Polynomial Transformations

The Polynomial datum transformation methods use a collection of parameters which define a high order mathematical function for transforming between two horizontal datums. These equations are usually created by local and regional geodetic authorities. They generally provide high accuracy transformations, but are limited to specific areas of use. In many cases, the accuracy of these transformations is around 1 meter.

For a detailed description of generalized polynomial datum transformations, please refer to the IOGP guidance notes. These are freely available from http://www.epsg.org

Note: Polynomial datum transformations are generally computed for specific areas of use. Since the derivations of the formula are based on a limited number of reference points, using the transformation for data outside of the pre-determined envelopes may cause unreliable results.

Affine Transform

Given a set of Control points and their subsequent Observed points, the corresponding Affine Transform is generated:

x_in, y_in : input point coordinates

x_out, y_out : output point coordinates

x = (x_in + Xshift_in) * scale_in

y = (y_in + Yshift_in) * scale_in

x' = (Xx * x) + (Xy * y) + X1

y' = (Yx * x) + (Yy * y) + Y1

x_out = (x' * scale_out) + Xshift_out

y_out = (y' * scale_out) + Yshift_out

The Affine math transform has the following parameters:

Parameter Name	Parameter String
input point scale	scale_in
input x-coordinate shift	Xshift_in
input y-coordinate shift	Yshift_in
output point scale	scale_out
output x-coordinate shift	Xshift_out
output y-coordinate shift	Yshift_out
x' x coefficient	Xx
x' y coefficient	Xy
x' delta	X1
y' x coefficient	Yx
y' y coefficient	Yy
y' delta	Y1

Second Order Polynomial Transform

Given a set of Control points and their subsequent Observed points, the corresponding Second Order Polynomial Transform is generated:

x_in, y_in : input point coordinates

x_out, y_out : output point coordinates

x = (x_in + Xshift_in) * scale_in

y = (y_in + Yshift_in) * scale_in

x' = (Xx * x) + (Xy * y) + X1 + (Xxy * x * y) + (Xx2 * x * x) + (Xy2 * y * y)

y' = (Yx * x) + (Yy * y) + Y1 + (Yxy * x * y) + (Yx2 * x * x) + (Yy2 * y * y)

x_out = (x' * scale_out) + Xshift_out

y_out = (y' * scale_out) + Yshift_out

The Second Order Polynomial math transform has the following parameters:

Parameter Name	Parameter String
input point scale	scale_in
input x-coordinate shift	Xshift_in
input y-coordinate shift	Yshift_in
output point scale	scale_out
output x-coordinate shift	Xshift_out
output y-coordinate shift	Yshift_out
x' x coefficient	Xx
x' y coefficient	Xy
x' delta	X1
x' (x * y) coefficient	Xxy
x' x2 coefficient	Xx2
x' y2 coefficient	Xy2
y' x coefficient	Yx
y' y coefficient	Yy
y' delta	Y1
y' (x * y) coefficient	Yxy
y' x2 coefficient	Yx2
y' y2 coefficient	Yy2

Third Order Polynomial Transform

Given a set of Control points and their subsequent Observed points, the corresponding Third Order Polynomial Transform is generated:

x_in, y_in : input point coordinates

x_out, y_out : output point coordinates

x = (x_in + Xshift_in) * scale_in

y = (y_in + Yshift_in) * scale_in

x' = (Xx * x) + (Xy * y) + X1 + (Xxy * x * y) + (Xx2 * x * x) + (Xy2 * y * y) + (Xx3 * x * x * x) + (Xx2y * x * x * y) + (Xxy2 * x * y * y) + (Xy3 * y * y * y)

y' = (Yx * x) + (Yy * y) + Y1 + (Yxy * x * y) + (Yx2 * x * x) + (Yy2 * y * y) + (Yx3 * x * x * x) + (Yx2y * x * x * y) + (Yxy2 * x * y * y) + (Yy3 * y * y * y)

x_out = (x' * scale_out) + Xshift_out

y_out = (y' * scale_out) + Yshift_out

The Third Order Polynomial math transform has the following parameters:

Parameter Name	Parameter String
input point scale	scale_in
input x-coordinate shift	Xshift_in
input y-coordinate shift	Yshift_in
output point scale	scale_out
output x-coordinate shift	Xshift_out
output y-coordinate shift	Yshift_out
x' x coefficient	Xx
x' y coefficient	Xy
x' delta	X1
x' (x * y) coefficient	Xxy
x' x2 coefficient	Xx2
x' y2 coefficient	Xy2
x' x3 coefficient	Xx3
x' (x2 * y) coefficient	Xx2y
x' (x * y2) coefficient	Xxy2
x' y3 coefficient	Xy3
y' x coefficient	Yx
y' y coefficient	Yy
y' delta	Y1
y' (x * y) coefficient	Yxy
y' x2 coefficient	Yx2
y' y2 coefficient	Yy2
y' x3 coefficient	Yx3
y' (x2 * y) coefficient	Yx2y
y' (x * y2) coefficient	Yxy2
y' y3 coefficient	Yy3

Fourth Order Polynomial Transform

Given a set of Control points and their subsequent Observed points, the corresponding Fourth Order Polynomial Transform is generated:

x_in, y_in : input point coordinates

x_out, y_out : output point coordinates

x = (x_in + Xshift_in) * scale_in

y = (y_in + Yshift_in) * scale_in

x_out = (x' * scale_out) + Xshift_out

y_out = (y' * scale_out) + Yshift_out

The Fourth Order Polynomial math transform has the following parameters:

