RConics (version 1.1.2)

arcLengthEllipse: Arc length of an ellipse

Description

This function computes the arc length of an ellipse centered in \((0,0)\) with the semi-axes aligned with the \(x\)- and \(y\)-axes. The arc length is defined by the points \(1\) and \(2\). These two points do not need to lie exactly on the ellipse: the \(x\)-coordinate of the points and the quadrant where they lie define the positions on the ellipse used to compute the arc length.

Usage

arcLengthEllipse(p1, p2 = NULL, saxes, n = 5)

Value

The length of the shortest arc of the ellipse defined by the points 1 and 2.

Arguments

p1

a \((2 \times 1)\) vector of the Cartesian coordinates of point \(1\).

p2

a \((2 \times 1)\) vector of the Cartesian coordinates of point \(2\) (optional).

saxes

a \((2 \times 1)\) vector of length of the semi-axes of the ellipse.

n

the number of iterations used in the numerical approximation of the incomplete elliptic integral of the second kind.

Details

If the coordinates p2 of the point \(2\) are omitted the function arcLengthEllipse computes the arc length between the point \(1\) and the point defined by \((0,b)\), \(b\) beeing the minor semi-axis.

See Also

pEllipticInt

Examples

p1 <- c(3,1)
p2 <- c(0,2)

# Ellipse with semi-axes: a = 5, b= 2
saxes <- c(5,2)

# 1 iteration
arcLengthEllipse(p1,p2,saxes,n=1)

# 5 iterations
arcLengthEllipse(p1,p2,saxes,n=5)

# 10 iterations
arcLengthEllipse(p1,p2,saxes,n=10)