versine

The versine values for the a perfect circular curve would have the same number.

Or to find versine of a given constant radius curve, we can use:

The relationship of versine, chord and radius is derived from the Pythagorean theorem.

Based on the versine values, the radius of that circular curved track can be approximated to:

For a straight line, the versine of any chord is zero, so this measurement characterizes the straightness of the curve.

The Hallade method is use the chord to continuously measure versine in an overlapping pattern along the curve.

The versine of the chord, which is equal to this measured offset value can be calculated using the approximation of:

The term versine is also sometimes used to describe deviations from straightness in an arbitrary planar curve, of which the above circle is a special case.

Defined the versine (utkrama-jya).

There are several related functions, most notably the haversine, half the versine, known in the haversine formula of navigation.

Illustrated this way, the sine is vertical (rectus) while the versine is flipped on its side (versus); both are distances from C to the circle.

The formulas could equally be written in terms of any multiple of the haversine, such as the older versine function (twice the haversine).

Various other permutations on these identities are possible: for example, some early trigonometric tables used not sine and cosine, but sine and versine).

This figure also illustrates the reason why the versine was sometimes called the sagitta, Latin for arrow, from the Arabic usage sahem of the same meaning.

The trigonometric functions of sine and versine, from which it was trivial to derive the cosine, were used by the mathematician, Aryabhata, in the late 5th century.

He was also the first to specify sine and versine (1 cos x) tables, in 3.75 intervals from 0 to 90 , to an accuracy of 4 decimal places.

The sagitta (also known as the versine) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the arc of the circle.

Given a chord between two points in a curve, the perpendicular distance v from the chord to the curve (usually at the chord midpoint) is called a versine measurement.

The versine of a circular curve for a known length of chord is the perpendicular distance between the mid-point of the chord and the mid-point of its arc.

If the arc ADB is viewed as a "bow" and the chord AB as its "string", then the versine CD is clearly the "arrow shaft".

So I treated H-above-G, ten klicks, as a versine, applied the haversine rule and got four degrees thirty-seven minutes or two hundred seventy-seven kilometers to the theoretical horizon.

They used the words jya for sine, kojya for cosine, utkrama-jya for versine, and otkram jya for inverse sine.

The sagitta may also be calculated from the versine function, for an arc that spans an angle of , and coincides with the versine for unit circles:

If the curve needs to be of a desired constant radius, which will usually be determined by physical obstructions and the degree of cant which is permitted, the versine can be calculated for the desired radius using this approximation.

As θ goes to zero, versin(θ) is the difference between two nearly equal quantities, so a user of a trigonometric table for the cosine alone would need a very high accuracy to obtain the versine, making separate tables for the latter convenient.