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.
(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: Arc (geometry), Arc Length
Famous quotes containing the words length of, length, arc and/or parabola:
“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)
“Rivers must have been the guides which conducted the footsteps of the first travelers. They are the constant lure, when they flow by our doors, to distant enterprise and adventure; and, by a natural impulse, the dwellers on their banks will at length accompany their currents to the lowlands of the globe, or explore at their invitation the interior of continents.”
—Henry David Thoreau (1817–1862)
“You say that you are my judge; I do not know if you are; but take good heed not to judge me ill, because you would put yourself in great peril.”
—Joan Of Arc (c.1412–1431)
“Along a parabola life like a rocket flies,
Mainly in darkness, now and then on a rainbow.”
—Andrei Voznesensky (b. 1933)