angle of a point P with respect to a unit sphere, in the mathematics convention. In this image, r equals 4/6, θ equals 90°, and φ equals 30°. In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is. The other way to get this range is from the cone by itself. By first converting the equation into cylindrical coordinates and then into spherical coordinates we get the following, z = r ρ cos φ = ρ sin φ 1 = tan φ ⇒ φ = π 4 z = r ρ cos ⁡ φ = ρ sin ⁡ φ 1 = tan ⁡ φ ⇒ φ = π 4

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For the second part (ii), by exploiting a connection recently mentioned in between the bounds and cubature rules, we can rely on known results for cubature rules on the unit sphere to show tightness of the bounds. Organization of the paper In Sect. 2 we recall some previously known results that are most relevant to this paper NEW UPPER BOUNDS ON SPHERE PACKINGS I 691 The density ∆ of a packing is defined to be the fraction of space covered by the balls in the packing. Density is not necessarily well-defined for patho-logical packings, but in those cases one can take a lim sup of the densities for since a unit sphere has volume.

Let S be the closed surface that bounds the portion of the unit sphere x2 + y2 + x2 = 1 in the first octant. Namely, S comprises of the spherical portion x2 + y2 + 22 = 1, x,y,z > 0 and the portion of the three coordinate planes that are contained in the unit sphere and have positive x,y,z coordinates Unit sphere packings are the classical core of (discrete) geometry. We survey old as well new results giving an overview of the art of the matters. The emphases are on the following five topics: the contact number problem (generalizing the problem of kissing numbers), lower bounds for Voronoi cells (studying Voronoi cells from volumetric point. Then solve for z to find z = r*cos (ϕ). For x and y, we first have to find the component in the xy-plane, then use θ to solve for the two coordinates. The component of r in the xy-plane, which I'll refer to as R, is given by sin (ϕ) = R/r. Solving for R gives R = r*sin (ϕ)

New bounds on the number of unit spheres that can touch a

  1. This gives the upper half of a sphere. We wish to find the volume under this top half, then double it to find the total volume. The region we need to integrate over is the circle of radius \(a\), centered at the origin. Polar bounds for this equation are \(0\leq r\leq a\), \(0\leq\theta\leq2\pi\)
  2. Lower Bounds on Mean Curvature of Closed Curves Contained in Convex Boundaries Greg McNulty, Robert King, Haijian Lin, Sarah Mall August 13, 2004 Abstract We investigate a geometric inequality that states that in R2, the mean of the unit sphere is the same as the angle made between the line through th
  3. Spherical Discrepancy Minimization and Algorithmic Lower Bounds for Covering the Sphere Chris Jones Matt McPartlon June 14, 2020 Abstract.
  4. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): New upper bounds are given for the maximum number, 7m, of nonoverlapping unit spheres that can touch a unit sphere in n-dimensional Euclidean space, for n < 24. In particular it is shown that 78 = 240 and ~~~ = 196560. The problem of finding the maximum number, 7S, of billiard balls that can touch another billiard.
  5. This is an easy surface integral to calculate using the Divergence Theorem: ∭ E d i v ( F) d V = ∬ S = ∂ E F → ⋅ d S. However, to confirm the divergence theorem by the direct calculation of the surface integral, how should the bounds on the double integral for a unit ball be chosen? Since, d i v ( F →) = 0 in this case, hence, it's.

scene graph the Bounds object moves with the object that references it. I set up a simple program (below) that puts a sphere in the scene graph and then moves it by one unit. The bounds of the sphere both before and after the move is at the same location Abstract: It is known that the unit sphere, centered at the origin in Rn, has a dense set of points with rational coordinates. We give an elementary proof of this fact that includes explicit bounds on the complexity of the coordi-nates: for every point v on the unit sphere in Rn;. I am trying to write a function that generates a random walk bound by the unit sphere, resulting in the last location of the random walk given by a vector. In trying to figure out where my function is breaking, I have tried piecing it out like this: norm_vec <- function (x) sqrt (sum (x^2)) sphere_start_checker <- function () { repeat { start.

