Geometry+ Parabola
Parabola is a class of curves on a plane with the following defining properties.
For each curve of this class (that is, for each parabola) there is one specific point called its focus and a specific straight line called its directrix not going through a focus, such that this parabola consists of all points on a plane (or is a locus of all points on a plane) equidistant from its focus and directrix.
As we see, the position of a focus point F and a directrix d (should not go through focus F) uniquely identifies a parabola.
Obviously, a midpoint of a perpendicular from focus F onto directrix d belongs to a parabola defined by them.
From this point we can draw a parabola point by point maintaining equality of the distance from each point to both a focus and a directrix.
Choosing an X-axis perpendicular to a directrix with an origin of coordinates 0 at midpoint between a focus and a directrix, we will have the following picture of a parabola on a coordinate plane.
We can derive an equation that defines this parabola using the main characteristic property of every point on this parabola to be equidistant from focus and a directrix.The distance from any point P(x,y) on a parabola to a focus F that has coordinates (c,0) can be calculated using the known formula of a distance between two points.
The distance from point P to a directrix is calculated along a perpendicular from point P to a directrix that is parallel to X-axis.
If the distance between a focus and a directrix is 2c, as on a drawing above, the equality of the distances to a focus and a directrix is
√(x−c)²+(y−0)² = x−(−c)
We can simplify this as follows
(x−c)²+(y−0)² = (x+c)²
y² = (x+c)²−(x−c)²
y² = 4c·x
The only parameter that defines the shape of a parabola is c (half of a distance between a focus and a directrix), which is called the focal distance (or focal length) of a parabola.
The perpendicular from a focus to a directrix is an axis of symmetry of a parabola.
The midpoint of this perpendicular is called a vertex of a parabola.
Another item of interest is parabola's focal width. This is the length of a segment drawn through a focus parallel to a directrix with its endpoints being its intersections with a parabola.
Using a formula of a parabola y²=4c·x we determine that if x=c then y²=4c² and y=2c, which is a half of a segment described above.
Therefore, parabola's focal width is 4c.
Let's derive a formula r=r(θ) for a parabola in polar coordinates with an origin at its focus and base axis perpendicular to a directrix.
The distance from point (r,θ) to a focus is, obviously, r.The distance from this point to a directrix is
AB = AF + FB = 2c+r·cos(θ).
From this the equation of a parabola in polar coordinates is
r = 2c/[1−cos(θ)]
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