BorlandTalk.com Borland discussion newsgroups   Archives   FAQ   Search   Memberlist   Usergroups   Register   Profile   Log in to check your private messages   Log in  Even rand distribution on part of a sphere surface...          BorlandTalk.com Forum Index -> Delphi Graphics View previous topic :: View next topic   Author Message Paul Nicholls Guest Posted: Fri Sep 10, 2004 3:19 am    Post subject: Even rand distribution on part of a sphere surface... Hi all, if I have a normal vector, N, on the surface of a sphere, is there an easy way of generating random points around the vector within an angle constraint, A, from the vector? Each point needs to be on the surface of the sphere... This means each random point needs to be clustered within the 'cone' formed around the vector by the angle, but on the sphere surface only... Thanks in advance, Paul Nicholls (Delphi 5/6 Professional) "The plastic veneer of civilization is easily melted in the heat of the moment" - Paul Nicholls [email]paul-nicholls (AT) hotmail (DOT) com[/email] Replace '-' with '_' to reply Back to top Marcus F. Guest Posted: Fri Sep 10, 2004 3:51 am    Post subject: Re: Even rand distribution on part of a sphere surface... "Paul Nicholls" <.> wrote Quote: Hi all, if I have a normal vector, N, on the surface of a sphere, is there an easy way of generating random points around the vector within an angle constraint, A, from the vector? Each point needs to be on the surface of the sphere... This means each random point needs to be clustered within the 'cone' formed around the vector by the angle, but on the sphere surface only... Without lack of generality, assume the vector N=[0,0,1]. Generate two uniformly distributed random numbers, "z" between cos(A) and 1, "phi" between 0 and 2*Pi. Then your random points are at [ (1-z^2)*cos(Phi), (1-z^2)*sin(Phi), z]. Back to top Paul Nicholls Guest Posted: Fri Sep 10, 2004 4:07 am    Post subject: Re: Even rand distribution on part of a sphere surface... "Marcus F." <a@b.com> wrote Quote: "Paul Nicholls" <.> wrote Hi all, if I have a normal vector, N, on the surface of a sphere, is there an easy way of generating random points around the vector within an angle constraint, A, from the vector? Each point needs to be on the surface of the sphere... This means each random point needs to be clustered within the 'cone' formed around the vector by the angle, but on the sphere surface only... Without lack of generality, assume the vector N=[0,0,1]. Generate two uniformly distributed random numbers, "z" between cos(A) and 1, "phi" between 0 and 2*Pi. Then your random points are at [ (1-z^2)*cos(Phi), (1-z^2)*sin(Phi), z]. I'm not sure what you meant about "Without lack of generality", and how does the resulting point change if the vector N is <> [0,0,1]? Cheers, Paul. Back to top Nils Haeck Guest Posted: Fri Sep 10, 2004 5:17 am    Post subject: Re: Even rand distribution on part of a sphere surface... Quote: I'm not sure what you meant about "Without lack of generality", and how does the resulting point change if the vector N is <> [0,0,1]? Just multiply the random vector with a matrix (3x3) which has the 3rd column the vector N (scaled to unity), the second column a vector Q that is perpendicular to N (also unity length), and the 1st column a vector P is the crossproduct of both. This is a orthonormal transformation. Something like: [Px Qx Nx] M = [Py Qy Ny] [Pz Qz Nz] So with Q dot N = 0 (just choose any Q that works) Q cross N = P Make sure that |Q| = 1 and |N| = 1, then |P| = 1 automatically You can verify that: [0] [Nx] M * [0] = [Ny] [1] [Nz] Just like that you multiply each Ri with M Ri-newframe = M * Ri-oldframe where Ri-oldframe is the random vector created with Marcus' method, and Ri-newframe is the one in the frame you want relative to [Nx, Ny, Nz] Hope that helps, Nils www.simdesign.nl Back to top Paul Nicholls Guest Posted: Fri Sep 10, 2004 5:33 am    Post subject: Re: Even rand distribution on part of a sphere surface... "Nils Haeck" <n.haeckno (AT) spamchello (DOT) nl> wrote Quote: I'm not sure what you meant about "Without lack of generality", and how does the resulting point change if the vector N is <> [0,0,1]? Just multiply the random vector with a matrix (3x3) which has the 3rd column the vector N (scaled to unity), the second column a vector Q that is perpendicular to N (also unity length), and the 1st column a vector P is the crossproduct of both. This is a orthonormal transformation. SNIP Thanks Nils, this makes sense :-) Cheers, Paul. 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