| Title: | Fit Data to Any Conic Section |
|---|---|
| Description: | Fit data to an ellipse, hyperbola, or parabola. Bootstrapping is available when needed. The conic curve can be rotated through an arbitrary angle and the fit will still succeed. Helper functions are provided to convert generator coefficients from one style to another, generate test data sets, rotate conic section parameters, and so on. References include Nikolai Chernov (2014) "Fitting ellipses, circles, and lines by least squares" <https://people.cas.uab.edu/~mosya/cl/>; A. W. Fitzgibbon, M. Pilu, R. B. Fisher (1999) "Direct Least Squares Fitting of Ellipses" IEEE Trans. PAMI, Vol. 21, pages 476-48; N. Chernov, Q. Huang, and H. Ma (2014) "Fitting quadratic curves to data points", British Journal of Mathematics & Computer Science, 4, 33-60; N. Chernov and H. Ma (2011) "Least squares fitting of quadratic curves and surfaces", Computer Vision, Editor S. R. Yoshida, Nova Science Publishers, pp. 285-302. |
| Authors: | Carl Witthoft [aut, cre], Jose Gama [ctb], Nikolai Chernov [ctb] |
| Maintainer: | Carl Witthoft <[email protected]> |
| License: | LGPL-3 |
| Version: | 1.2.1 |
| Built: | 2024-11-20 06:23:26 UTC |
| Source: | CRAN |
Fit data to an ellipse, hyperbola, or parabola. Bootstrapping is available when needed. The conic curve can be rotated through an arbitrary angle and the fit will still succeed. Helper functions are provided to convert generator coefficients from one style to another, generate test data sets, rotate conic section parameters, and so on. References include Nikolai Chernov (2014) "Fitting ellipses, circles, and lines by least squares" <https://people.cas.uab.edu/~mosya/cl/>; A. W. Fitzgibbon, M. Pilu, R. B. Fisher (1999) "Direct Least Squares Fitting of Ellipses" IEEE Trans. PAMI, Vol. 21, pages 476-48; N. Chernov, Q. Huang, and H. Ma (2014) "Fitting quadratic curves to data points", British Journal of Mathematics & Computer Science, 4, 33-60; N. Chernov and H. Ma (2011) "Least squares fitting of quadratic curves and surfaces", Computer Vision, Editor S. R. Yoshida, Nova Science Publishers, pp. 285-302.
The DESCRIPTION file:
| Package: | fitConic |
| Title: | Fit Data to Any Conic Section |
| Version: | 1.2.1 |
| Date: | 2023-08-28 |
| Authors@R: | c(person(given = "Carl", family = "Witthoft", role = c("aut","cre"), email= "[email protected]") ,person(given="Jose", family= "Gama", role = "ctb"), person(given="Nikolai", family = "Chernov", role = "ctb")) |
| Description: | Fit data to an ellipse, hyperbola, or parabola. Bootstrapping is available when needed. The conic curve can be rotated through an arbitrary angle and the fit will still succeed. Helper functions are provided to convert generator coefficients from one style to another, generate test data sets, rotate conic section parameters, and so on. References include Nikolai Chernov (2014) "Fitting ellipses, circles, and lines by least squares" <https://people.cas.uab.edu/~mosya/cl/>; A. W. Fitzgibbon, M. Pilu, R. B. Fisher (1999) "Direct Least Squares Fitting of Ellipses" IEEE Trans. PAMI, Vol. 21, pages 476-48; N. Chernov, Q. Huang, and H. Ma (2014) "Fitting quadratic curves to data points", British Journal of Mathematics & Computer Science, 4, 33-60; N. Chernov and H. Ma (2011) "Least squares fitting of quadratic curves and surfaces", Computer Vision, Editor S. R. Yoshida, Nova Science Publishers, pp. 285-302. |
| Imports: | pracma |
| License: | LGPL-3 |
| NeedsCompilation: | no |
| Packaged: | 2023-08-28 14:36:40 UTC; cgw |
| Author: | Carl Witthoft [aut, cre], Jose Gama [ctb], Nikolai Chernov [ctb] |
| Maintainer: | Carl Witthoft <[email protected]> |
| Repository: | CRAN |
| Date/Publication: | 2023-08-28 17:40:12 UTC |
Index of help topics:
AtoG A Set Of Functions To Convert Among Various
Conic-Section-Defining Parameter Sets.
