Wednesday, May 18, 2016

Geometry: The Sagitta in Circles

Geometry:  The Sagitta in Circles

When the word Sagitta is mentioned, we would associate sagitta with a small constellation between the constellations Aquila and Cygnus.  Sagitta is one of the smallest constellations in astronomy.  In mythology, the Sagitta refers to an arrow, involved in a number of myths:  (1) the arrow Hercules used to kill an eagle to free Prometheus, (2) the arrow used by Apollo to avenge Asclepius’ death, and (3) the arrow Eros used to shoot Zeus to make him fall in love with Ganymede. 

But did you know that sagitta was referred to a length in geometry.  The sagitta is the length from the center of a circular arc to its base (defined by the circle’s chord). 



Variables: 
r = radius of the circle
c = chord length
s = sagitta
θ = angle between radius and line that connects center to end-chord line (see the diagram above)

Derivation:  Calculating the length of the sagitta

Given radius and angle:

cos θ = (r – s) / r
r * cos θ = r – s
s = r – r cos θ
s = r * (1 – cos θ)

Example:  r = 5, θ = 60°

s = 5 * (1 – cos 60°)
s = 2.5

Given radius and chord length:

(r – s)^2 + (c/2)^2 = r^2
(r – s)^2 = r^2 – (c/2)^2
r – s = √(r^2 – (c/2)^2)
s = r - √(r^2 – (c/2)^2)

Example:  c = 10, r = 6

s = 6 - √(6^2 – (10/2)^2)
s ≈ 2.68338

Sources:

“Sagitta (geometry)”.  Wikipedia.  https://en.wikipedia.org/wiki/Sagitta_(geometry)  Retrieved May 6, 2016

“Sagitta Constellation”  Constellation Guide.  Constellations:  A Guide to the Night Sky  http://www.constellation-guide.com/constellation-list/sagitta-constellation/   Retrieved May 18, 2016


This blog is property of Edward Shore, 2016.

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