The basic idea is to simulate d independent standardized normal variables, project them radially onto the unit sphere, and then adjust their distance to the origin appropriately. You can find this algorithm in many textbooks (Devroye, 1986; Fishman, 1996), but Harman and Lacko (2010) summarize the process nicely For nitegraphs: hierarchies ofsemide nite upper bounds. (Lov asz-Schrijver 1991, Lasserre 2001, Laurent 2007) Forin nitegraphs: Generalization of Lasserre's hierarchy (de Laat-Vallentin 2015), related to the previous 2-point (Delsarte-Goethals-Seidel 1977) and 3-point bounds (Bachoc-Vallentin 2008). A unitary design is a collection of unitary matrices that approximates the entire unitary group, much like a spherical design approximates the entire unit sphere. We use irreducible representations of the unitary group to find a general lower bound on the size of a unitary t-design in U(d), for any d and t Proof. Let Pbe a sphere packing of unit spheres in R nwith density . Consider S 1 R, a n 1 dimensional sphere of radius R2[1;2], in Rn. It can be located such that it contains at least Rn center points of spheres in Pwhile none of them is concentric with Sn 1 R. This is the case as for a uniformly random location E E # centerpoints of Pin Sn 1. A 3D sphere is a 3-hypersphere and the unit sphere is a collection of points a distance of 1 from a fixed central point. The unit hypersphere is the next dimension up: a 4-hypersphere with a collection of points (x, y, u, v) so that x 2 + y 2 + u 2 + v 2 = 1. Add a fourth dimension to the unit sphere, and you get the unit hypersphere

the bounds (3) for the unit sphere: after showing in Section 3.1 that the convergence rate is in O(1=r2) we prove in Section 3.2 that the analysis is tight for nonzero linear polynomials. 4. Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere and its implementation in Matlab, provides numerical results and gives a sketch of the proof of the bounds on the diameter of regions. A companion paper gives details of the proof. Key words. sphere, partition, area, diameter, zone AMS subject classications. 11K38, 31-04, 51M15, 52C99, 74G65 1. Introduction. For dimension , the unit sphere.

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  1. In this section we will define the spherical coordinate system, yet another alternate coordinate system for the three dimensional coordinate system. This coordinates system is very useful for dealing with spherical objects. We will derive formulas to convert between cylindrical coordinates and spherical coordinates as well as between Cartesian and spherical coordinates (the more useful of the.
  2. Alternative method 1. An alternative method to generate uniformly disributed points on a unit sphere is to generate three standard normally distributed numbers X, Y, and Z to form a vector V = [ X, Y, Z]. Intuitively, this vector will have a uniformly random orientation in space, but will not lie on the sphere
  3. Example 15.8.6: Setting up a Triple Integral in Spherical Coordinates. Set up an integral for the volume of the region bounded by the cone z = √3(x2 + y2) and the hemisphere z = √4 − x2 − y2 (see the figure below). Figure 15.8.9: A region bounded below by a cone and above by a hemisphere. Solution
  4. Calculate surface integral where and S is the portion of the unit sphere in the first octant with outward orientation. 0. Hint. Use . Calculating Mass Flow Rate. Let represent a velocity field (with units of meters per second) of a fluid with constant density 80 kg/m 3
  5. ators in ε-approximations, under the ∥∥∞norm, with (√ 32⌈log 2d⌉/ε)2⌈log 2 d⌉. Based on this, rational approximation
  6. g bounds for error-correcting codes, and use it to prove upper bounds for the density of sphere packings, which are the best bounds known at least for di-mensions 4 through 36

The silhouette of the 3D sphere is the curve in 3D whose perspective projection into screen space is the 2D ellipse that exactly bounds the projection of the sphere. The silhouette is a circle in 3D (note that the silhouette circle has radius less than r, since less than half of the sphere is ever visible under perspective projection.) Th Sphere Packings Give an Explicit Bound for the Besicovitch Covering Theorem By John M. Sullivan ABSTRACT. We show that the number of disjointed families needed in the Besi-covitch Covering Theorem equals the number of unit spheres that can be packed into a ball of radius five, with one at the center, and get estimates on this number. 1. A sphere with radius 1 occupies a volume of (4/3)*π, which is about 4.18. A cube whose sides touch this sphere has each side of length 2, to give a volume of 8. The probability of a point chosen from a uniform random distribution in the cube being outside the sphere is (8-4.18)/8, which is 0.48. [return

