Conics

A conic or conic section is a plane curve that can be obtained by intersecting a cone with a plane that does not go through the vertex of the cone. There are three possibilities, depending on the relative position of the cone and the plane. If no line of the cone is parallel to the plane, the intersection is a closed curve, called an ellipse. If one line of the cone is parallel to the plane, the intersection is an open curve whose two ends are asymptotically parallel; this is called a parabola. Finally, there may be two lines in the cone parallel to the plane; the curve in this case has two open pieces, and is a hyperbola.

In mathematics, parametric equations bear slight similarity to functions: they allow one to use arbitrary values, called parameters, in place of independent variables in equations, which in turn provide values for dependent variables. A simple kinematical example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion. Abstractly, a relation is given in the form of an equation, and it is shown also to be the image of functions from items such as R. It is therefore somewhat more accurately defined as a parametric representation. It is part of regular parametric representation.

Converting a set of parametric equations to a single equation involves solving one of the equations usually the simplest of the two for the parameter. Then the solution of the parameter is substituted into the remaining equation, and the resulting equation is usually simplified. It should be noted that the parameter is never present when the equation is in singular form; it must cancel out during conversion. Or, the process put simply: the simultaneous equations need to be solved for the parameter, and the result will be one equation. Additional steps need to be performed if there are restrictions on the value of the parameter.

The arc length of a circle is the distance from one point on the circumference to another point on the circumference, traveling along the edge of the circle. Because we know that the measure of the central angle is equal to the arc it intercepts, by dividing that measure by 360 degrees, we find out what fraction of the circumference that the arc covers. The polar coordinate systems are coordinate systems in which a point is identified by a distance from some fixed feature in space and one or more subtended angles. They are the most common systems of coordinates. The term polar coordinates often refer to circular coordinates two-dimensional. Other commonly used polar coordinates are cylindrical coordinates and spherical coordinates both three-dimensional.

Kepler believed firmly in the Copernican system. In retrospect, the reason that the orbit of Mars was particularly difficult was that Copernicus had correctly placed the Sun