JmatrixLMA Calculate a Jacobian Matrix
Residuals.ellipse Calculate Residual Error For Current
Coefficients
Residuals.hyperbola Calculate Residual Error For Current
Coefficients
bootEllipse Simple, Medium-Quality Ellipse Fitting Function
bootHyperbola A Function to Attempt a Crude Fit of Data to a
Hyperbola
createConic Create A Conic Section Dataset Based on
Parameter Set
doWeights Function to Apply Weights to Data
fhyp Internal Functions to Perform Bootstrap Fitting
Operations
fitConic Fit Data to A Conic Section Curve
fitConic-package Fit Data to Any Conic Section
fitParabola Fit Data to Parabola
rotateA Rotate Conic Section Equation Parameters Or A
Dataset, With Respect To X-Y Axes.
The main function is fitConic .
Carl Witthoft [aut, cre], Jose Gama [ctb], Nikolai Chernov
[ctb], based on code provided in the references and in conicfit::fit.conicLMA()
Maintainer: Carl Witthoft <[email protected]>
https://www.mathworks.com/matlabcentral/answers/80541 for the RANSAC-style search to fit rotated parabolas. https://math.stackexchange.com/questions/426150 for detailed ellipse parametric equations. https://math.stackexchange.com/questions/2800817 for "focus/directrix/eccentricity" information https://people.cas.uab.edu/~mosya/cl/ and the folks referred to there, for fitConicLMA . https://en.wikipedia.org/wiki/Ellipse for several parameter conversion formulas
A. W. Fitzgibbon, M. Pilu, R. B. Fisher, "Direct Least Squares Fitting of Ellipses", IEEE Trans. PAMI, Vol. 21, pages 476-480 (1999)
Halir R, Flusser J (1998) Proceedings of the 6th International Conference in Central Europe on Computer Graphics and Visualization, Numerically stable direct least squares fitting of ellipses (WSCG, Plzen, Czech Republic), pp 125132.
AtoG Convert from full quadratic "ABCDEF" to focus, axis, angle "hvab theta" parameters.
GtoA Convert from "hvab theta" to "ABCDEF" parameters.
parab3toA Simple conversion from a + bx + cx^2 to "ABCDEF" parameters.
FEDtoA Convert focus, eccentricity, and directrix to "ABCDEF" parameters.
AtoG(parA, tol = 1e-06) GtoA(parG, conicType = c("e", "h")) parab3toA(ADF, theta = 0) FEDtoA(focus = c(0, 0), directrix = c(1, 0, 1), eccentricity = 0.5)AtoG(parA, tol = 1e-06) GtoA(parG, conicType = c("e", "h")) parab3toA(ADF, theta = 0) FEDtoA(focus = c(0, 0), directrix = c(1, 0, 1), eccentricity = 0.5)
parA |
The six coefficients in the quadratic Ax^2 + Bxy + Cy^2 +Dx + Ey +F = 0 |
tol |
A small value, used to check whether small coefficient values might be actually zero. See "Details." |
parG |
a five-element vector "h,v,a,b,theta" . See "Details" for the standard equation form for this. |
conicType |
Because the 'hvab' equation has a sign difference for ellipses vs. hyperbolas, it is necessary to indicate which kind of input is intended. See "Details." |
focus |
location of the conic sections focus. |
directrix |
the 3-element directrix. |
eccentricity |
the eccentricity of the conic section. |
ADF |
The A,D,F coeffients in the standard quadratic. Thus, the x^2 term, the x term, and the constant term. |
theta |
An angle by which the entire parabola is to be rotated. |
The tol input for AtoG checks two conditions. First, is B practically zero, in which case B is set to exactly zero, implying no rotation of the conic section. Second, is B^2 - 4*A*C almost zero, implying that the conic is probably a parabola, and conversion to 'hvab' form is not useful.
