;| WIKI-3PTTOBULGE
Returns the bulge of an arc defined by three points, PNT1, PNT2, and PNT3
If point 2 is nil, returns 0.
Edit the source code for this function at
3pttobulge (AutoLISP function)
External function references:
WIKI-ASIN
WIKI-TAN
Theory:
/\
/ \
/ \
/ \
/ delta \
/ \
/ R
/ \
/ \
/ \
/ \
/ \
/ __chord___ ----- 1
3 ------ | *
. ANG2 B _.*
- 2 - ._|..-
Bulge = 2*B/chord
Bulge = tan(delta/4)
Theory (in classic triangle geometry terms):
We want to find the radius, R.
Points 1, 2, and 3 form a triangle with sides 1, 2, and 3 opposite them.
A unique circle passes through the three points.
Sides 1, 2, and 3 are all chords of the circle.
Side 2 (opposite point 2) is the chord of this segment of the circle.
It happens to be true that angle 2 doubled plus delta equals 360 or 2pi.
This is intuitive when point 2 is at the midpoint of the circle segment.
Consider how angle 2 acts in the following cases:
Delta Angle 2
0 180
90 135
180 90
270 45
360 0
But it is also true when point 2 is not at the midpoint.
Therefore delta = 2*pi - 2*ang2 = 2*(pi-ang2) = 2*(ang1 + ang3)
and delta/2=pi-ang2 (equation 1)
A line from the center of the circle is a perpendicular bisector of any of the three lines
Therefore (side2/2)/R=sin(delta/2). (equation 2)
We'd like to get R just in terms of side2 and ang2. Luckily we have this trig identity,
sin(pi-theta)=sin(theta), (equation 3)
so we can say from equations 1 and 3 that sin(delta/2)=sin(pi-ang2)=sin(ang2) (equation 4)
and from equations 2 and 4 that (side2/2)/R=sin(ang2). (equation 5)
Solving for R we get R=side2/(2*sin(ang2)). (equation 6)
That gives us R in terms of things we know.
In fact, from the law of sines (a/sin(A)=b/sin(B)=c/sin(C)), since 2R=side2/sin(ang2),
in geometry triangle terms (where "a" is a triangle side and "A" is its opposite angle),
R=a/(2*sin(A)) for any of the three points/angles/opposite sides
|;
(DEFUN
WIKI-3PTTOBULGE (PNT1 PNT2 PNT3 / ANG1 ANG2 ANG3 BULGE CHORD DELTA DELTA1 R)
(COND
((NOT PNT2) 0)
(T
(SETQ
CHORD
(DISTANCE PNT1 PNT3)
ANG2
(- (ANGLE PNT2 PNT1) (ANGLE PNT2 PNT3))
;;We use the theory above to write an expression for R
;;using chord and ang2
;;Sin of ang2 (and thus R) will be negative if the arc is clockwise.
R
(/ CHORD (* 2 (SIN ANG2)))
DELTA1
(* 2 (WIKI-ASIN (/ CHORD (* 2 R))))
ANG1
(ABS (- (ANGLE PNT1 PNT3) (ANGLE PNT1 PNT2)))
ANG1
(ABS
(IF (> ANG1 PI)
(- ANG1 (* 2 PI))
ANG1
)
)
ANG3
(ABS (- (ANGLE PNT3 PNT1) (ANGLE PNT3 PNT2)))
ANG3
(ABS
(IF (> ANG3 PI)
(- ANG3 (* 2 PI))
ANG3
)
)
DELTA
(* 2 (+ ANG1 ANG3))
BULGE
(* (IF (MINUSP R)
-1
1
)
(WIKI-TAN (/ DELTA 4.0))
)
)
)
)
)