Wrong result for best fit plane to set of points w

Question

I have a set of (x, y, z) points for which I need to find the plane that best fits them. A plane is defined by its coefficients as:

a*x + b*y + c*z + d = 0

or equivalently:

A*X +B*y + C = z

The second equation is just a re-write of the first.

I'm using the method developed in this gist, which is a translation to Python from the Matlab code given in this answer. The method finds the coefficients to define the plane equation that best fits the set of points.

The issue is that I am able to come up with a set of coefficients that gives a better fit to that set of points.

To define "better", I calculate the sum of absolute distances of each point to the given plane, following the math given here. A smaller value, means a "better" fit, since the points are then on average closer to the plane.

The MWE is below. As can be seen, the hand-picked coefficients result in a smaller sum of absolute distance values (~155.89), than using the "best" coefficients found by the method described above (~158.78).

What am I missing here?

---

MWE

import numpy as np
import scipy.linalg


def sum_dist_2_plane(x, y, z, a, b, c, d):
    """
    Sum of the absolute values of the distances to a plane, given by the
    a,b,c,d coefficients, for the set of points defined by x,y,z.
    """
    return np.sum(abs(a*x + b*y + c*z + d)/np.sqrt(a**2+b**2+c**2), axis=0)


# Some xyz points.
xyz = np.array([[1.1724546888698482, 0.67037911349217505, 1.6014525241637045], [2.0029440384631063, 1.2163076402918147, -1.1082409593302032], [-0.87863180025363918, 1.261853987259635, 1.1598532675831237], [0.42789396045777467, 0.67325845732274703, 1.1421266649135475], [1.366142552248496, 1.0959456367043121, -1.6046393305927751], [-2.1595534005011485, -2.2582441035518794, -1.0663372184011806], [2.1104543583371633, -2.3711560770628917, 0.33077589412150843], [1.1974640975387107, 1.2100068141421523, 0.71395322259985505], [0.44492797840962123, 0.51098686422493145, 0.23383900276620295], [-2.0810094204638281, -2.11327958929372, -1.0758230448163033], [1.1655230345226737, 2.3777304002844968, -1.5663228128649394], [0.90952208156596781, 0.84978064084217519, 1.5986081506274985], [1.2951624720758836, 1.2231899029278033, 1.6154291293114866], [0.97545563477882025, 1.1844143994262264, 0.25292733170194026], [2.0281659385206012, 1.3370146330231019, 1.1961575550766028], [-1.9843445684092424, -0.012247402159192651, -2.0732736152121092], [1.0852175044560746, 1.8083916604163963, 0.27402181385868829], [-0.97983337631837208, 1.1032503818628847, 1.1579341604311182], [2.5033961310304029, 1.5628354191569325, -0.60785250636200061], [0.84123393662217383, 1.6169587554844618, -0.66116704633280676], [-1.8572657771039134, 0.043103553120073364, -2.0779545355975415], [2.6979128603518787, 1.70987170366249, -0.59306759275995091], [1.898614831265683, -2.9411794973775129, 1.7095862940118209], [0.81052668401212824, 0.89107411631439926, 1.597589407046101], [-2.0466083174114331, 0.14841369250699468, -1.120794708199135], [2.7004384737959648, 1.3616954868011328, 1.2294957766312749], [2.5373220833750385, 1.7067484497548233, 0.32345763726774379], [0.42025310188487158, 0.25762913945011717, -2.5899822318304473], [1.0425582222020597, 1.2902156453507225, 1.1638276333984123], [1.8492329386150801, 1.369745208770941, -1.1101559957041474], [-1.9685282554587256, -0.053725287173628226, 0.26827797508054374], [2.1798881190450285, 1.2454661605758286, -1.5732113885771071], [2.097212096433736, -2.9271738140601462, -0.56568133063870363], [-4.0108387171254396, -0.95559594599890008, 1.7588521192455815], [1.1558287640906737, 0.84330421357278096, 1.1565989504480143], [-2.9571643443632118, -2.847346163285049, 1.3087401683271338], [1.8592900784537116, 1.3952561066549967, 0.28365423946831214], [-3.4841441062982867, -3.0501496018162109, -0.48161393173162992], [2.5524429115550777, 0.62723764313314334, 0.29882336571990464], [2.2267279436912251, -3.8561674586606758, 1.3393813829669483], [2.1214758016437449, -0.20203416631090113, -1.5903243997743601], [0.14882165322179747, 0.4127883227210779, 0.23115527212661391], [1.2042041122995621, 1.2013226392201846, -0.2014020012510187], [-0.91807770884292583, 1.1176994160488214, -2.5723612427329385], [1.910565457302241, 1.1857852625952567, -1.5853233609652335], [1.0660312416826301, 1.3594393638452948, 0.71483235729161265], [0.65109075860726373, 0.58395151990229632, 1.590486638605114], [2.0967121651174518, 3.5121496638531586, 0.85481080660772335], [1.1484000297535542, 0.93256813649663772, 0.25125672956252743], [-1.7670514601312102, 0.17479726844255272, 0.26097336908379276], [-0.38814151285133675, -1.36837872393391, -2.0916940966530149], [1.5825758742579219, -0.34854211056693962, 0.2556641250097158], [2.586881293405797, -4.371974479474976, -2.3458559556297445], [0.22496107684878977, 0.26917053206799602, -0.69280100767942088], [-0.92198332953292639, 5.3103622894708327, 1.4344469946544294], [1.5669967464035819, -0.13527817891479368, 1.6081806927677107], [-0.56872000311273319, -1.9823395333139691, -2.5517609300755879], [-3.7708737466313824, -3.2863308845331081, 1.3928734104180975], [0.26086111146896701, 0.91063726352187491, -2.1025221562973897], [4.3490818342473947, 1.7969605233982313, -0.94470942930075807], [0.8202509554992351, 1.6178074457637883, -0.66148472916848533], [-1.5947972211483237, 0.18933818654144918, -0.20453683465790107], [0.9736103155058905, 1.4905334895713331, -2.0806647444063202], [1.2838541958241105, 2.0842224244281931, -0.17045822168000058], [3.7985716232291624, 