Parabola - Length of An Arc of A Parabola

Length of An Arc of A Parabola

If a point X is located on a parabola which has focal length and if is the perpendicular distance from X to the axis of symmetry of the parabola, then the lengths of arcs of the parabola which terminate at X can be calculated from and as follows, assuming they are all expressed in the same units.

This quantity, is the length of the arc between X and the vertex of the parabola.

The length of the arc between X and the symmetrically opposite point on the other side of the parabola is

The perpendicular distance, can be given a positive or negative sign to indicate on which side of the axis of symmetry X is situated. Reversing the sign of reverses the signs of and without changing their absolute values. If these quantities are signed, the length of the arc between any two points on the parabola is always shown by the difference between their values of

This can be useful, for example, in calculating the size of the material needed to make a parabolic reflector or parabolic trough.

This calculation can be used for a parabola in any orientation. It is not restricted to the situation where the axis of symmetry is parallel to the y-axis.

(Note: In the above calculation, the square-root, must be positive. The quantity ln(a), sometimes written as loge(a), is the natural logarithm of a, i.e. its logarithm to base "e".)

Read more about this topic:  Parabola

Famous quotes containing the words length of, length, arc and/or parabola:

    The light that was shadowed then
    Was seen to be our lives,
    Everything about us that love might wish to examine,
    Then put away for a certain length of time, until
    The whole is to be reviewed, and we turned
    Toward each other, to each other.
    John Ashbery (b. 1927)

    It is the vice of our public speaking that it has not abandonment. Somewhere, not only every orator but every man should let out all the length of all the reins; should find or make a frank and hearty expression of what force and meaning is in him.
    Ralph Waldo Emerson (1803–1882)

    Male urination really is a kind of accomplishment, an arc of transcendance. A woman merely waters the ground she stands on.
    Camille Paglia (b. 1947)

    Along a parabola life like a rocket flies,
    Mainly in darkness, now and then on a rainbow.
    Andrei Voznesensky (b. 1933)