Upper bounds for packings of spheres of several radii. We give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies. tance from the center of some sphere. Thus, a covering of the space is achieved if each sphere center is encompassed by a sphere of unit radius and the density of this covering (2)2−d Minkowski (1905) [ln(√ 2)d]2−d Davenport and Rogers (1947) (2d)2−d Ball (1992) TABLE 1. Dominant asymptotic behavior of lower bounds on φL max for large d F or d > 1, N > 2, there is a partition F S (d, N) of the unit sphere S d into N regions, with each re gion R ∈ F S ( d, N ) having ar ea Ω /N and Euclidean diameter bounded above b The bounds represent different things in each case. An axis-aligned bounding box, or AABB for short, is a box aligned with coordinate axes and fully enclosing some object. Because the box is never rotated with respect to the axes, it can be defined by just its center and extents, or alternatively by min and max points Abstract. We give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies through the use of semidefinite programming

circles on the normalized unit sphere. This algorithm not only bounds the search space but treats the finite vanishing points and the vanishing points at infinity with the same way. The experimental results on synthetic and real data show good performance of this algorithm. Keyword volume of sphere is the measure of space that can be occupied by a sphere. If we draw a circle on a sheet of paper, take a circular disc, paste a string along its diameter and rotate it along the string. This gives us the shape of a sphere. The unit of volume of a sphere is given as the (unit)3. The metric units of volume are cubic meters or cubi sphere packing, and study these bounds numerically to prove the best bounds known1 for sphere packing in dimensions 4 through 36. In dimensions 8 and 24, our bounds are very close to the densities of the known packings: they are too high by factors of 1.000001 and 1.0007071 in dimensions 8 and 24, respectively. (The best bounds previously known. because the unit sphere is K 2 and not a one dimensional circular graph. In higher dimensions, graphs are geometric if all vertices have the same dimension and every unit sphere is a sphere type graph. The smallest two dimensional one is the octahe-dron, where b 0 = b 2 = 1;b 1 = 0 and ˜(G) = b 0 b 1+b 2 = v e+f= 6 12+8 = 2 The size of the maximum independent set (MIS) in a graph G is called the independence number.The size of the minimum connected dominating set (MIN-CDS) in G is called the connected domination number.The aim of this paper is to determine two better upper bounds of the independence number; dependent on the connected domination number for a unit ball graph

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approximation bounds based on a bi-linear SDP relaxation and SOS techniques. When the unit sphere in (1.1) is replaced by a simplex, De Klerk, Laurent and Parrilo [7] proposed some polynomial time approximation schemes (PTASs) based on P¶olya's theorem or rational grid points, and proved some approximation bounds. De Klerk and Pasechnik [6. [12] Q. T. LE GiA AND I. H. SLOAN, The uniform norm of hyperinterpolation on the unit sphere in an arbitrary number of dimensions, Constr. Approx., 17 (2001), pp. 249-265. [13] P. LEOPARDI, Diameter bounds for equal area partitions of the unit sphere, in preparation, 2006 Lower bounds on lattice sieving and information set decoding. Elena Kirshanova Thijs Laarhoven. 3rd PQC Standardization Conference. June 4, 2021 To appear at Crypto'2 graphs for which unit spheres of a graph satisfy properties familiar to unit spheres in Rd. In that case, the results look more similar to differential geometry [8]. The curvature for three-dimensional graphs for example is zero everywhere and positive sectional curvature everywhere leads to definite bounds on the diameter of the graph

The bounds are obtained by a new method for obtaining good polynomials required in a linear programming bound due to Delsarte, Goethals and Seidel. (24) points on the unit sphere in the space. We study the convergence rate of a hierarchy of upper bounds for polynomial minimization problems, proposed by Lasserre (SIAM J Optim 21(3):864-885, 2011), for the special case when the feasible set is the unit (hyper)sphere. The upper bound at level r∈ N of the hierarchy is defined as the minimal expected value of the polynomial over all. adshelp[at]cfa.harvard.edu The ADS is operated by the Smithsonian Astrophysical Observatory under NASA Cooperative Agreement NNX16AC86

Coxeter proposed upper bounds on k(n) in 1963 [10]; for n =4,5,6, 7, and 8 these bounds were 26, 48, 85, 146, and 244, respectively. Coxeter's bounds are based on the conjecture that equal size spherical caps on a sphere can be packed no denser than packing where the Delaunay triangulation wit the unit sphere Sd 1. We are interested in computing an -approximation y2Qd for x, that is exactly on Sd 1 and has low bit size. We revise lower bounds on rational approximations and provide explicit, spherical instances. We prove that oating-point numbers can only pro-vide trivial solutions to the sphere equation in R2 and R3. Moreover, we.