2.5292902540646183, -0.022070946178700979], [1.175697191763003, 0.70063646974704663, 0.24808027552254686], [1.7834118390535998, 1.2937296781793448, -0.1818232448888395], [1.1281441478154344, 0.89641394438231292, 1.6040641573676311], [-2.0118889302553362, 2.7916846393274373, -0.57683324778643197], [-0.5995803308341846, -2.2434949940054554, 0.2835440401850704], [0.32077033536702831, -0.95844872063257081, -1.6245015133016167], [0.81357199339193753, 1.5540883407880133, -0.19956720143058249], [0.62611590692268004, 2.5129849486626958, -0.62767513959140331], [1.3018663649626585, 0.92514176013041427, 0.71042211390030729], [-0.72715254964437737, -2.3705643250823436, -0.63320562968051775], [1.9172742234794142, -2.8680592171367834, -1.9965843559235594], [-0.7108415762295921, -2.2783943434144658, -0.63767826146936812], [1.968546542650037, -2.8305910089272146, -0.11154135958968681], [-3.1492524087194655, -2.8503098024243823, -0.049957063615551078], [-4.0600431110777313, -0.97891479243488955, -0.055962425569617835], [-3.3752702254780629, 5.7587998072406652, 2.0459797674238658], [-1.9855135921592455, 2.7466682542750638, -0.58034791274582886], [2.033073141968945, 1.5208650449610079, -0.16592183863411947], [-1.0379089220195949, -4.7336396164389383, 0.0045652508195388464], [0.059579198580756186, 0.50654688886459498, -0.69144595015375643], [2.1785293390435458, -2.67576518666927, -2.4787451249989232], [2.1096278381494935, -0.41668256763302775, -2.5482230530414327], [2.898772426390924, 1.9762337520130302, 1.2619960149795091], [0.95620776766155502, 1.4639884373148864, -0.19976180368861662], [0.78751831482788348, 1.6888070662998231, -1.1280318812973462], [0.75574071441925506, -0.89893698883953688, -0.21651308186821439], [-0.26825101547751962, -3.4496728096007274, 1.7066486428460195], [1.6690385240329706, -0.49893224975237227, -0.66401176702524367], [-0.28877792353045606, 1.5139628395303639, 0.25314013342428154], [0.33435105972001761, 0.72567663189581422, -2.5862147225048417], [-0.29757422904759573, 1.5866751937867298, -0.6682501010682671], [2.7581055173587461, -3.973585217996157, 0.0036824743223959899], [-3.4344275379769509, -3.089933175898083, 0.44457796620464052], [-2.9394415977285413, -2.6122275577950083, 1.2944549102942418], [2.0038460695984823, 1.515512638618338, -1.5731231727332897], [2.206216953170296, 1.4688891052013793, -1.5661966567970254], [-1.035208468220836, 4.4666436487176657, 0.89858770640569929], [-2.0039938640838546, 0.24894412179006209, -1.1220951191237916], [-3.9104727661324539, -0.70689702779279451, 1.2978242803460915], [1.7290487193475563, 1.2850859351795931, -0.18395259620439219], [1.1198244545179541, 1.7335817969585154, -0.18776435816536718], [0.32239533364835676, 0.2896168073626299, -1.1602117002106667], [0.36649393980823192, 0.28244286109766281, -0.69190114531475189], [0.71629324271161154, 0.62574841994964003, 1.1448784055936088], [-0.65109499789331204, -1.3933343864454197, -2.0884024350786063], [0.97046822380567643, 1.5321191441287463, -0.19744980702830617], [-0.9585141324426697, 1.3494884330155692, 1.610936445675776], [0.9615111008482673, 2.4535668843530907, -1.0939899554364985], [-1.0667872216702354, 0.9585914740866075, 1.6038639420443772], [1.8021244106955299, 1.1320598433704154, 1.1820726259869971], [-0.060098920604716666, 0.46839599864404674, 2.0277692055269654], [0.1721690681247055, 0.33837718694053642, 1.137078044079125], [-1.5964760388322969, 0.29775223476696611, 1.1626558382504655], [2.233093222044507, -2.8349614127699461, 0.36052101139762271], [1.9257633093026034, -2.5325763598899247, -1.5360887301240496], [1.116293873468281, 0.82698434754975214, -2.5739062165349651], [1.1781306304855363, 0.67917370389645249, 1.6017135739225736], [-1.8600651472693519, 0.078727875114422086, 1.6184578422253679], [-1.43994317003447, 0.13431327308359137, 2.0472930703748276], [0.84521838040660946, 0.63970047924770745, -2.100345751420285], [1.7661749989776647, -0.37651847162651875, -2.0797840873592222], [0.83547092354865804, 1.7219104152802622, 0.2661115369175846], [1.8300570222025725, -0.28592323411250137, 1.6180934388285593], [-0.62076647836845089, -0.99191053757063119, -1.1486388713745725], [-1.6239006006253158, 0.41366361326031414, 0.2574990624750626], [0.89195815704237569, 2.2004172385784, -0.17400231396826626], [0.36791088305589931, 0.36096348396301231, -2.5897662606427687], [0.073648763901347059, 0.19675260582587464, -2.1107265203482299], [2.161140531872539, -2.842373820387067, 0.35775402140617274], [-2.0416416353442859, -4.4051625504298446, 0.0054589213454931951], [-2.0525396585901774, 3.6758248479033888, -2.4231570023949089], [-0.96441167578601306, -4.6667609706070516, -0.0032107139968431397], [-1.8689820843196163, 0.021432805852950151, 0.26440433366338567], [-0.15613351765730205, -1.0964152703770347, 1.5952653951331826], [-0.91084152695600051, 1.2388514346844914, 1.1598544561959656], [0.94699177145572266, 1.2276340276860185, 2.0505581774713733], [-0.8929399989505632, 1.2806485400811793, -0.20595242802870217], [1.2023125342023806, 2.3477287603163717, -1.5668539565738087], [1.1651535046949058, 1.3836371788871575, 0.26217241277176129], [-1.0929407572158512, 1.3887078738113698, -0.19910861560325088], [-0.76452840903206265, 1.4237410113821392, -1.6090659495628117], [-1.5594385646555604, 0.1455415355638911, 1.1607640518832483], [-0.59734981961340872, -1.2800366176149909, 1.6032259368271653], [1.2325774703556955, 0.80804053623212702, 0.25109224401040819], [1.177240124012167, 0.90163100927998241, -1.1405108476689563]])
x, y, z = xyz[:, 0], xyz[:, 1], xyz[:, 2]

