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In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the ...
Coordinate surfaces of the three-dimensional parabolic coordinates. The red paraboloid corresponds to τ=2, the blue paraboloid corresponds to σ=1, and the yellow half-plane corresponds to φ=-60°. The three surfaces intersect at the point P (shown as a black sphere) with Cartesian coordinates roughly (1.0, -1.732, 1.5).
Analogous to confocal ellipses and hyperbolas, the plane can be covered by an orthogonal net of parabolas, which can be used for a parabolic coordinate system. The net of confocal parabolas can be considered as the image of a net of lines parallel to the coordinate axes and contained in the right half of the complex plane by the conformal map w ...
Orthogonal trajectories are used in mathematics, for example as curved coordinate systems (i.e. elliptic coordinates) and appear in physics as electric fields and their equipotential curves. If the trajectory intersects the given curves by an arbitrary (but fixed) angle, one gets an isogonal trajectory .
Real algebraic geometry is the study of real algebraic varieties. The fact that the field of the real numbers is an ordered field cannot be ignored in such a study. For example, the curve of equation x 2 + y 2 − a = 0 {\displaystyle x^{2}+y^{2}-a=0} is a circle if a > 0 {\displaystyle a>0} , but has no real points if a < 0 {\displaystyle a<0} .
For example, to study the equations of ellipses and hyperbolas, the foci are usually located on one of the axes and are situated symmetrically with respect to the origin. If the curve (hyperbola, parabola , ellipse, etc.) is not situated conveniently with respect to the axes, the coordinate system should be changed to place the curve at a ...
Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple. In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates.
The Euclidean plane R 2 is embedded in the real projective plane by adjoining a line at infinity (and its corresponding points at infinity) so that all the lines of a parallel class meet on this line. On the other hand, starting with the real projective plane, a Euclidean plane is obtained by distinguishing some line as the line at infinity and ...