Parameter Name	Parameter String
input point scale	scale_in
input x-coordinate shift	Xshift_in
input y-coordinate shift	Yshift_in
output point scale	scale_out
output x-coordinate shift	Xshift_out
output y-coordinate shift	Yshift_out
x' x coefficient	Xx
x' y coefficient	Xy
x' delta	X1
x' (x * y) coefficient	Xxy
x' x2 coefficient	Xx2
x' y2 coefficient	Xy2
x' x3 coefficient	Xx3
x' (x2 * y) coefficient	Xx2y
x' (x * y2) coefficient	Xxy2
x' y3 coefficient	Xy3
x' x4 coefficient	Xx4
x' (x3 * y) coefficient	Xx3y
x' (x2 * y2) coefficient	Xx2y2
x' (x * y3) coefficient	Xxy3
x' y4 coefficient	Xy4
y' x coefficient	Yx
y' y coefficient	Yy
y' delta	Y1
y' (x * y) coefficient	Yxy
y' x2 coefficient	Yx2
y' y2 coefficient	Yy2
y' x3 coefficient	Yx3
y' (x2 * y) coefficient	Yx2y
y' (x * y2) coefficient	Yxy2
y' y3 coefficient	Yy3
y' x4 coefficient	Yx4
y' (x3 * y) coefficient	Yx3y
y' (x2 * y2) coefficient	Yx2y2
y' (x * y3) coefficient	Yxy3
y' y4 coefficient	Yy4

Fifth Order Polynomial Transform

Given a set of Control points and their subsequent Observed points, the corresponding Fifth Order Polynomial Transform is generated:

x_in, y_in : input point coordinates

x_out, y_out : output point coordinates

x = (x_in + Xshift_in) * scale_in

y = (y_in + Yshift_in) * scale_in

x' = (Xx * x) + (Xy * y) + X1 + (Xxy * x * y) + (Xx2 * x * x) + (Xy2 * y * y) + (Xx3 * x * x * x) + (Xx2y * x * x * y) + (Xxy2 * x * y * y) + (Xy3 * y * y * y) + (Xx4 * x * x * x * x) + (Xx3y * x * x * x * y) + (Xx2y2 * x * x * y * y) + (Xxy3 * x * y * y * y) + (Xy4 * y * y * y * y) + (Xx5 * x * x * x * x * x) + (Xx4y * x * x * x * x * y) + (Xx3y2 * x * x * x * y * y) + (Xx2y3 * x * x * y * y * y) + (Xxy4 * x * y * y * y * y) + (Xy5 * y * y * y * y * y)

y' = (Yx * x) + (Yy * y) + Y1 + (Yxy * x * y) + (Yx2 * x * x) + (Yy2 * y * y) + (Yx3 * x * x * x) + (Yx2y * x * x * y) + (Yxy2 * x * y * y) + (Yy3 * y * y * y) + (Yx4 * x * x * x * x) + (Yx3y * x * x * x * y) + (Yx2y2 * x * x * y * y) + (Yxy3 * x * y * y * y) + (Yy4 * y * y * y * y) + (Yx5 * x * x * x * x * x) + (Yx4y * x * x * x * x * y) + (Yx3y2 * x * x * x * y * y) + (Yx2y3 * x * x * y * y * y) + (Yxy4 * x * y * y * y * y) + (Yy5 * y * y * y * y * y)

x_out = (x' * scale_out) + Xshift_out

y_out = (y' * scale_out) + Yshift_out

The Fifth Order Polynomial math transform has the following parameters:

Parameter Name	Parameter String
input point scale	scale_in
input x-coordinate shift	Xshift_in
input y-coordinate shift	Yshift_in
output point scale	scale_out
output x-coordinate shift	Xshift_out
output y-coordinate shift	Yshift_out
x' x coefficient	Xx
x' y coefficient	Xy
x' delta	X1
x' (x * y) coefficient	Xxy
x' x2 coefficient	Xx2
x' y2 coefficient	Xy2
x' x3 coefficient	Xx3
x' (x2 * y) coefficient	Xx2y
x' (x * y2) coefficient	Xxy2
x' y3 coefficient	Xy3
x' x4 coefficient	Xx4
x' (x3 * y) coefficient	Xx3y
x' (x2 * y2) coefficient	Xx2y2
x' (x * y3) coefficient	Xxy3
x' y4 coefficient	Xy4
x' x5 coefficient	Xx5
x' (x4 * y) coefficient	Xx4y
x' (x3 * y2) coefficient	Xx3y2
x' (x2 * y3) coefficient	Xx2y3
x' (x * y4) coefficient	Xxy4
x' y5 coefficient	Xy5
y' x coefficient	Yx
y' y coefficient	Yy
y' delta	Y1
y' (x * y) coefficient	Yxy
y' x2 coefficient	Yx2
y' y2 coefficient	Yy2
y' x3 coefficient	Yx3
y' (x2 * y) coefficient	Yx2y
y' (x * y2) coefficient	Yxy2
y' y3 coefficient	Yy3
y' x4 coefficient	Yx4
y' (x3 * y) coefficient	Yx3y
y' (x2 * y2) coefficient	Yx2y2
y' (x * y3) coefficient	Yxy3
y' y4 coefficient	Yy4
y' x5 coefficient	Yx5
y' (x4 * y) coefficient	Yx4y
y' (x3 * y2) coefficient	Yx3y2
y' (x2 * y3) coefficient	Yx2y3
y' (x * y4) coefficient	Yxy4
y' y5 coefficient	Yy5