# Best-fit linear plane, for the Eq: z = a*x + b*y + c.
# See: https://gist.github.com/amroamroamro/1db8d69b4b65e8bc66a6
A = np.c_[x, y, np.ones(xyz.shape[0])]
C, _, _, _ = scipy.linalg.lstsq(A, z)

# Coefficients in the form: a*x + b*y + c*z + d = 0.
a, b, c, d = C[0], C[1], -1., C[2]

# Sum of absolute distances of each point to this plane.
print sum_dist_2_plane(x, y, z, a, b, c, d)

# Hand-picked coefficients.
a, b, c, d = 0.28, -0.14, 0.95, 0.

# Sum of absolute distances of each point to this plane.
print sum_dist_2_plane(x, y, z, a, b, c, d)

Answer 1

The formula for the plane can be written as

z_plane = a*x + b*y + d

The vertical z-distance from the point to the plane is given by

|z_plane - z| = |a*x + b*y + d - z|

scipy.linalg.lstsq minimizes the square of the sum of these distances.

def zerror(x, y, z, a, b, d):
    return (((a*x + b*y + d) - z)**2).sum()

Indeed, the parameters returned by scipy.linalg.lstsq yield a smaller zerror than the hand-picked values:

In [113]: zerror(x, y, z, C[0], C[1], C[2])
Out[113]: 245.03516402045813

In [114]: zerror(x, y, z, 0.28, -0.14, 0.)
Out[114]: 323.81785779708787

---

The formula

gives the perpendicular distance between the point (x_0, y_0, z_0) and the plane, ax + by + cz + d = 0.

