R中心给定3分——求解联立方程

Question

lets say I have 3 binary points (5,0),(0,5),(-5,0) and I want to find a point equidistant from those 3 point (in short find center of the circle passing through those 3 points). I know from geometry that if my answer is (a,b) then I can find distance between the (a,b) and 3 points and equate them and then solve 3 simultaneous equations. How can I quickly do this in R? I know equations will be linear and all square terms will cancel out.

_____________________________update1

I tried searching google for how to solve linear equations in R. But didnt get good results as all the links expect me to provide LHS coefficients and RHS value for all 3 equations. But I dont have RHS. I have to take 2 equations at a time and move terms to find RHS. Is there any R package that will do this for me?

Answer 1

You've probably moved on from now, but I'm going to add to this thread for posterity sake. I suppose people were just mad that you didn't pose an attempt? I stumbled upon this post looking for help and some of the respondents up there were completely unhelpful. I tried your approach with not much success for my application and wrote the following function, circlefit(), to perform a least squares approximation of the points along the arc if supplied a dataframe with column 1= X, and column2=Y. I'm pretty sure that the code above failed in my application because I had >100 of points along a "fuzzy" edge so my application was more attuned to a least squares approach.

cheers!

x_x<-c(0,0.5, 1, 1.5, 2, 2.5, 3)
y_x<-c(0,.25, 1, 2.25, 4, 6.25, 9)
df<-as.data.frame(cbind(x_x, y_x))


circlefit<-function (df){
  names(df)<-c("X", "Y")

  #find mean x y so we can cacluate the difference of each X, Y from its respective         mean
  xmean<-mean(df$X)
  ymean<-mean(df$Y)
  #adds needed columns for summations required to perform least squares
  mat2<-df%>%
    mutate(a=X-xmean)%>%
    mutate(b=Y-ymean)%>%
    mutate(aa=a^2)%>%
    mutate(ab=a*b)%>%
    mutate(bb=b^2)%>%
    mutate(aaa=a^3)%>%
    mutate(abb=a*b^2)%>%
    mutate(baa=b*a^2)%>%
    mutate(bbb=b^3)
  #column sums for construction of linear system of equations
  Saa<-sum(mat2$aa)
  Sab<-sum(mat2$ab)
  Sbb<-sum(mat2$bb)
  Saaa<-sum(mat2$aaa)
  Sbbb<-sum(mat2$bbb)
  Sabb<-sum(mat2$abb)
  Sbaa<-sum(mat2$baa)
  #linear stystem of equations
  sums_row1<-c(Saa, Sab)
  sums_row2<-c(Sab, Sbb)

  sum_mat<-as.matrix(rbind(sums_row1, sums_row2), nrow=2)
  #gauss elimination ratio
  gauss_ratio<-sum_mat[1,2]/sum_mat[1,1]
  #new eliminated row
  elim_row2<-sums_row2-(sums_row1*gauss_ratio)

  #initial (A,B)
  Ac<-0.5*(Saaa+Sabb)
  Bc<-0.5*(Sbbb+Sbaa)


  #result of Bc after elimination
  elim_Bc<-Bc-(gauss_ratio*Ac)

  #final deviation of (A, B) from mean
  fin_Bc<-elim_Bc/elim_row2[2]
  fin_Ac<-(Ac-(fin_Bc*sum_mat[1,2]))/sum_mat[1,1]

  #center of least squares fit of circle (xc,yc)
  Xc<-xmean+fin_Ac
  yc<-ymean+fin_Bc

  alpha<-fin_Ac^2+fin_Bc^2+((Saa+Sbb)/nrow(mat2))

  #radius of circle
  radius<-sqrt(alpha)

  #temporarily stores circle parameters, names them and then puts them in the globalEnv
  circle_parms<-c(Xc, yc, radius)
  names(circle_parms)<-c("Xc", "Yc", "Radius")
  circle_parms<<-circle_parms

  #generates a ggplot of the the input data and the approximated circle; puts plot in the globalEnv as circleplot
  circleplot<<-ggplot(df, aes(x=X, y=Y))+geom_point()+
    geom_point(aes(x=Xc, y=yc), color="Red", size=4)+theme(aspect.ratio = 1)

  #defines a circle fit function such that it can be added to the circleplot above; this function is available in globalEnv
  gg_circle <<- function(r, xc, yc, color="blue", fill=NA, ...) {
    x <- xc + r*cos(seq(0, pi, length.out=100))
    ymax <- yc + r*sin(seq(0, pi, length.out=100))
    ymin <- yc + r*sin(seq(0, -pi, length.out=100))
    annotate("ribbon", x=x, ymin=ymin, ymax=ymax, color=color, fill=fill, ...)
  }
  #adds the fit circle to the data.
  circleplot+gg_circle(r=radius, xc=Xc, yc=yc)
} 

circlefit(df)
circle_parms

Answer 2

I used the link that was given in comments. My code is as below

#finding circles center
p3=c(0,5,5,0,-5,0)#coordinates of point in (x1,y1,x2,y2,x3,y3) format


mat1=matrix(c(p3[1]^2+p3[2]^2,p3[2],1,p3[3]^2+p3[4]^2,p3[4],1,p3[5]^2+p3[6]^2,p3[6],1),nrow=3,ncol=3,byrow=TRUE)
mat2=matrix(c(p3[1],p3[1]^2+p3[2]^2,1,p3[3],p3[3]^2+p3[4]^2,1,p3[5],p3[5]^2+p3[6]^2,1),nrow=3,ncol=3,byrow=TRUE)
mat3=matrix(c(p3[1],p3[2],1,p3[3],p3[4],1,p3[5],p3[6],1),nrow=3,ncol=3,byrow=TRUE)
mat1
mat2
mat3


xcenter=det(mat1)/(2*det(mat3))
ycenter=det(mat2)/(2*det(mat3))
radius=sqrt((p3[1]-xcenter)^2+(p3[2]-ycenter)^2)