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2020-11-06

b and

→ B is the midpoint of FC, so its y coordinate is zero, thus it lies on the x axis.

{\displaystyle {\vec {f}}_{0}} =

As of 4/27/18. A further generalization is given by the Veronese variety, when there is more than one input variable.

The above proof and the accompanying diagram show that the tangent BE bisects the angle ∠FEC. ⋅ y f In the United States, vertical curves in roads are usually parabolic by design. y B 4 is uniquely determined by three points {\displaystyle y=x^{2}} {\displaystyle y=-f} (opening to the right) has the polar representation. . 0 is the radius of the osculating circle at the vertex. x = The triple-angle formula

1 A proof of this sentence can be inferred from the proof of the. P 2

It effectively proves the line BE to be the tangent to the parabola at E if the angles α are equal. , = It fits several other superficially different mathematical descriptions, which can all be proved to define exactly the same curves. The presence of air resistance, for example, always distorts the shape, although at low speeds, the shape is a good approximation of a parabola. The axis of symmetry always passes through the

In general, the two vectors P Proof: can be performed for the unit parabola 2 2 The axis of symmetry of a parabola is a vertical line that divides the parabola into two congruent halves. and

+

As stated above in the lead, the focal length of a parabola is the distance between its vertex and focus.

The intersection of an upright cone by a plane {\displaystyle \sigma }

They both define curves of exactly the same shape. 0 = The P α = Proof: straight forward calculation for the unit parabola This includes the point F, which is not mentioned above. P defined by three points α l The correctness of this construction can be seen by showing that the x coordinate of It has a chord DE, which joins the points where the parabola intersects the circle. − The point B is the midpoint of the line segment FC.

which has its vertex at the origin, opens upward, and has focal length f (see preceding sections of this article).

p

{\displaystyle {\color {green}x},} The implicit equation of a parabola is defined by an irreducible polynomial of degree two: such that are parallel to the axis of the parabola.). =

2

{\displaystyle p}

{\displaystyle \;t\cdot t-t^{2}=0\;} {\displaystyle y=x^{2}}

) It can easily be shown that the parallelogram has twice the area of the triangle, so Archimedes' proof also proves the theorem with the parallelogram. {\displaystyle y=x^{2}} x 2 x {\displaystyle \sigma } , x

2 {\displaystyle V=(0,0)}

a

+ 2

+ The graph of a ∥ Otherwise, if there are two generatrices parallel to the intersecting plane, the intersection curve will be a hyperbola (or degenerate hyperbola, if the two generatrices are in the intersecting plane).

Let three tangents to a parabola form a triangle.

Remark 1: The 2-points–2-tangents property of a parabola is an affine version of the 3-point degeneration of Pascal's theorem. d

From this, y =