arcLengthEllipse {RConics}

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)

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.

Value

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

Source

Van de Vel, H. (1969). On the series expansion method for Computing incomplete elliptic integrals of the first and second kinds, Math. Comp. 23, 61-69.

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)