The "hvab" form for describing an ellipse or a hyperbola looks like [Center(1:2), Axes(1:2)/2] angle A, to fill the equation
((x-h)cosA +(y-v)sinA)^2/a^2 + ((x-h)sinA-(y-v)cosA)^2/b^2 = 1 The length of the axes are 2*a, 2*b .
A discussion of the focus/directrix/eccentricity form of a conic section is rather lengthy and not presented here. One short introduction can be found at https://en.wikipedia.org/wiki/Conic_section#Eccentricity,_focus_and_directrix
for AtoG,
parG |
c(h,v,a,b,theta) |
exitCode |
a value used in fitConic. 1,2, or 3 for ellipse, hyperbola, parabola |
conicType |
matching exitCode with a char "e", "h", or "p" |
for GtoA
parA |
the ABCDEF coefficients of the general quadratic |
exitCode |
a value used in fitConic. 1,2, or 3 for ellipse, hyperbola, parabola |
conicType |
matching exitCode with a char "e", "h", or "p" |
for FEDtoA, the ABCDEF coefficients of the general quadratic
for parab3toA,
parA |
the ABCDEF coefficients of the general quadratic |
exitCode |
always numeric 3, a value used in fitConic |
conicType |
always char "p" |
.
This function generates a half-decent fit to the source data. It is intended only for internal use, to bootstrap the higher-quality fitConic function.
bootEllipse(x, y = NULL, ...)bootEllipse(x, y = NULL, ...)
x |
vector of x-values, or a Nx2 array of x and y values. In the latter case, the input |
y |
vector of y-values. |
... |
possible other arguments to be passed to future upgrades |
This can be used as a Q&D ellipse fitting algorithm, but is intended only for internal use by fitConic, providing that function with an initial estimate for the ellipse's defining parameter set.
parA |
6-element set with estimate of the "ABCDEF" coefficients for the general quadratic equation |
centroid |
estimate of the ellipse's centroid |
Carl Witthoft, [email protected]
This is a revision of the function EllipseDirectFit in package conicfit by Jose Gama, with minor upgrades. Original MATLAB code by: Nikolai Chernov https://www.mathworks.com/matlabcentral/fileexchange/22684-ellipse-fit-direct-method A. W. Fitzgibbon, M. Pilu, R. B. Fisher, "Direct Least Squares Fitting of Ellipses", IEEE Trans. PAMI, Vol. 21, pages 476-480 (1999) Halir R, Flusser J (1998) Proceedings of the 6th International Conference in Central Europe on Computer Graphics and Visualization, Numerically stable direct least squares fitting of ellipses (WSCG, Plzen, Czech Republic), pp 125132.
This function is not intended for direct use. It attempts to generate an approximate fit of a data set to a hyperbola, returning a parameter set for use in intializing the main function conicFit .
bootHyperbola(x, y = NULL, maxiter = 10000, ...)bootHyperbola(x, y = NULL, maxiter = 10000, ...)
x |
vector of x-values, or a Nx2 array of x and y values. In the latter case, the input |
y |
vector of y-values. |
maxiter |
A 'safety' limiter on the number of iterations to try before giving up. |
... |
possible other arguments to be passed to future upgrades |
parA |
the new 6-parameter set defining the non-rotated conic. |
parAr |
the new 6-parameter set defining the rotated conic. |
theta |
the angle of rotation between ParA and ParAr |
fitdat |
the information returned from |
Carl Witthoft, [email protected]
Given a vector of x-values and a parameter set defining a conic section, produce an array of x- and y- values, optionally with noise added, for the specified conic section.
