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A linear equation in line coordinates has the form al + bm + c = 0, where a, b and c are constants. Suppose (l, m) is a line that satisfies this equation.If c is not 0 then lx + my + 1 = 0, where x = a/c and y = b/c, so every line satisfying the original equation passes through the point (x, y).
If a coordinate curve is a straight line, it is called a coordinate line. A coordinate system for which some coordinate curves are not lines is called a curvilinear coordinate system. [13] Orthogonal coordinates are a special but extremely common case of curvilinear coordinates. A coordinate line with all other constant coordinates equal to ...
A Cartesian coordinate system in two dimensions (also called a rectangular coordinate system or an orthogonal coordinate system [8]) is defined by an ordered pair of perpendicular lines (axes), a single unit of length for both axes, and an orientation for each axis. The point where the axes meet is taken as the origin for both, thus turning ...
a coordinate line, a linear coordinate dimension; In the context of determining parallelism in Euclidean geometry, a transversal is a line that intersects two other lines that may or not be parallel to each other. For more general algebraic curves, lines could also be: i-secant lines, meeting the curve in i points counted without multiplicity, or
This produces a variation on the definition, namely the projective plane is defined as the set of lines in that pass through the origin and the coordinates of a non-zero element (,,) of a line are taken to be homogeneous coordinates of the line. These lines are now interpreted as points in the projective plane.
Radial lines (those running through the pole) are represented by the equation =, where is the angle of elevation of the line; that is, = , where is the slope of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line = perpendicularly at the point (,) has the equation = ().
The mapping from 3D to 2D coordinates is (x′, y′) = ( x / w , y / w ). We can convert 2D points to homogeneous coordinates by defining them as (x, y, 1). Assume that we want to find intersection of two infinite lines in 2-dimensional space, defined as a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0.
The international standard convention —that east is positive—is consistent with a right-handed Cartesian coordinate system, with the North Pole up. A specific longitude may then be combined with a specific latitude (positive in the northern hemisphere ) to give a precise position on the Earth's surface.