       RE: Re: Generating Two Unit Orthogonal Vectors to a 3D Vector

• To: mathgroup at smc.vnet.net
• Subject: [mg36413] RE: [mg36377] Re: [mg36352] Generating Two Unit Orthogonal Vectors to a 3D Vector
• From: "DrBob" <drbob at bigfoot.com>
• Date: Fri, 6 Sep 2002 03:16:42 -0400 (EDT)
• Reply-to: <drbob at bigfoot.com>
• Sender: owner-wri-mathgroup at wolfram.com

I timed Daniel's three solutions and Gary's one (plus a couple of my own
a little later):

perps1[v_] := If[v[] ==
v[] == 0, {{1, 0, 0}, {0, 1, 0}}, {{v[], -v[[
1]], 0}, Cross[v, {v[], -v[], 0}]}]
perps2[v_] := With[{vecs = NullSpace[{v}]}, {vecs[[
1]], vecs[] - (vecs[].vecs[])*vecs[]}]
perps2C = Compile[{{v, _Real, 1}}, Module[{vecs = NullSpace[{v}]}, {
vecs[], vecs[] - (vecs[].vecs[])*vecs[]}]]
helzer[v : {a1_, a2_,
a3_}] := With[{w = First[Sort[{{a2, -a1, 0}, {a3,
0, -a1}, {0, a3, -a2}}, OrderedQ[{Plus @@ Abs[#2], Plus @@ \
Abs[#1]}] &]]}, {w, Cross[v, w]}]

vecs = Table[Random[], {10000}, {3}];
Timing[perps1 /@ vecs; ]
Timing[perps2 /@ vecs; ]
Timing[perps2c /@ vecs; ]
Timing[helzer /@ vecs; ]

{1.7350000000000012*Second,   Null}
{0.5619999999999994*Second,   Null}
{0.219 Second, Null}
{2.7349999999999994*Second,   Null}

I made a small change to Daniel's perps1, and got a solution as fast as
perps2c, WITHOUT compiling.  Compiling tripled the speed again, so
"treatC" is the fastest solution I've seen so far.

treat[{a_, b_, c_}] :=
If[a == b == 0, {{1, 0, 0}, {0, 1, 0}}, {{b, -a, 0}, {a*c, b*c, -a^2 -
b^2}}]
treatC = Compile[{{v, _Real, 1}},
If[v[] == v[] == 0,
{{1, 0, 0}, {0, 1, 0}},
{{v[], -v[], 0},
{v[]*v[], v[]*v[],
-v[]^2 - v[]^2}}]]
vecs = Table[Random[],     {10000}, {3}];
Timing[perps2c /@ vecs; ]
Timing[helzer /@ vecs; ]
Timing[treat /@ vecs; ]
Timing[treatC /@ vecs;]

{0.2190000000000083*Second,   Null}
{2.7339999999999947*Second,   Null}
{0.25*Second, Null}
{0.07800000000000296*Second,   Null}

None of these solutions reliably return normalized vectors.

Bobby Treat

-----Original Message-----
From: Daniel Lichtblau [mailto:danl at wolfram.com]
To: mathgroup at smc.vnet.net
Subject: [mg36413] [mg36377] Re: [mg36352] Generating Two Unit Orthogonal Vectors
to a 3D Vector

David Park wrote:
>
> There are many cases in graphics, and otherwise, where it is useful to
> obtain two orthogonal unit vectors to a given vector. I know a number
of
> ways to do it, but they all seem to be slightly inelegant. I thought I
would
> pose the problem to MathGroup. Who has the most elegant Mathematica
> routine...
>
> OrthogonalUnitVectors::usage = "OrthogonalUnitVectors[v:{_,_,_}] will
return
> two unit vectors orthogonal to each other and to v."
>
> You can assume that v is nonzero.
>
> David Park
> djmp at earthlink.net

Some possibilities:

perps1[v_] := If [v[]==v[]==0,
{{1,0,0},{0,1,0}},
{{v[],-v[],0}, Cross[v,{v[],-v[],0}]}
]

perps2[v_] := With[{vecs=NullSpace[{v}]},
{vecs[], vecs[] - (vecs[].vecs[])*vecs[]}
]

This appears to be 2-3 times faster than perps1 for vectors of machine
reals. I get another factor of 2 using Compile, which is appropriate for
e.g. graphics use.

perps2C = Compile[{{v,_Real,1}},
Module[{vecs=NullSpace[{v}]},
{vecs[], vecs[] - (vecs[].vecs[])*vecs[]}
]]

In:= vecs = Table[Random[], {10000}, {3}];

In:= Timing[p2 = Map[perps1C,vecs];]
Out= {0.49 Second, Null}

This is on a 1.5 GHz processor.

Daniel Lichtblau
Wolfram Research

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