---

You could minimize the perpendicular distance to the plane using scipy.optimize.minimize (see minimize_perp_distance below).

import math
import numpy as np
import scipy.optimize as optimize
import matplotlib.pyplot as plt
import mpl_toolkits.mplot3d.axes3d as axes3d
np.random.seed(2016)

mean = np.array([0.0,0.0,0.0])
cov = np.array([[1.0,-0.5,0.8], [-0.5,1.1,0.0], [0.8,0.0,1.0]])
xyz = np.random.multivariate_normal(mean, cov, 50)
x, y, z = xyz[:, 0], xyz[:, 1], xyz[:, 2]

def minimize_z_error(x, y, z):
    # Best-fit linear plane, for the Eq: z = a*x + b*y + c.
    # See: https://gist.github.com/amroamroamro/1db8d69b4b65e8bc66a6
    A = np.c_[x, y, np.ones(x.shape)]
    C, resid, rank, singular_values = np.linalg.lstsq(A, z)

    # Coefficients in the form: a*x + b*y + c*z + d = 0.
    return C[0], C[1], -1., C[2]

def minimize_perp_distance(x, y, z):
    def model(params, xyz):
        a, b, c, d = params
        x, y, z = xyz
        length_squared = a**2 + b**2 + c**2
        return ((a * x + b * y + c * z + d) ** 2 / length_squared).sum() 

    def unit_length(params):
        a, b, c, d = params
        return a**2 + b**2 + c**2 - 1

    # constrain the vector perpendicular to the plane be of unit length
    cons = ({'type':'eq', 'fun': unit_length})
    sol = optimize.minimize(model, initial_guess, args=[x, y, z], constraints=cons)
    return tuple(sol.x)

initial_guess = 0.28, -0.14, 0.95, 0.
vert_params = minimize_z_error(x, y, z)
perp_params = minimize_perp_distance(x, y, z)

def z_error(x, y, z, a, b, d):
    return math.sqrt((((a*x + b*y + d) - z)**2).sum())

def perp_error(x, y, z, a, b, c, d):
    length_squared = a**2 + b**2 + c**2
    return ((a * x + b * y + c * z + d) ** 2 / length_squared).sum() 

def report(kind, params):
    a, b, c, d = params
    paramstr = ','.join(['{:.2f}'.format(p) for p in params])
    print('{:7}: params: ({}), z_error: {:>5.2f}, perp_error: {:>5.2f}'.format(
        kind, paramstr, z_error(x, y, z, a, b, d), perp_error(x, y, z, a, b, c, d)))

report('vert', vert_params)
report('perp', perp_params)
report('guess', initial_guess)

X, Y = np.meshgrid(np.arange(-3.0, 3.0, 0.5), np.arange(-3.0, 3.0, 0.5))
fig = plt.figure()
ax = fig.gca(projection='3d')

def Z(X, Y, params):
    a, b, c, d = params
    return -(a*X + b*Y + d)/c

ax.plot_surface(X, Y, Z(X, Y, initial_guess), rstride=1, cstride=1, alpha=0.3, color='magenta')
ax.plot_surface(X, Y, Z(X, Y, vert_params), rstride=1, cstride=1, alpha=0.3, color='yellow')
ax.plot_surface(X, Y, Z(X, Y, perp_params), rstride=1, cstride=1, alpha=0.3, color='green')
ax.scatter(x, y, z, c='r', s=50)
plt.xlabel('X')
plt.ylabel('Y')
ax.set_zlabel('Z')
ax.axis('equal')
ax.axis('tight')
plt.show()

The code above computes the parameters which minimize vertical distance from the plane and perpendicular distance from the plane. We can then compute the total error:

vert   : params: (0.94,0.52,-1.00,0.10), z_error:  2.63, perp_error:  3.21
perp   : params: (-0.68,-0.39,0.63,-0.06), z_error:  9.50, perp_error:  2.96
guess  : params: (0.28,-0.14,0.95,0.00), z_error:  5.22, perp_error: 52.31

Notice that the vert_params minimize z_error, but perp_params minimize perp_error.

The magenta plane corresponds to the initial_guess, the yellow plane corresponds to the vert_params and the green plane corresponds to the perp_params.