equation, so they are frequently described by parametric equations: where: x, y, and z are all functions of the independent variable t which ranges over the real numbers. (x0, y0, z0) is any point on the line. a, b, and c are related to the slope of the line, such that the vector (a, b, c) is parallel to the line. Conic sections In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section – though it may be degenerate, and all conic sections arise in this way. The equation will be of the form As scaling all six constants yields the same locus of zeros, one can consider conics as points in the five-dimensional projective space The conic sections described by this equation can be classified using the discriminant If the conic is non-degenerate, then: if , the equation represents an ellipse; if and , the equation represents a circle, which is a special case of an ellipse; if , the equation represents a parabola; if , the equation represents a hyperbola; if we also have , the equation represents a rectangular hyperbola. Quadric surfaces A quadric, or quadric surface, is a 2-dimensional surface in 3-dimensional space defined as the locus of zeros of a quadratic polynomial. In coordinates , the general quadric is defined by the algebraic equation Quadric surfaces include ellipsoids (including the sphere), paraboloids, hyperboloids, cylinders, cones, and planes. Distance and angle In analytic geometry, geometric notions such as distance and angle measure are defined using formulas. These definitions are designed to be consistent with the underlying Euclidean geometry. For example, using Cartesian coordinates on the plane, the distance between two points (x1, y1) and (x2, y2) is defined by the formula which can be viewed as a version of the Pythagorean theorem. Similarly, the angle that a line makes with the horizontal can be defined by the formula where m is the slope of the line. In three dimensions, distance is given by the generalization of the Pythagorean theorem: while the angle between two vectors is given by the dot product. The dot product of two Euclidean vectors A and B is defined by where θ is the angle between A and B. Transformations Transformations are applied to a parent function to turn it into a new function with similar characteristics. The graph of is changed by