MZ Circle

1
MZ Circle

• Bala Muralikrishnan
• Dept of MEES, UNCC

2
Minimum Zone Circle

• All data points must be contained within two
  concentric circles such that the difference in
  radii between them is the smallest

ri R acos(?i)bsin (?i)
Zone 2h
ri R h acos(?i)bsin (?i)
ri R -h acos(?i)bsin (?i)
3
MZ circle implementation

• Arbitrarily choose four points.
• Fit a limacon through the points
• r1 Rh acos(?1)bsin (?1)
• r2 R-h acos(?2)bsin (?2)
• r3 Rh acos(?3)bsin (?3)
• r4 R -h acos(?4)bsin (?4)
• Solve for (R,h,a,b)
• If h is negative, the starting assumption of
  point one lying
• outside the circle is wrong.

4

• If no other data point lies furthur radially from
  the limacon either outside or inside, we have a
  solution
• Else, identify the point lying farthest radially
  either outside or inside
• Replace it by one of the 4 points currently used,
  but making sure the alternating rule is obeyed.

5
Alternating Rule

• Alternating Rule says that the 4 points that
  now lie either on the inner or outer circle
  they must alternate I.e, if the 1st lies on the
  inner circle, the 2nd must lie on the outer
  circle etc

6
HINTS
7
Minimum Zone circle

• Function R,a,b,h mzcircle(Y)
• Arbitrarily select 4 points to start
• Fit a limacon through the four points
• Set flag 1
• While flag 1
• Find the point that lies farthest away from the
  circles
• If there is no such point
• Flag 0
• Else
• replace one of the points with the new point
  using the alternating rule
• fit a limacon through the new points
• End
• End

8
functions

• You will need two functions
• Function R,a,b,h Limaconfit(p1,p2,p3,p4,Y,thet
  a)
• Function p1,p2,p3,p4 alternatingrule(p1,p2,p3,
  p4,index,h,deviation)

9
Limacon fit function

• function R,a,b,h limaconfit(p1,p2,p3,p4,Y,thet
  a)
• solve the system of 4 equations in 4 variables
• R h acos(theta(p1)) bsin(theta(p1))
  r(p(1))
• R h acos(theta(p2)) bsin(theta(p2))
  r(p(2))
• R h acos(theta(p3)) bsin(theta(p3))
  r(p(3))
• R - h acos(theta(p4)) bsin(theta(p4))
  r(p(4))
• four equations and four unknowns

10
Alternating rule

• Function p1,p2,p3,p4 alternatingrule(p1,p2,p3,
  p4,index,h,deviation)
• Five cases to consider
• If (index lt p1)
• ???
• Elseif (index lt p2)
• ???
• Elseif (index lt p3)
• ???
• Elseif (index lt p4)
• ???
• Else (index gt p4)
• ???
• End

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If (index lt p1) with hgt 0, deviation gt 0
deviation is the distance of index from the
mean circle deviation gt0 or lt0
p1
index
p2
p4
p3
12
If (index lt p1), with hgt 0, deviation lt 0
deviation is the distance of index from the
mean circle deviation gt0 or lt0
p1
p2
index
p4
p3
13
If (index lt p1) with hlt 0, deviation gt 0
deviation is the distance of index from the
mean circle deviation gt0 or lt0
p2
index
p1
p3
p4
14
If (index lt p1) with hlt 0, deviation lt 0
deviation is the distance of index from the
mean circle deviation gt0 or lt0
p1
p2
index
p4
p3
15
Alternating rule

• Function p1,p2,p3,p4 alternatingrule(p1,p2,p3,
  index,h,deviation)
• Five cases to consider
• If (index lt p1)
• If (deviation gt0 hgt0)
• p1 index
• Elseif (deviation gt 0 hlt 0)
• Elseif (deviation lt 0 hgt 0)
• Elseif (deviation lt 0 hlt 0)
• end
• Elseif (index lt p2)
• ???
• Elseif (index lt p3)
• ???
• Elseif (index lt p4)
• ???
• Else (index gt p4)
• End