5.12. Planetary orbit [2]. The expression z = az2 + bry +cy2 + dr + ey + j is known as a quadratic form. The set of points (x,y), where z = 0, is a conic section. It can be an ellipse, a parabola, or a hyperbola, depending on the sign of the discriminant b -4ac. Circles and lines are special cases. The equation z = 0 can be normalized by dividing the quadratic form by any nonzero coefficient. For example, if f 0, we can divide all the other coefficients by f and obtain a quadratic form with the constant term equal to one. You can use the MATLAB meshgrid and contour functions to plot conic sections. Use meshgrid to create arrays X and Y. Evaluate the quadratic form to produce Z. Then use contour to plot the set of points where Z is zero (X,Y] meshgrid(xmin : deltax: xmax,ymin : deltay:ymax); = contour(X,Y,z, [0 0]) A planet follows an elliptical orbit. Here are ten observations of its position in the (x, y) plane: 26 Chapter 5. Least Squares x=[1.02 .95 .87 .77 .67 .56 .44 .30 .16 .011% y = [0.39 .32 .27 .22 .18 .15 .13 .12 .13 .15],; (a) Determine the coefficients in the quadratic form that fits these data in the least squares sense by setting one of the coefficients equal to one and solving a 10-by-5 overdetermined system of linear equations for the other five coefficients. Plot the orbit with x on the r-axis and y on the y-axis. Superimpose the ten data points on the plot. 5.12. Planetary orbit [2]. The expression z = az2 + bry +cy2 + dr + ey + j is known as a quadratic form. The set of points (x,y), where z = 0, is a conic section. It can be an ellipse, a parabola, or a hyperbola, depending on the sign of the discriminant b -4ac. Circles and lines are special cases. The equation z = 0 can be normalized by dividing the quadratic form by any nonzero coefficient. For example, if f 0, we can divide all the other coefficients by f and obtain a quadratic form with the constant term equal to one. You can use the MATLAB meshgrid and contour functions to plot conic sections. Use meshgrid to create arrays X and Y. Evaluate the quadratic form to produce Z. Then use contour to plot the set of points where Z is zero (X,Y] meshgrid(xmin : deltax: xmax,ymin : deltay:ymax); = contour(X,Y,z, [0 0]) A planet follows an elliptical orbit. Here are ten observations of its position in the (x, y) plane: 26 Chapter 5. Least Squares x=[1.02 .95 .87 .77 .67 .56 .44 .30 .16 .011% y = [0.39 .32 .27 .22 .18 .15 .13 .12 .13 .15],; (a) Determine the coefficients in the quadratic form that fits these data in the least squares sense by setting one of the coefficients equal to one and solving a 10-by-5 overdetermined system of linear equations for the other five coefficients. Plot the orbit with x on the r-axis and y on the y-axis. Superimpose the ten data points on the plot