Mid Point Ellipse Algorithm

 Mid Point Ellipse Algorithm

Lets start with the general definition of Ellipse that is:

"the set of all points, the sum of whose distances from two fixed points is a constant.”

In Mathematical term,

d₁ + d₂ = constant

For an ellipse with center point (h,k), the standard rectangular expression can be written as

{ (x-h)/a }² + { (y-k)/b }² = 1

So, lets assume a ellipse with center at origin ((h,k)= (0,0)), having major and minor axis a and b respectively,
By putting all this value in above equation, we get

(x/a)² + (y/b)² = 1

On solving:

 (x/a)² + (y/b)² - 1 = 0

In first region, the equation will be:
  
f(x, y) = a²x² + b²y² - a²b²


The mid points will be
F(X_mid, Y_mid) = (X_k+1 , Y_k – 1/2)

having following properties:    

1. F(X_mid, Y_mid) < 0, when midpoints will lie inside the ellipse boundary.
2. F(X_mid, Y_mid) = 0, when midpoints will lie on ellipse boundary.
3. F(X_mid, Y_mid) > 0, when midpoints will lie outside the ellipse boundary. 

Now, put this midpoints value in first region equation:
F(X_mid , Y_mid) = b²(X_k+1)² + a²(Y_k - 1/2)² – a²b²

On solving:
P’_k = b²(X_k+1)² + a²(Y_k² + ¼ - Y_k) – a²b²
P’_k = b²(X_k² + 1 + 2X_k) + a²(Y_k² + ¼ -Y_k) – a²b²

and
P’_k+1 = b²(X_k+1² + 1 + 2X_k+1) + a²(Y_k+1² + ¼ -Y_k+1) – a²b²

Initial decision parameter of Region-1 is (0 , b), hence putting this value in above equation:
P’o=b²(0+1+0) + a²(b² + ¼ -b) – a²b²
P’o = b² + a²b² + a²/4 –a²b – a²b²
P’o= b² + a²/4 – a²b
P’o= a²/4 – a²b + b²

This is the initial decision parameter of region 1.

Now, substracting P'_k from P'_k+1
P’_k+1 – P’_k = b²(X_k+1² – X²_k) +2b²(X_k+1-X_k) + a²(Y_k+1² – Y_k²) + a²(Y_k+1 - Y_k)
P’_k+1 – P’_k = b²(X_k+1 – X_k)(X_k+1 + X_k) +2b²(X_k+1-X_k) + a²(Y_k+1 – Y_k)(Y_k+1 + Y_k) -a²(Y_k+1 - Y_k)

If P is negative: 
X_k+1 – X_k = 1
Y_k+1 – Y_k = 0

hence
P’_k+1 = P’_k + 2b² X_k+1 + b²

This is the decision parameter for less than zero of region 1.

If P is positive: 
X_k+1 – X_k = 1
Y_k+1 – Y_k = -1

hence
P’_k+1 = P’_k +b²(X_k+1 + X_k) + 2b² + a²(-1)(Y_k+1 + Y_k) –a²(-1)
P’_k+1 = P’_k + 2b² X_k+1 + b² – 2a² Y_k+1

This is the decision parameter for greater than zero of region 1

Over Region R²
Now lets solve the derivation for region 2

F(X,Y) = b²X² + a²Y² – a²b²
F(X_mid , Y_mid) = (X_k+1/2 , Y_k – 1)
F(X_mid , Y_mid) = b²(X_k + 1/2)² + a²(Y_k - 1)² – a²b²

hence
P’’_k = b²(X_k + 1/2)² + a²(Y_k - 1)² – a²b²
P’’_k = b²(X²_k + 1/4 + X_k) + a²(Y²_k + 1 -2Y_k) – a²b²
P’’_k+1 = b²(X_k+1² + 1/4 + X_k+1) + a²(Y_k+1² + 1 -2Y_k+1) – a²b²

Initial decision parameter of Region R² (X₀ , Y₀):
 P’’₀ = b²(x₀ + 1/2)² + a²( Y₀ - 1)² – a²b²

This is the initial decision parameter of region R².

Now, substracting P'_k from P'_k+1:
P’_k+1 – P’_k = b²(X²_k+1 – X_k²) +b²(X_k+1 - X_k) + a²(Y_k+1² – Y²_k) -2a²(Y_k+1 - Y_k)
P’_k+1 – P’_k = b²(X_k+1 – X_k)(X_k+1 + X_k) +b²(X_k+1-X_k) + a²(Y_k+1 – Y_k)(Y_k+1 + Y_k) -2a²(Y_k+1 - Y_k)

If P is positive: 
X_k+1 – X_k = 0
Y_k+1 – Y_k = -1

Hence
P’’_k+1 = P’’_k – 2a² Y_k+1 + a²

This is the decision parameter for greater than zero of region 2.

If P is negative:
X_k+1 – X_k = 1
Y_k+1 – Y_k = -1

Hence
P’’_k+1 = P’’_k + 2b² X_k+1 -2a² Y_k+1 + a²

This is the decision parameter for less than zero of region 2.

X²/a² + Y²/b² – 1 = 0 b²X² + a²Y² – a²b²

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