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angle azimuth called characteristic roots characteristic vector coefficients Compute cone coordinate axes coordinate system corresponding curve cylinder determined diagonal elements diagonal matrix direction cosines direction numbers ellipsoid equation f(x equation of formula fcth Find an equation given plane gnomonic projection homogeneous coordinates hyperbola hyperboloid ILLUSTRATIVE EXAMPLES ith row jth column Lemma line joining line segment linear combination linearly independent nonsingular matrix nonzero vectors obtain origin orthogonal matrix orthogonal transformation paraboloid parametric equations perpendicular Pi and P2 planar rotations plane sections planes of symmetry planes parallel points Pi polynomial projective transformation proved the following quadratic form quadric surface real numbers rectangular coordinates regulus result rotation of axes scalar multiple set of direction set of equations sphere spherical coordinates square matrix surface of revolution symmetric matrix tangent plane Theorem tion translation of axes unit vector vertex zero