"In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system."
This topic combines algebra with geometry, and covers different systems of coordinates, how to graph shapes using coordinates, and how to use coordinate geometry to solve problems.
Cartesian Coordinates: The foundation of coordinate geometry, including how to plot and locate points on a two-dimensional plane using x and y coordinates.
Distance Formula: The formula used to calculate the distance between two points on a coordinate plane.
Midpoint Formula: The formula used to calculate the midpoint between two points on a coordinate plane.
Slope: The measure of the "steepness" of a line, or rate of change between two points on a line.
Slope-Intercept Form: A linear equation where y is written as a function of x in terms of slope and y-intercept.
Point-Slope Form: A linear equation where y is written in terms of the slope and a given point on the line.
Parallel and Perpendicular Lines: Lines that are either parallel or perpendicular to each other in relation to their slope.
Conic Sections: Shapes that can be formed by intersecting a plane with a cone at different angles, including circles, ellipses, parabolas, and hyperbolas.
Transformations: Changes in the position, size, or shape of a geometric figure, including reflections, translations, rotations, and dilations.
Polar Coordinates: A system of coordinates that locate points on a plane based on their distance and angle from an origin point.
Vectors: Quantities that have both magnitude and direction, often used in representing a line segment or movement from one point to another on a plane.
Three-Dimensional Coordinates: A way of plotting points and describing space in three dimensions, using x, y, and z coordinates.
Distance Between Point and Line: A formula used to calculate the shortest distance between a point and a line on a coordinate plane.
Equation of a Circle: A formula and graph that represent all the points on a plane that are equidistant from a given center point.
Equation of an Ellipse: A formula and graph that represent the shape of an oval or stretched circle on a coordinate plane.
Equation of a Parabola: A formula and graph that represent the shape of a curve on a coordinate plane that opens either upwards or downwards.
Equation of a Hyperbola: A formula and graph that represent the shape of a curve on a coordinate plane with two separate arms that extend outwards.
Cartesian Coordinates: A system where points are located using two perpendicular axes.
Polar Coordinates: A system where points are located using a distance from the origin and an angle.
Cylindrical Coordinates: A system that uses three parameters to locate points in space: distance from the origin, angle from the x-axis, and height.
Spherical Coordinates: A system that uses three parameters to locate points in space: radial distance from the origin, polar angle from the x-axis, and azimuthal angle from the y-axis.
Barycentric Coordinates: A system of coordinates that describes a point on a triangle in terms of the ratio in which it divides the three sides of the triangle.
Homogeneous Coordinates: A system of coordinates where a point in two-dimensional space is represented by three numbers instead of two.
Trilinear Coordinates: A system of coordinates used to describe a point on a triangle in terms of the ratios of its distances from the triangle's sides.
Affine Coordinates: A system of coordinates used in affine geometry that uses vectors to locate points.
Projective Coordinates: A system of coordinates used in projective geometry where all points on a line are considered to be equal.
Logarithmic Coordinates: A system of coordinates in which curves are represented by straight lines, and intersections are represented by addition.
Screen Coordinates: A system of coordinates used to locate points on a computer screen or other digital display.
Absolute Coordinates: A system of coordinates that refers to a fixed point of reference, such as the origin.
Relative Coordinates: A system of coordinates that refers to a point of reference that is not fixed, but rather is relative to another point.
"This contrasts with synthetic geometry."
"Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight."
"It is the foundation of most modern fields of geometry, including algebraic, differential, discrete, and computational geometry."
"Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles, often in two and sometimes three dimensions."
"Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space."
"Analytic geometry can be explained more simply: it is concerned with defining and representing geometric shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations."
"That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom."
"Also known as coordinate geometry or Cartesian geometry..."
"...used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight."
"...used in aviation, rocketry, space science, and spaceflight."
"Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles..."
"Often in two and sometimes three dimensions."
"It is the foundation of most modern fields of geometry, including algebraic, differential, discrete, and computational geometry."
"It is concerned with defining and representing geometric shapes in a numerical way..."
"...extracting numerical information from shapes' numerical definitions and representations."
"That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom."
"This contrasts with synthetic geometry."
"Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space."
"That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom."