Re: fit plane to xz axis of data

*To*: mathgroup at smc.vnet.net*Subject*: [mg89522] Re: [mg89515] fit plane to xz axis of data*From*: "Szabolcs HorvÃt" <szhorvat at gmail.com>*Date*: Thu, 12 Jun 2008 02:55:38 -0400 (EDT)*References*: <200806110720.DAA15072@smc.vnet.net>

On Wed, Jun 11, 2008 at 10:20, will parr <willpowers69 at hotmail.com> wrote: > Dear Math Forum, > > I am having problems fitting a plane to some data. I am using the following procedure to fit the plane to my data (data is pasted at the bottom of this message): > > In[2]:= plane = Fit[data, {1, x, y}, {x, y}] > > Out[2]= 3.28723- 0.189001 x - 0.0874557 y > > Then the following to display the points and plane: > > Show[ListPointPlot3D[data, Boxed -> False, Axes -> True, > AxesEdge -> {{-1, -1}, {-1, -1}, {-1, -1}}, > AxesLabel -> {"x", "y", "z"}], > Plot3D[plane, {x, Min[data[[All, 1]]], Max[data[[All, 1]]]}, {y, > Min[data[[All, 2]]], Max[data[[All, 2]]]}, > PlotStyle -> Directive[Green, Opacity[0.5]], > Mesh -> None]] > > This works fine at fitting the plane in the xy axis, but I want to fit the plane to the xz axis. Does anyone know how this is done? You just need to permute the coordinates: Replace[data, {x_, y_, z_} :> {x, z, y}, 1] Or, if necessary, you can do an "isotropic" fitting (that prefers no axis) by calculating the distances from the plane. The original fitting attempt showed that the plane will not pass through {0,0,0} so we can use the equation a x + b y + c z == 1. The distance-squared of point {x,y,z} from this plane is (a x + b y + c z - 1)^2/(a^2 + b^2 + c^2), so the fitting can be done like this: In[2]:= planei = a x + b y + c z /. Last@NMinimize[ Expand@Total[ Function[{x, y, z}, (a x + b y + c z - 1)^2] @@@ data]/( a^2 + b^2 + c^2), {a, b, c}] Out[2]= 0.100048 x + 0.0136288 y + 0.0112116 z In[3]:= plane = Fit[data, {1, x, y}, {x, y}] Out[3]= 3.28723- 0.189001 x - 0.0874557 y In[4]:= plane2 = Fit[Replace[data, {x_, y_, z_} -> {x, z, y}, 1], {1, x, z}, {x, z}] Out[4]= 8.12817- 0.817828 x - 0.225109 z In[5]:= {{xa, xb}, {ya, yb}, {za, zb}} = {Min[#], Max[#]} & /@ Transpose[data] Out[5]= {{7.08299, 11.4614}, {-5.66284, 4.59986}, {-1.83404, 4.49073}} In[6]:= Show[ ContourPlot3D[planei == 1, {x, xa, xb}, {y, ya, yb}, {z, za, zb}, ContourStyle -> Directive[Blue, Opacity[.5]], Mesh -> None], Plot3D[plane, {x, xa, xb}, {y, ya, yb}, PlotStyle -> Directive[Green, Opacity[0.5]], Mesh -> None], ContourPlot3D[plane2 == y, {x, xa, xb}, {y, ya, yb}, {z, za, zb}, ContourStyle -> Directive[Red, Opacity[.5]], Mesh -> None], ListPointPlot3D[data, PlotStyle -> Black], PlotRange -> All, AxesLabel -> {"x", "y", "z"} ] Your points don't seem to lie on a plane, so the three approaches give give different results.

**References**:**fit plane to xz axis of data***From:*will parr <willpowers69@hotmail.com>