The homogenous representation of a circle is given by x 2 </msup> + y

Craig Mendoza 2022-06-24 Answered
The homogenous representation of a circle is given by x 2 + y 2 + 2 g x z + 2 f y z + c z 2 = 0 (or, equivalently, if we set z = 1, x 2 + y 2 + 2 g x + 2 f y + c = 0). Now, given 3 points (in a homogenous form), we can solve a system of linear equations and retrieve the unknowns f, g and c.
This is all very nice (because of linear algebra), but what do these unknowns actually represent with respect to the circle? Which of these numbers represent the x and y coordinates of a circle and which one represents the radius?
Apparently, f and g would be the x and y coordinate of the center of the circle? Why is that the case? I would like to see a proof/derivation of it. Also, what is the radius then?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Braylon Perez
Answered 2022-06-25 Author has 34 answers
0 = x 2 + y 2 + 2 g x + 2 f y + c = x 2 + 2 g x + g 2 + y 2 + 2 f y + f 2 + ( c g 2 f 2 ) = ( x + g ) 2 + ( y + f ) 2 ( f 2 + g 2 c )
This equation says that the squared distance of the point ( x , y ) from the point ( g , f ) is f 2 + g 2 c, which describes a circle centered at ( g , f ) with radius f 2 + g 2 c .
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-06-10
Determine whether the given set S is a subspace of the vector space V.
A. V=P5, and S is the subset of P5 consisting of those polynomials satisfying p(1)>p(0).
B. V=R3, and S is the set of vectors (x1,x2,x3) in V satisfying x16x2+x3=5.
C. V=Rn, and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.
D. V=C2(I), and S is the subset of V consisting of those functions satisfying the differential equation y″−4y′+3y=0.
E. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=5.
F. V=Pn, and S is the subset of Pn consisting of those polynomials satisfying p(0)=0.
G. V=Mn(R), and S is the subset of all symmetric matrices
asked 2021-09-14

The coefficient matrix for a system of linear differential equations of the form y′=Ay has the given eigenvalues and eigenspace bases. Find the general solution for the system.
λ1=2{[431]},λ2=2{[120],[231]}

asked 2022-07-28

X-3Z= -3

2X+KY-Z= -2

X+2Y-KZ= 1

asked 2022-04-29

3.1 where 1 is repeating

asked 2021-09-30

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given row echelon form. Solve the system by back substitution. Assume that the variables are namedx1,x2, from left to right.
[137101400001]

asked 2021-05-05
The reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use x,y;x,y; or x,y,z;x,y,z; or x1,x2,x3,x4 as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution. [1001−40][
asked 2021-08-10

The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. [101230012340]
image

New questions