createConic(x, param, conicType, ranFun = NULL, noise = 1, seedit = NULL, tol = 1e-06)createConic(x, param, conicType, ranFun = NULL, noise = 1, seedit = NULL, tol = 1e-06)
x |
Vector of (real) values |
param |
Either a 6-value set representing the standard quadratic form Ax^2 + Bxy + Cy^2 +Dx + Ey +F = 0 or a 5-value set representing the "hvab,theta" form ((x-h)cosA +(y-v)sinA)^2/a^2 + ((x-h)sinA-(y-v)cosA)^2/b^2 = 1 . In the latter case the value |
conicType |
Either the character "e" for ellipse or "h" for hyperbola. Only required if the "hvab,theta" form is used in |
ranFun |
If random noise is to be added to the calculated y-values, provide a vectorized function which takes a single input (x). See Details. |
noise |
Optional argument to multiply the output of |
seedit |
Optional argument to set a starting seed for |
tol |
A (small) value used to decide whether various parameter terms are so small that they should be zero. This is used to facilitate distinguishing, e.g., parabolas from hyperbolas. |
When supplied ranFun is used as follows.
y <- y + ranFun(y)*noise . Make sure any function supplied fits that form (no other input argument required; only a vector returned).
An N x 2 array of the x,y pairs. Warning: since there are often two possible y-values for a given x-value (these being quadratic equations), the array does contain duplicate x-values. This may "annoy" some other packages' functions which don't allow that sort of repeated value. If this presents a problem, I'd recommend applying a very small amount of noise to the x-values in this output.
Carl Witthoft <[email protected]>
# create noisy ellipse parGr <- c(-2.3,4.2,5,3,pi/4) xe <-seq(-8,9,by=.05) elipGrn <- createConic(xe, parGr, 'e',ranFun=rnorm, noise=0.25) elipGr <- createConic(xe, parGr, 'e') plot(elipGrn, pch='.',cex = 4, asp = TRUE) #, xlim = c(-5,8), ylim = c(0,7)) lines(elipGr,col='green')# create noisy ellipse parGr <- c(-2.3,4.2,5,3,pi/4) xe <-seq(-8,9,by=.05) elipGrn <- createConic(xe, parGr, 'e',ranFun=rnorm, noise=0.25) elipGr <- createConic(xe, parGr, 'e') plot(elipGrn, pch='.',cex = 4, asp = TRUE) #, xlim = c(-5,8), ylim = c(0,7)) lines(elipGr,col='green')
This function applies an integer weight set to an array of (x,y) data points. It normally is only called from fitConic but can be applied directly to a dataset if desired.
doWeights(XY, weights)doWeights(XY, weights)
XY |
A Nx2 array of data representing (x,y) pairs |
weights |
A vector of weights the same length as the number of rows in XY. At this time, only nonnegative integer values are allowed. Doubles are rounded and negative values are set to zero. A zero weight will remove the matching data value from the dataset. |
A new Nx2 array. Basically, each row in the input XY is repeated weights[j] times.
..
Carl Witthoft <[email protected]>
These functions are not intended for external use.
fhyp and fhypopt support the parent function bootHyperbola by providing functions for optimize to use.
The functions costparab costparabxy similarly provide functions for optim to use inside the function fitParabola .
fhyp(xy, b3, Ang) costparabxy(theta, xy) costparab(theta, xy)fhyp(xy, b3, Ang) costparabxy(theta, xy) costparab(theta, xy)
xy |
A Nx2 array of data |
b3 |
Three of the parameters describing a hyperbola. These three are the "other parameters" fed to |
Ang |
The initial angle of rotation, also optimized during the process. |
theta |
The angle of rotation of the parabola for this run of |
various combinations of "cost" values, i.e. Figure of Merit, to determine optimal set of coefficients, along with datasets where necessary. ..
