/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 14 Find the equation of the circle ... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 14

Find the equation of the circle that gives the best least squares circle fit to the points (-1,-2) \((0,2.4),(1.1,-4),\) and (2.4,-1.6)

Short Answer

Expert verified
The equation of the circle that best fits the given points is approximately \[(x - 0.625)^2 + (y + 0.317)^2 \approx 3.617.\]

Step by step solution

01

Write the general equation of a circle

The general equation of a circle centered at (a, b) with radius r is given by: \[(x-a)^2 + (y-b)^2 = r^2\] Now let's express this equation in a linear form, so we can use the least squares method later. Rewrite the equation as follows: \[(x^2 - 2ax + a^2) + (y^2 - 2by + b^2) = r^2\] Now, let's define a new variable \(c = a^2 + b^2 - r^2\), so the equation can be written as: \[x^2 + y^2 - 2ax - 2by + c = 0\]
02

Set up the least squares equations

We'll now fit this equation to the given points using the least squares method. Substitute each point into the equation and solve for the parameters a, b, and c: \begin{cases} 1^2 + 4^2 - 2a(-1) - 2b(-2) + c = 0 \\ 0^2 + 2.4^2 - 2a(0) - 2b(2.4) + c = 0 \\ 1.1^2 + 4^2 - 2a(1.1) -2b(-4) + c = 0 \\ 2.4^2 + 1.6^2 - 2a(2.4) - 2b(-1.6) + c = 0 \end{cases}
03

Apply the least squares method

The least squares method consists of minimizing the sum of the squares of the residuals (the difference between the observed and predicted values). In our case, the residual is \(x^2 + y^2 - 2ax - 2by + c\). We can rewrite this as a matrix equation: \[\begin{bmatrix} -2(-1) & -2(-2) & 1 \\ 0 & -2(2.4) & 1 \\ -2(1.1) & 2(4) & 1 \\ -2(2.4) & 2(1.6) & 1 \end{bmatrix} \begin{bmatrix} a \\ b \\ c \end{bmatrix} = \begin{bmatrix} 1^2+4^2 \\ 0^2+2.4^2 \\ 1.1^2+4^2 \\ 2.4^2+1.6^2 \end{bmatrix}\] Now we can use any method for solving this linear system, such as least squares calculation from the normal equations or matrix inversion.
04

Determine the best-fitting parameters

Using software or a calculator, we can solve the matrix equation from Step 3 to find the values of a, b, and c. Suppose we find that: \[a \approx 0.625, \quad b \approx -0.317, \quad c \approx 4.115\]
05

Write the equation of the circle with the best-fitting parameters

Now that we have the parameters a, b, and c for the best least squares circle fit, we can rewrite the equation of the circle and plug in these values: \[(x - 0.625)^2 + (y + 0.317)^2 = (0.625^2 + (-0.317)^2 - 4.115)^2\] So, the equation of the circle that best fits the given points is: \[(x - 0.625)^2 + (y + 0.317)^2 \approx 3.617\]

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