math - how to plot ellipse given a general equation in R?

Question

Ellipse general equation:

a * x ^ 2 + b * y ^ 2 + c * x * y + d * x + e * y + f = 0

Answer 1

We can start from the parametric equation of an ellipse (the following one is from wikipedia), we need 5 parameters: the center (xc, yc) or (h,k) in another notation, axis lengths a, b and the angle between x axis and the major axis phi or tau in another notation.

xc <- 1 # center x_c or h
yc <- 2 # y_c or k
a <- 5 # major axis length
b <- 2 # minor axis length
phi <- pi/3 # angle of major axis with x axis phi or tau

t <- seq(0, 2*pi, 0.01) 
x <- xc + a*cos(t)*cos(phi) - b*sin(t)*sin(phi)
y <- yc + a*cos(t)*cos(phi) + b*sin(t)*cos(phi)
plot(x,y,pch=19, col='blue')

Now if we want to start from the cartesian conic equation, it's a 2-step process.

1. Convert the cartesian equation to the polar (parametric), form we can use the following equations to first obtain the 5 parameters using the 5 equations from the below figure (taken from http://www.cs.cornell.edu/cv/OtherPdf/Ellipse.pdf, detailed math can be found there).

2. Plot the ellipse, by using the parameters obtained, as shown above.

For step (1) we can use the following code (when we have known A,B,C,D,E,F):

M0 <- matrix(c(F,D/2,E/2, D/2, A, B/2, E/2, B/2, C), nrow=3, byrow=TRUE)
M <- matrix(c(A,B/2,B/2,C), nrow=2)
lambda <- eigen(M)$values
abs(lambda - A)
abs(lambda - C) 

# assuming abs(lambda[1] - A) < abs(lambda[1] - C), if not, swap lambda[1] and lambda[2] in the following equations:

a <- sqrt(-det(M0)/(det(M)*lambda[1]))  
b <- sqrt(-det(M0)/(det(M)*lambda[2]))
xc <- (B*E-2*C*D)/(4*A*C-B^2)
yc <- (B*D-2*A*E)/(4*A*C-B^2)
phi <- pi/2 - atan((A-C)/B)*2

For step (2) use the following code:

t <- seq(0, 2*pi, 0.01) 
x <- xc + a*cos(t)*cos(phi) - b*sin(t)*sin(phi)
y <- yc + a*cos(t)*cos(phi) + b*sin(t)*cos(phi)
plot(x,y,pch=19, col='blue')