Parametric Curve Fit

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Description

Polynomial fitting of parametrized points on 2D curves, also requiring to meet some points exactly.

Usage

1
curvefit(u, x, y, n, U = NULL, V = NULL)

Arguments

u

the parameter vector.

x, y

x-, y-coordinates for each parameter value.

n

order of the polynomials, the same in x- and y-dirction.

U

parameter values where points will be fixed.

V

matrix with two columns and lemgth(U) rows; first column contains the x-, the second the y-values of those points kept fixed.

Details

This function will attempt to fit two polynomials to parametrized curve points using the linear least squares approach with linear equality constraints in lsqlin. The requirement to meet exactly some fixed points is interpreted as a linear equality constraint.

Value

Returns a list with 4 components, xp and yp coordinates of the fitted points, and px and py the coefficients of the fitting polynomials in x- and y-direction.

Note

In the same manner, derivatives/directions could be prescribed at certain points.

See Also

circlefit, lsqlin

Examples

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##  Approximating half circle arc with small perturbations
N <- 50
u <- linspace(0, pi, N)
x <- cos(u) + 0.05 * randn(1, N)
y <- sin(u) + 0.05 * randn(1, N)
n <- 8
cfit1 <- curvefit(u, x, y, n)
## Not run: 
plot(x, y, col = "darkgray", pch = 19, asp = 1)
xp <- cfit1$xp; yp <- cfit1$yp
lines(xp, yp, col="blue")
grid()
## End(Not run)

##  Fix the end points at t = 0 and t = pi
U <- c(0, pi)
V <- matrix(c(1, 0, -1, 0), 2, 2, byrow = TRUE)
cfit2 <- curvefit(u, x, y, n, U, V)
## Not run: 
xp <- cfit2$xp; yp <- cfit2$yp
lines(xp, yp, col="red")
## End(Not run)

## Not run: 
##  Archimedian spiral
n <- 8
u <- linspace(0, 3*pi, 50)
a <- 1.0
x <- as.matrix(a*u*cos(u))
y <- as.matrix(a*u*sin(u))
plot(x, y, type = "p", pch = 19, col = "darkgray", asp = 1)
lines(x, y, col = "darkgray", lwd = 3)
cfit <- curvefit(u, x, y, n)
px <- c(cfit$px); py <- c(cfit$py)
v <- linspace(0, 3*pi, 200)
xs <- polyval(px, v)
ys <- polyval(py, v)
lines(xs, ys, col = "navy")
grid()
## End(Not run)

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