Visualization of the Sagitta with Pythagorean theorem relationship

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the sagitta (sometimes abbreviated as sag) of a

circular arc A circular arc is the arc of a circle between a pair of distinct points. If the two points are not directly opposite each other, one of these arcs, the minor arc, subtends an angle at the center of the circle that is less than radians (180 ...

is the distance from the

midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dim ...

of the arc to the

of its chord. It is used extensively in architecture when calculating the arc necessary to span a certain height and distance and also in optics where it is used to find the depth of a spherical mirror or lens. The name comes directly from

Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...

''sagitta'', meaning an "

arrow An arrow is a fin-stabilized projectile launched by a bow. A typical arrow usually consists of a long, stiff, straight shaft with a weighty (and usually sharp and pointed) arrowhead attached to the front end, multiple fin-like stabilizers c ...

Formulas

In the following equations,

s

denotes the sagitta (the depth or height of the arc),

r

equals the

radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...

of the

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

, and

l

the length of the chord spanning the base of the arc. As

\tfrac12l

and

r - s

are two sides of a

right triangle A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle i ...

with

r

as the

hypotenuse In geometry, a hypotenuse is the side of a right triangle opposite to the right angle. It is the longest side of any such triangle; the two other shorter sides of such a triangle are called '' catheti'' or ''legs''. Every rectangle can be divided ...

, the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

gives us :

r^2 = \left(\tfrac12l\right)^2 + \left(r-s\right)^2.

This may be rearranged to give any of the other three: :

r &= \frac = \frac+\frac. \end

The sagitta may also be calculated from the

versine The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'', :

s\approx \frac.

Alternatively, if the sagitta is small and the sagitta, radius, and chord length are known, they may be used to estimate the arc length by the formula :

a\approx l+\frac\approx l+\frac,

where is the length of the arc; this formula was known to the Chinese mathematician

Shen Kuo Shen Kuo (; 1031–1095) or Shen Gua, courtesy name Cunzhong (存中) and Art name#China, pseudonym Mengqi (now usually given as Mengxi) Weng (夢溪翁),Yao (2003), 544. was a Chinese polymath, scientist, and statesman of the Song dynasty (960� ...

, and a more accurate formula also involving the sagitta was developed two centuries later by

Guo Shoujing Guo Shoujing (, 1231–1316), courtesy name Ruosi (), was a Chinese astronomer, hydraulic engineer, mathematician, and politician of the Yuan dynasty. The later Johann Adam Schall von Bell (1591–1666) was so impressed with the preserved astro ...

Applications

Architects, engineers, and contractors use these equations to create "flattened" arcs that are used in curved walls, arched ceilings, bridges, and numerous other applications. The sagitta also has uses in physics where it is used, along with chord length, to calculate the radius of curvature of an accelerated particle. This is used especially in

bubble chamber A bubble chamber is a vessel filled with a superheated transparent liquid (most often liquid hydrogen) used to detect electrically charged particles moving through it. It was invented in 1952 by Donald A. Glaser, for which he was awarded th ...

experiments where it is used to determine the momenta of decay particles. Likewise historically the sagitta is also utilised as a parameter in the calculation of moving bodies in a centripetal system. This method is utilised in Newton's Principia.

References

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External links

Calculating the Sagitta of an Arc
Architectural terminology Geometric measurement

Formulas

Applications

See also

References

External links