ARC Formulas

[David Bethel]

In plain Autolisp, I'm needing to calculate some arc info:

 

I have:

 

1) A known ( fixed ) chord length AB ( 84" )

2) A specified segment height ED ( from 1" to 42" )

 

I looking for the ARC radius OA / OB and the central angle AOB

 

 

I have a lot of ARC formulas but not these 2. Thanks -David

[Lee Mac]

(defun rad ( chrd segh )
 (/ (+ (* segh segh) (/ (* chrd chrd) 4.)) (* 2. segh))
)

(defun ang ( chrd segh )
 (* 2. (asin (/ chrd (* 2. (rad chrd segh)))))
)

(defun asin ( x )
 (if (<= (abs x) 1.0)
   (if (equal (abs x) 1.0 1e-
     (* x pi 0.5)
     (atan (/ x (sqrt (- 1.0 (* x x)))))
   )
 )
)

[David Bethel]

Thanks !

 

I was leaning to the perpendiculars of AD and BD to find the center. Kind of a klulge to say the least. -David

[Lee Mac]

Mine uses rearrangements of the identities shown on this (badly drawn) diagram:

 

 

s = segment height
r = radius
c = chord length
a = angle

----------------------------------------------
Radius
----------------------------------------------
s = r - sqrt(r^2 - (c/2)^2)

(r - s)^2 = r^2 - c^2/4

r^2 - 2rs + s^2 = r^2 - c^2/4

r = (s^2 + c^2/4) / 2s

----------------------------------------------
Angle
----------------------------------------------
c/2 = r sin (a/2)

a/2 = arcsin (c/2r)

a = 2 arcsin (c/2r)

[David Bethel]

Thanks, I'll bookmark this one

 

And the final production use is the radius top on the end cabinets. -David

[eldon]

You might not want to know this, but the length ED is also known as the Sagitta, from the Latin word for arrow. (It looks like the arrow in the bow AB).

 

This length ED is also called a Versine.

 

All these names help when searching the web for information

[David Bethel]

Yes,

 

I saw that on 1728.com where I snatched the image from. But then I'm not that good at Latin . -David

[Lee Mac]

This length ED is also called a Versine.

 

This image explains quite a lot regarding the trig functions:

 

http://upload.wikimedia.org/wikipedia/commons/9/9d/Circle-trig6.svg

[David Bethel]

Another reason to bookmark this 1

 

secant squared + cosecant squared = tangent + cotangent squared