"…a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space."
Understanding the coordinate system, Cartesian plane, and polar coordinates.
Cartesian Coordinates: The system of plotting points on a plane by using two perpendicular coordinate lines, usually called the x-axis (horizontal) and y-axis (vertical).
Polar Coordinates: The system of plotting points on a plane using an angle and a distance from a fixed point, usually called the origin.
Distance Formula: The formula used to find the distance between two points on a plane using their x and y coordinates.
Midpoint Formula: The formula used to find the midpoint between two points on a plane using their x and y coordinates.
Slope Formula: The formula used to find the slope of a line passing through two points on a plane.
Parallel and Perpendicular Lines: The concept of lines that are parallel or perpendicular to each other on a plane.
Equation of a Line: The formula used to represent a line on a plane using its slope and y-intercept.
Slope-Intercept Form: The form of a linear equation that expresses the y-coordinate in terms of the slope and y-intercept.
Standard Form: The form of a linear equation that expresses both x and y coordinates in terms of the coefficients of the x and y variables.
Point-Slope Form: The form of a linear equation that expresses the slope and one point on the line to find the equation.
Parallelism and Perpendicularity of Lines: The rules used to determine if two lines are parallel or perpendicular.
Reflections and Rotations: The transformations of a figure or shape with respect to a particular point or axis.
Transformations of Coordinate Systems: The process of using a coordinate system to transform shapes and figures.
Equations of Circles: The formula used to find the equation of a circle on a plane using the distance between the center and the radius.
Conic Sections: The shapes formed when intersecting a plane with a cone at various angles and distances.
Distance Between Point and Line: The formula used to find the shortest distance between a point and a line on a plane.
Formulas of Ellipses, Hyperbolas, and Parabolas: The formula used to identify the equations of conic sections such as ellipses, hyperbolas, and parabolas.
Three-Dimensional Coordinate Systems: The system of plotting points in a three-dimensional space.
Equations of Planes: The formula used to identify the equations of a plane in a three-dimensional space.
Distance Between Two Planes: The formula used to find the shortest distance between two non-parallel planes in a three-dimensional space.
Cartesian coordinates: A set of coordinates in which an ordered pair (x, y) represents a point on a 2-dimensional plane.
Polar coordinates: A set of coordinates in which a point is represented by an ordered pair (r, θ) where "r" represents the distance from the origin to the point and "θ" represents the angle between the positive x-axis and a line connecting the origin and the point.
Cylindrical coordinates: A set of coordinates in which a point in 3-dimensional space is represented by an ordered triple (r, θ, z) where "r" represents the distance from the origin to a point in a plane perpendicular to the z-axis, "θ" represents the angle between the positive x-axis and a line connecting the origin and the projection of the point on that plane, and "z" represents the height of the point above the xy-plane.
Spherical coordinates: A set of coordinates in which a point in 3-dimensional space is represented by an ordered triple (r, θ, φ) where "r" represents the distance from the origin to the point, "θ" represents the angle between the positive x-axis and a line connecting the origin and the point projected onto the xy-plane, and "φ" represents the angle between the positive z-axis and the line connecting the origin and the point.
Homogeneous coordinates: A set of coordinates in which a point in n-dimensional space is represented by an ordered set of n+1 coordinates. These are used in computer graphics where affine transformations (such as translation, rotation, scaling, and shearing) can be represented as matrix multiplications.
Projective coordinates: A set of coordinates in which a point in 2-dimensional plane is represented by an ordered triple (x, y, z) where x, y and z are not all zero. When two projective points have the same ratio of x to z and y to z, they represent the same point in the Cartesian plane.
Barycentric coordinates: A set of coordinates used to describe a point inside a triangle. The coordinates are ratios of the distances from the point to the sides of the triangle.
Tangent coordinates: A set of coordinates used to describe a point on a curve. Instead of the usual Cartesian coordinates (x, y), each point is represented by the intersection of the curve and its tangent line at that point.
Bi-polar coordinates: A set of coordinates used to describe a point in a plane, in which the coordinates are based on two focal points (foci) rather than a single origin. A point is represented by two distances: the distance from each focus to the point.
Elliptic coordinates: A set of coordinates used to describe a point in a plane, in which the coordinates are based on two focal points and the distance between them. A point is represented by two distances: the distance from each focus to the point, and the distance between the two foci.
"…coordinates…uniquely determine the position of the points or other geometric elements..."
"The order of the coordinates is significant…"
"They are sometimes identified by their position in an ordered tuple and sometimes by a letter."
"The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers…"
"The coordinates are taken to be real numbers in elementary mathematics…"
"The coordinates…may be complex numbers…"
"The coordinates…may be elements of a more abstract system such as a commutative ring."
"The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa…"
"The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry."
"...geometry…"
"...points or other geometric elements…"
"…on a manifold such as Euclidean space."
"The coordinates are taken to be real numbers in elementary mathematics…"
"The coordinates are sometimes identified…by a letter."
"The use of a coordinate system allows problems in geometry to be translated into problems about numbers…"
"The coordinates…may be complex numbers or elements…"
"The coordinates…may be elements of a more abstract system such as a commutative ring."
"A coordinate system is a system that uses… coordinates to uniquely determine the position of the points or other geometric elements…"
"The use of a coordinate system allows problems in geometry to be translated into problems about numbers…"