Recently, Schmutz provided an divide-&-conquer approach on the sphere equation, using Diophantine approximation by continued fractions, to derive points in Q d ∩ S d − 1 for a point on the unit sphere S d − 1. The main theorem bounds the denominators in ε-approximations, under the ∥ ∥ ∞ norm, with (√ 32 ⌈ log 2 d ⌉ / ε) 2. We study the convergence rate of a hierarchy of upper bounds for polynomial minimization problems, proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp. 864-885], for the special case when the feasible set is the unit (hyper)sphere. The upper bound at level r of the hierarchy is defined as the minimal expected value of the polynomial over all probability distributions on the sphere, when the.

integration - Triple integral in a sphere - Mathematics

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One is the Separating Axis Theorem. But this only works on convex meshes. This is why Unity actually has a tick box to perform collision detection on a mesh with convex mesh only. You'll actually notice that the IGeom interface in that framework has a method called Project The result is a PAC-bound that is tighter when the base learner adapts quickly, which is precisely the goal of meta-learning. We show that our bound provides a tighter guarantee than other bounds on a toy non-convex problem on the unit sphere and a text-based classification example In Proposition 1, we deliver lower bounds of the regularizing (φ, ψ)-admissible constants C φ and C ψ, depending on the dimension of the latent space d and on the levels (L φ, L ψ) of approximately angle preserving property. These bounds ensure the non-emptiness of the sets M C φ and M C ψ The Hamming or sphere-packing bound gave an upper bound on the size (or rate) of codes, these bounds in terms of identifying the largest possible distance for asymptotically good binary of pairwise far-apart unit vectors in Euclidean space, and then use a geometric argument for the.

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of how to put n non-overlappingspherical caps of maximum equal radius on the unit sphere. 2.1. An Upper Limit. The following inequality of L. Fejes-T6th [4J gives a useful upper bound on dn : THEOREM 2.1. Given n > 2 points on the surface ofthe unit sphere, there always exist two having spherical distance _1{cot 2 W-l} Q ~cos 2 where n 1r w. Theorem 17.5 (Isoperimetry Theorem for Sphere) Let A ⊂ Sn−1 be a measurable sub- set of the unit sphere Sn−1 and let C be a spherical cap on the sphere Sn−1 of the same surface area as A, then for all > 0, µ(A ) ≥ µ(C ). Or equivalently, among all (measurable) patches on the unit sphere, the spherical cap has th

Calculus III - Triple Integrals in Spherical Coordinate

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Given a unit sphere, a spherical triangle on the surface of the sphere is defined by the. Spherical trigonometry (5,650 words) New bounds on the number of unit spheres that can touch a unit sphere in n dimensions. Journal of Combinatorial Theory. A26: 210-214. CiteSeerX 10 Each non-zero point in Rd identifies one closest point x on the unit sphere Sd-1. We are interested in computing an ε-approximation y ∈ Qd for x, that is exactly on Sd-1 and has low bit size. We revise lower bounds on rational approximations and provide explicit, spherical instances. We prove that floating-point numbers can only provide trivial solutions to the sphere equation in R2 and R3 First construction: Inverse gnomonic projection. Our first construction is based on lower bounds for the Szemerédi-Trotter problem (e.g., see the first two posts of this series). It shows that points and unit spheres in can yield incidences. For the requested values of and , consider a set of points and a set of lines, both in and with Usage spheres = pack([opt]) Packs 3D spheres (default) or 2D circles with the given options: dimensions — Can either be 3 (default) for spheres, or 2 for circles; bounds — The normalized bounding box from -1.0 to 1.0 that spheres are randomly generated within and clip to, default 1.0; packAttempts — Number of attempts per sphere to pack within the space, default 50 Since by definition, we have a d-dimensional graph G, the unit sphere S(p) at a point is a (d−1)-dimensional graph. While Puiseux type formulas allow to discretize curvature in two dimensions, the lack of a natural second order differ- Schoenberg-Myers bounds show that there are only finitely many d ≥ 2 dimensiona

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an integral over the unit sphere S2 = fx 2R3: kxk 2 = 1g (1) I[f] Z S2 f(x)d = Z 2ˇ 0 Z ˇ 0 f('; )sin'd'd ; where f : S2!R and the second integral is in the spherical coordinates repre-sentation. The case of vectorial functions is a trivial extension from this scalar problem. Often the precise form of f is either too complicated to. Recently A. Schrijver derived new upper bounds for binary codes using semidefinite programming. In this paper we adapt this approach to codes on the unit sphere and we compute new upper bounds for the kissing number in several dimensions. In particular our computations give the (known) values for the cases n = 3, 4, 8, 24 sphere. For large d, almost all the volume of the cube is located outside the sphere. 1.2.2 Volume and Surface Area of the Unit Sphere For xed dimension d, the volume of a sphere is a function of its radius and grows as rd. For xed radius, the volume of a sphere is a function of the dimension of the space dimensional sphere, the set of points in R4 that are one unit away from the origin. By doing so, we establish a geometric setting for electrodynamics in positive curvature. When applied to a vector field, the Biot-Savart operator behaves like a magnetic field; we display suitable electric fields so tha Let R n, S n− 1 (x, r)⊂ R n, x∈ R n be the n− dimensional Euclidean space and sphere of radius r with the center in x. Denote S n− 1∆= S n− 1 (0, 1). Let B n (x, r)⊂ R n be the (closed) ball of radius r with the center in x. We say that (finite) se

OpenGL rendering: All vertices move to bounds of unit spher

The following theorems show upper and lower bounds on the length of arcs in the Delaunay graph on a d-sphere. Theorem 3. Given the Delaunay graph D(P) of a set Pof n>2 points distributed uniformly and independently at random in a unit d-sphere, with probability at least 1 , for 0 <<1, there is no arc abb2D(P), a;b2P, such that A d(1; (a;b. sphere is the sphere plus the region it bounds, speci ed as jX Cj r. An in nite single-sided cone with vertex V, axis ray with origin at V and unit-length direction A, and cone angle 2(0;ˇ=2) is de ned by the set of points X such that the vector X V forms an angle with A Bounds of Eigenvalues 243 as ‚ ! +1, where!n is the volume of the unit solid ball in Rn, and N(‚) is the number of eigenvalues less than or equal to ‚, multiplicity counted.Courant's nodal domain theorem states that the number of nodal domains of the k-th eigen- function is less than or equal to k+1.These theorems are of fundamental impor In fact, for a sphere of radius \(r\), as \(d \to infty\), almost all the volume is contained in an annulus of width \(r/d\) near the boundary of the sphere. And since the volume of the unit sphere goes to 0 while the volume of unit sphere is constant at 1 while \(d\) goes to infinity, essentially all the volume is contained in the corners.

>C + xy plane bounds the portion of the unit sphere S

The surface of the sphere is always perpendicular to its outward motion. Therefore the surface area is the derivative of volume. Differentiate r n to get an extra factor of n. If a n is the surface area then a n = nv n . In other words, the surface area of the unit hypersphere is volume times dimension Over the Unit Sphere Vijay Bhattiprolu Mrinalkanti Ghosh† Venkatesan Guruswami‡ Euiwoong Lee§ Madhur Tulsiani ¶ November 18, 2016 Abstract We consider the following basic problem: given an n-variate degree-d homogeneous poly-nomial f with real coefficients, compute a unit vector x 2Rn that maximizes jf(x)j. Beside It shows that calculating ex,ey,ez (extents) takes twice as long for bounds-bounds compared to sphere-bounds. The center of the sphere was originally at (0,0,0), but I changed that to ensure it was't giving the sphere an unfair advantage. I think both tests include the same number of additions/subtractions when calculating extents (6) a sphere. Also the next de nitions are inductive: a d-graph is a graph for which every unit sphere is a (d 1)-sphere; a d-graph with boundary is a graph for which every unit sphere is a (d 1)-sphere or (d 1)-ball; a d-ball is a d-graph with boundary for which the boundary is a (d 1)-sphere. Let w k(G m) denote the number of K k+1 subgraphs in. For a sphere centered at a point (x o,y o,z o) the equation is simply (x - x o) 2 + (y - y o) 2 + (z - z o) 2 = r 2. If the expression on the left is less than r 2 then the point (x,y,z) is on the interior of the sphere, if greater than r 2 it is on the exterior of the sphere. A sphere may be defined parametrically in terms of (u,v) x = x o + r.

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points around the resulting circles (1-spheres). Given a unit 2-sphere and an angle of separation. of (say) 30 degrees, we can slice the sphere into. polar points and intermediate circles at the. equator and at 30 and 60 degrees north and south. latitude. No point on any one slice can be closer bounds can also be derived from analysis in the Grassmann manifold directly. Let B(δ) denote a metric ball of radius δ in G n,p (L). The sphere packing bounds can be derived from the volume of B(δ) [3]. The exact volume formula for a B(δ) in G n,p (L) with p = 1 and L = C is derived in [4]. An asymptotic volume formula for a B(δ) in The GU_Detail::polyIsoSurface() method evaluates the callback function densityFunction() inside the bounding box specified by bounds. A mesh of polygons is created and saved to sphere.bgeo. Try experimenting by changing the implementation of densityFunction() and see what kind of surfaces you can create