给定R中的一般方程，如何绘制椭圆? [英] how to plot ellipse given a general equation in R?

问题描述

椭圆一般公式:

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

推荐答案

我们可以从ellipse的parametric公式开始(以下一个来自维基百科)，我们需要5个参数:center (xc, yc)在另一种表示法中，轴长度a, b和x轴与主轴phi或tau之间的夹角在另一种表示法中.

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')

现在，如果要从cartesian conic公式开始，则需要两个步骤.

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

1. 将cartesian公式转换为polar(parametric)，形成形式，我们可以使用以下公式首先使用下图中的5个方程式获得5个参数(取自 http://www.cs.cornell.edu/cv/OtherPdf/Ellipse.pdf ，可以在此处找到详细的数学信息.)

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).

使用获得的参数绘制椭圆，如上所示.

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

对于步骤(1)，我们可以使用以下代码(当我们知道A,B,C,D,E,F时):

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

对于步骤(2)，请使用以下代码:

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')

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