Arc relationships

A knowledge of arc relationships is required when working with curves in AutoCAD design and programming.

Definitions

r=radius

delta=included angle

tan(angle)=sin(angle)/cos(angle)

Units

delta in radians

=delta in degrees*pi/180

=delta in degrees/57.2957786666 approx.

delta in degrees

=delta in radians*180/pi

=delta in radians*57.2957786666 approx.

Relationships

For all these relationships, angles are in radians. Angle 2 is the angle of triangle 123 at Point 2

Arc length=r*delta

Chord=r*tan(delta/2)

Bulge=tan(delta/4)

=b/(chord/2)

=2b/chord

$= \tan \left[ \frac{1}{2}\arcsin \left( \frac{chord}{2R} \right) \right]$

Tangent

=r*tan(delta/2)

=r*sin(delta/2)/cos(delta/2)

Delta

=4*arctan(bulge)

R

=chord/(2*sin(delta/2)))

=arc length/delta

$= \frac{chord}{2\sin \left(angle2 \right)}$

Angle 2 (even when point 2 is not at midpoint of arc)

=(2*pi-delta)/2

=pi-(angle1+angle3)

=arcsin(chord/(2*R))