You are asked to fit a circle with unknown radius r but known center (cx, cy)...
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You are asked to fit a circle with unknown radius r but known center (cx, cy) to n points (x, y), ie [1, n]. Points on a circle satisfy: (x x) + (y - c) = = r. (Xn. Yn) (x, y) r=? (x2,32) (Cx, Cy) (X4,Y4) (X3,Y3) (a) Specify the over-constrained system of linear equations in matrix/vector form. If this is not possible, briefly explain why. (b) Complete the following if it is possible to solve the system: i. How many points n are needed for the system to be over-constrained? Why? ii. Give the quadratic-error function for this least-squares estimator in matrix/vec- tor form. You need to specify the dimension of your matrices and vectors and their elements. iii. Differentiate this error function (in matrix/vector form) and set it equal to zero to obtain the normal equations. iv. Solve for r. Show that the least-squares solution for the radius r is simply the average distance of each point to the center. You are asked to fit a circle with unknown radius r but known center (cx, cy) to n points (x, y), ie [1, n]. Points on a circle satisfy: (x x) + (y - c) = = r. (Xn. Yn) (x, y) r=? (x2,32) (Cx, Cy) (X4,Y4) (X3,Y3) (a) Specify the over-constrained system of linear equations in matrix/vector form. If this is not possible, briefly explain why. (b) Complete the following if it is possible to solve the system: i. How many points n are needed for the system to be over-constrained? Why? ii. Give the quadratic-error function for this least-squares estimator in matrix/vec- tor form. You need to specify the dimension of your matrices and vectors and their elements. iii. Differentiate this error function (in matrix/vector form) and set it equal to zero to obtain the normal equations. iv. Solve for r. Show that the least-squares solution for the radius r is simply the average distance of each point to the center.
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a It is not possible to specify this system of equations in a matrixvector form because the equation... View the full answer
Related Book For
Statistics For Business And Economics
ISBN: 9780132745659
8th Edition
Authors: Paul Newbold, William Carlson, Betty Thorne
Posted Date:
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