Carl Witthoft <[email protected]>
This function fits data to an ellipse, hyperbola, or parabola. It can do so without any initial conditions, or can accept initial parameter values when known.
fitConic(X, Y = NULL, parInit = NULL, conicType = c("e", "h", "p"), weights = NULL, LambdaIni = 1, epsilonP = 1e-06, epsilonF = 1e-06, IterMAX = 20000)fitConic(X, Y = NULL, parInit = NULL, conicType = c("e", "h", "p"), weights = NULL, LambdaIni = 1, epsilonP = 1e-06, epsilonF = 1e-06, IterMAX = 20000)
X |
vector of x-values, or a Nx2 array of x and y values.
In the latter case, the input |
Y |
vector of y-values. |
parInit |
Optional. A vector either of six values representing an initial guess
at the "ABCDEF" coefficients of the quadratic, or five values
representing an initial guess at the "hvab,theta" coefficients.
In the latter case, a value of either "e" or "h" is required for
|
conicType |
If |
weights |
Optional vector of weights to apply to data. Must be same length as the input data. Only non-negative integer weights are allowed. See the Details section. |
LambdaIni |
A control parameter used in the fitting algorithm. Typically there is no reason to change from the default value. |
epsilonP |
A tolerance value to determine whether convergence has occurred. |
epsilonF |
A tolerance parameter for determining when to adjust lambda
away from the input value |
IterMAX |
A "safety" value to avoid loop thrashing when convergence isn't taking place. |
ParInit, when supplied is either a 6-value set representing the standard quadratic form Ax^2 + Bxy + Cy^2 +Dx + Ey +F = 0 or a 5-value set representing the "hvab,theta" form ((x-h)cosA +(y-v)sinA)^2/a^2 + ((x-h)sinA-(y-v)cosA)^2/b^2 = 1 . In the latter case the value conicType is required, because ellipses and hyperbolas have a different sign for the y-term.
In most cases, the bootstrapper tools work well enough to allow the main algorithm to fit to an ellipse or hyperbola. However, "knowledge is power." If you have a good idea approximately what the ParIni values are,
entering them will help avoid convergence to the wrong local minimum.
The algorithm branch which fits data to parabolas does not use or need
initialization, as it uses a RANSAC-type search to find
the best rotation angle, and then does a simple quadratic polynomial fit.
The weights input is restricted to nonnegative integers at this time. Doubles are rounded and negative values are set to zero. A zero weight will remove the matching data value from the dataset.
parA vector of the six "ABCDEF" coefficients
RSS 'root sum square' figure of merit describing the relative fit quality
iters number of iterations at convergence
exitCode 1 means ellipse, 2 means hyperbola, 3 means parabola.
If other values show up (possibly -1, 0, 4), most likely the
dataset led to a degenerate case such as a line fit.
Carl Witthoft <[email protected]>
https://people.cas.uab.edu/~mosya/cl/ for information on the original "LMA" fitting algorithm. https://math.stackexchange.com/questions/426150 and https://math.stackexchange.com/questions/2800817 for various related equations concerning conic sections. https://en.wikipedia.org/wiki/Ellipse for several parameter conversion formulas
##-create a hyperbola, add noise Ang = 0.42 #radians xh <- seq(-20,20,by=0.1) parAxyh <- c(0, 1, 0, -2, 4, -15 ) parAxyhr <- rotateA(parAxyh, Ang)$parA newxyr <-createConic(xh,parAxyhr) newxyrn <- createConic(xh,parAxyhr,ranFun=rnorm, noise= 0.05) plot(newxyr, t = 'l',asp=TRUE) points(newxyrn, pch = '.', cex = 3) # Now find the hyperbola for that dataset hypfitr <-fitConic(newxyrn, conicType = 'h') hypdatr <- createConic(xh, hypfitr$parA) lines(hypdatr, col='red')##-create a hyperbola, add noise Ang = 0.42 #radians xh <- seq(-20,20,by=0.1) parAxyh <- c(0, 1, 0, -2, 4, -15 ) parAxyhr <- rotateA(parAxyh, Ang)$parA newxyr <-createConic(xh,parAxyhr) newxyrn <- createConic(xh,parAxyhr,ranFun=rnorm, noise= 0.05) plot(newxyr, t = 'l',asp=TRUE) points(newxyrn, pch = '.', cex = 3) # Now find the hyperbola for that dataset hypfitr <-fitConic(newxyrn, conicType = 'h') hypdatr <- createConic(xh, hypfitr$parA) lines(hypdatr, col='red')
This function fits a data set to a parabola, including any rotation angle.
