Formula to Calculate Overlap of Two Arcs on a Circle

Date: 05/27/2005 at 15:55:22
From: Adnan
Subject: Intersecting arcs on a circle

Hi there,

I have a circle and I draw an arc from 0 to 45 degrees with a blue 
pen, and another from 40 to 60 degrees with a red pen.  Now I know 
that the arcs are intersecting from 40 to 45, but I want to construct 
a formula with the 4 angle entries and one Boolean output for the
intersection answer.  

One problem is that if one arc is from 340 to 30 (50 deg) and the 
other from 350 to 355, the 360 to 0 turn has to be calculated 
separately.  It is simple to calculate it with many "if" statements, 
but I need a one line formula.

Date: 05/29/2005 at 22:48:51
From: Doctor Peterson
Subject: Re: Intersecting arcs on a circle

Hi, Adnan.

You want a simple formula (presumably in some computer language or a
spreadsheet) that will be true if two arcs, for which the starting and
ending angles are given, overlap.

The calculation is made easier if we first standardize the angles.
Suppose we are given one arc counterclockwise from a to b degrees, and
another counterclockwise from c to d.  We can first move our point of
reference so that we measure 0 degrees at a; then the two arcs are

  0 to b-a
  c-a to d-a

Now we can make sure that these angles are between 0 and 360 degrees.
One way to do this is to use a "mod" function, which gives the
remainder between 0 and 360 after division by 360; we replace each
variable as follows:

  d = mod(d-a, 360)
  c = mod(c-a, 360)
  b = mod(b-a, 360)
  a = 0

If the given angles can be negative, you have to be careful with this,
as what is often called the "mod" function (e.g. "%" in C) doesn't do
what we need for negative inputs.  See this page if you need help with
that:

  What is Mod?
    http://mathforum.org/library/drmath/sets/select/dm_mod.html 

Now, we have something like this:

  a===============b
  +-----------+---+-------+----+
  0           c===========d    360

The only way the two arcs can NOT overlap is if a<b<c<d:

  a===========b
  +-----------+---+-------+----+
  0               c=======d    360

If they do overlap, then either a<c<b<d as in the first picture, or
d<c and the second arc wraps around:

  a===============b
  +-----------+---+-------+----+
  0===========d           c====360

This gives two ways to state the condition:

  not((b<c) and (c<d))

or

  (b<c) or (d<c)

If you need to get it all in one expression without modifying the
variables first, you would use

  (mod(b-a, 360) < mod(c-a, 360)) or (mod(d-a, 360) < mod(c-a, 360))

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/30/2005 at 02:48:48
From: Adnan
Subject: Thank you (Intersecting arcs on a circle)

Many thanks, Dr. Peterson.

It is now clear.  The main confusing thing was that I had to move the
point of reference to 0 degrees.  I love math, it has such simple
techniques to solve complicated problems, and thank goodness we have
doctors like you to show us the path.

Kind regards,

Adnan