A straight line passing through the midpoint of a segment and perpendicular to it is called the perpendicular bisector of the segment. The circle passing through the vertices of a triangle is called the circumcircle of the triangle. The center O[ of the circumcircle. called the circumcemer, is the point where the perpendicular bisectors of the sides of the triangle | . Curves on Plane 87 . Parametric equations of a curve. Parametric equations of a curve on the plane have the form x t y t where x and y are treated as the coordinates of some point A for each value of the variable parameter t. In general the variables x and y vary with t and the point A moves on the plane. Parametric equations play an important role in applied mathematics and mechanics where they are called the equations of motion of a mass point. The parameter t has the meaning of time. Remark 1. Eliminating the parameter t from equations we obtain an equation of the curve in the form . Remark 2. In different problems the variable parameter in equations may have different meanings. Example 11. The circle of radius a centered at the origin is described by the following parametric equations for the Cartesian system x a cos t y a sin t. By squaring these equations and by adding them we obtain the equation of the circle in the form x2 y2 a2. Example 12. The spiral of Archimedes a the hyperbolic spiral b and the logarithmic spiral c are described by the following equations in the polar coordinate system a p aO b p - c p a9. O The parametric equations for the Cartesian coordinates of these curves have the form a x aO cos O y aO sin O b a cos O a sin O c x a9 cos O y a9 sin O. In all three cases the variable parameter is the polar angle O. . Algebraic curves. The curves given by algebraic equations of the form Ax By C 0 Ax2 Bxy Cy2 Dx Ey F 0 Ax3 Bx2y Cxy2 Dy3 Ex2 Fxy Gy2 Hx ly K 0 in a rectangular Cartesian coordinate system are called algebraic curves on the plane. A curve given by an algebraic equation of degree n in a rectangular Cartesian coordinate system is called an nth-order algebraic curve. When passing from one rectangular Cartesian coordinate system to another the degree of the equation of an algebraic curve does not change . any nth-order algebraic curve remains such in any rectangular Cartesian .
