Cartesian geometry, also known as analytical geometry, is a field of mathematics that uses a coordinate system to define points and geometric objects on a plane. It is named after the French mathematician René Descartes, who was the first to use this method.
Key Concepts
- Coordinate System: A coordinate system is a system for determining the position of a point on a plane. The most common coordinate system is the Cartesian coordinate system, which uses two perpendicular axes (the x-axis and the y-axis) to define the position of a point.
- Points: A point is a location on a plane. It is defined by its coordinates, which are the distances from the point to the x-axis and y-axis.
- Lines: A line is a straight path that extends in both directions. In Cartesian geometry, a line can be defined by an equation in the form of (y = mx + b), where (m) is the slope of the line and (b) is the y-intercept.
- Curves: Curves are lines that are not straight. They can be defined by equations in various forms, such as quadratic equations, exponential equations, and logarithmic equations.
Examples
- Circle: A circle is a set of points that are all the same distance from a given point (the center). The equation of a circle with center ((h, k)) and radius (r) is ((x - h)^2 + (y - k)^2 = r^2).
- Parabola: A parabola is a curve that is symmetrical with respect to a line called the axis of symmetry. The equation of a parabola with vertex ((h, k)) and opening to the right is ((x - h)^2 = 4p(y - k)), where (p) is the distance from the vertex to the focus.
- Ellipse: An ellipse is a curve that is similar to a circle but is not perfectly round. The equation of an ellipse with center ((h, k)) and semi-major axis (a) and semi-minor axis (b) is (\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1).
Further Reading
For more information on Cartesian geometry, please visit our Geometry Resources page.