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# Long description

The figure consists of three parts labelled (a) to (c) that illustrate the transformation of co-ordinates in three types of planes. In part (a), a square horizontal plane is shown. The plane has the following attributes: flat (k equals 0), alpha plus beta plus gamma equals 180 degrees, and C equals 2 pi b. On the plane, two identical parallel lines are drawn and a point is marked on each line. These lines are labelled ‘initially parallel lines remain parallel’. Beside the lines, a circle is shown. The centre of the circle is denoted by a red dot. The radius of the circle is denoted by b, and its circumference is denoted by C. A triangle is inscribed in the circle. The three interior angles of the triangle are labelled alpha, beta, and gamma. In part (b), a spherical plane is shown. The plane has the following attributes: positive curvature (k greater than 0), alpha plus beta plus gamma greater than 180 degrees, and C less than 2 pi b. On the plane, two identical parallel lines are drawn and a point is marked on each line. The parallel lines appear to be converging. These lines are labelled ‘initially parallel lines converge’. Beside the lines, a circle is shown. The centre of the circle is denoted by a red dot. The radius of the circle is denoted by b, and its circumference is denoted by C. A triangle is inscribed in the circle. The three interior angles of the triangle are labelled alpha, beta, and gamma. The radius of the circle and the circle appear to be curved. The sides of the triangle appear to be curved outward. In part (c), a plane having two curvatures is shown. The horizontal plane shown in part (a) now bends upward about one longitudinal axis and bends downward about the other longitudinal axis. The plane has the following attributes: negative curvature (k less than 0), alpha plus beta plus gamma less than 180 degrees, and C greater than 2 pi b. On the plane, two identical parallel lines are drawn and a point is marked on each line. The parallel lines appear to be diverging. These lines are labelled ‘initially parallel lines diverge’. Beside the lines, a circle is shown. The centre of the circle is denoted by a red dot. The radius of the circle is denoted by b, and its circumference is denoted by C. A triangle is inscribed in the circle. The three interior angles of the triangle are labelled alpha, beta and gamma. The radius of the circle and the circle appear to be curved. The sides of the triangle appear to be curved inward.