fitParabola(x, y = NULL, searchAngle = c(-pi/2, pi/2), ...)fitParabola(x, y = NULL, searchAngle = c(-pi/2, pi/2), ...)
x |
vector of x-values, or a Nx2 array of x and y values. In the latter case, the input |
y |
vector of y-values. |
searchAngle |
Optional pair of angles, in radians, defining the limits of the search range to find the rotation angle of the parabola. Usually the default range |
... |
For possible future expansion to pass to additional features. |
fitParabola starts by doing a RANSAC-style search to find the optimum rotation angle. Once that is chosen, the data are rotated by that angle and a simple polynomial fit to the (rotated) vertical parabola is done.
vertex |
calculated vertex of the parabola |
theta |
angle of rotation relative to a vertical parabola |
parA |
the "ABCDEF" coefficients of the fitted parabola |
parQ |
the coefficients of the derotated parabola's simple quadratic polynomial, highest power first |
cost |
final value of the "cost" parameter used for optimization |
When the function fitConic is called with instructions to fit to a parabola, it passes the inputs to fitParabola and does nothing else. For parabolic data, then, either function will give the same result.
Carl Witthoft <[email protected]>
Some of the code is based on https://www.mathworks.com/matlabcentral/answers/80541
# Create vertical parabola with some noise parP <-c(.5,0,0,2,-1,4) xp <- seq(-5,5,by=0.05) partest <-createConic(xp,param = parP,ranFun = rnorm, noise = 1) plot(partest, pch= '.',asp=TRUE, cex=3) # rotate the data partestr <-xyrot(partest,theta = -.35) points(partestr,col='green',pch='.',cex=3) # do the fit parfit <-fitParabola(partestr) points(parfit$vertex,pch='X',col='blue') parout <- createConic(xp,parfit$parA) lines(parout,col='red')# Create vertical parabola with some noise parP <-c(.5,0,0,2,-1,4) xp <- seq(-5,5,by=0.05) partest <-createConic(xp,param = parP,ranFun = rnorm, noise = 1) plot(partest, pch= '.',asp=TRUE, cex=3) # rotate the data partestr <-xyrot(partest,theta = -.35) points(partestr,col='green',pch='.',cex=3) # do the fit parfit <-fitParabola(partestr) points(parfit$vertex,pch='X',col='blue') parout <- createConic(xp,parfit$parA) lines(parout,col='red')
Calculate the Jacobian matrix with the original dataset and the
current version of fitted data. This is not intended for
external use. It is called from fitConic
JmatrixLMA(XY, parA, XYproj)JmatrixLMA(XY, parA, XYproj)
XY |
The original input dataset |
parA |
The current set of ABCDEF quadratic equation coefficients. |
XYproj |
The current calculated dataset based on the latest iteration of the coefficient set. |
Res |
residuals based on the norm of XY - XYproj |
J |
matrix of values for each input data point corresponding to the terms in the general quadratic Ax^2 + Bxy + Cy^2 +Dx + Ey +F |
Carl Witthoft <[email protected]>
This is a copy of JmatrixLMA with some
validation steps added.
This function is not intended for external use. It is called from fitConic when iterating to find the best-fit ellipse.
Residuals.ellipse(XY, parG)Residuals.ellipse(XY, parG)
XY |
The x,y dataset |
parG |
The "G-parameter" set for the current iteration. |
RSS |
Figure of merit, the 'norm' of the difference between the input XY data and the output "XYproj" data generated. |
XYproj |
Calculated dataset to be used in generating the Jacobian matrix for the next iteration of |
Carl Witthoft <[email protected]>
This is a slightly modified (and debugged) version of Residuals.ellipse
This function is not intended for external use. It is called from fitConic when iterating to find the best-fit hyperbola.
Residuals.hyperbola(XY, parG)Residuals.hyperbola(XY, parG)
XY |
The x,y dataset |
parG |
The "G-parameter" set for the current iteration. |
RSS |
Figure of merit, the 'norm' of the difference between the input XY data and the output "XYproj" data generated. |
XYproj |
Calculated dataset to be used in generating the Jacobian matrix for the next iteration of |
Carl Witthoft <[email protected]>
This is a slightly modified (and debugged) version of Residuals.hyperbola
rotateA Takes as input "parA," the 6 values of the general quadratic Ax^2 + Bxy + Cy^2 +Dx + Ey +F = 0 , and applies a rotation angle to the coefficient set.
derotateA calculates the rotation angle required to change the conic section defined by 'parA' into one that is orthogonal to the cartesian axes.
xyrot is a simple function to rotate the coordinate system by theta.
rotateA(parA, theta) derotateA(parA, ACmin = 1e-05) xyrot(x, y = NULL, theta)rotateA(parA, theta) derotateA(parA, ACmin = 1e-05) xyrot(x, y = NULL, theta)
parA |
the 6 values of the general quadratic Ax^2 + Bxy + Cy^2 +Dx + Ey +F = 0 |
theta |
the angle, in radians, to rotate the conic section. |
ACmin |
A tolerance parameter for deciding that the product of parameters A and C is actually zero (in which case the type of conic section is more likely a parabola or a degenerate case) |
x |
Either a vector of x-coordinates or a Nx2 array of x and y coordinates, in which case the y-input is ignored |
y |
A vector of y-coordinates. |
derotateA uses the following standard formula to calculate the angle.
Derotate means to remove the xy term, i.e. force B = 0 . Some algebra shows that cot(2theta) = (A-C)/B and thus tan(2theta) = B/(A-C)
For xyrot, the internal xy.coords is used. If you enter only a vector for x and nothing for y, this will feed the new vectors 1:N for x and x-input for y to the rotator, which is probably not useful.
For derotateA,
parA |
the new 6-parameter set defining the derotated conic. |
theta |
the derived angle by which the parameter set was rotated |
For rotateA
parA |
the new 6-parameter set defining the rotated conic. |
theta |
the angle by which the parameter set was rotated |
For xyrot a Nx2 array of the x,y coordinates of the rotated data set.
Carl Witthoft, [email protected]
# make an ellipse and derotate it parGr <- c(-2.3,4.2,5,3,pi/4) xe <-seq(-8,9,by=.05) elipGr <- createConic(xe, parGr, 'e') plot(elipGr, t= 'l', asp = TRUE) # convert to ABCDEF form parAr <- GtoA(parGr,'e') elipAr <- createConic(xe,parAr$parA) points(elipAr,pch='.',col='red') # remove rotation angle parAd <- derotateA(parAr$parA) # returns theta = pi/4, how much the ellipse had been rotated by elipAd <-createConic(xe,parAd$parA) lines(elipAd) # rotate back parAdr <- rotateA(parAd$parA, parAd$theta) elipAdr <-createConic(xe,parAdr$parA) lines(elipAdr,lty=3, lwd = 3, col='green')# make an ellipse and derotate it parGr <- c(-2.3,4.2,5,3,pi/4) xe <-seq(-8,9,by=.05) elipGr <- createConic(xe, parGr, 'e') plot(elipGr, t= 'l', asp = TRUE) # convert to ABCDEF form parAr <- GtoA(parGr,'e') elipAr <- createConic(xe,parAr$parA) points(elipAr,pch='.',col='red') # remove rotation angle parAd <- derotateA(parAr$parA) # returns theta = pi/4, how much the ellipse had been rotated by elipAd <-createConic(xe,parAd$parA) lines(elipAd) # rotate back parAdr <- rotateA(parAd$parA, parAd$theta) elipAdr <-createConic(xe,parAdr$parA) lines(elipAdr,lty=3, lwd = 3, col='green')