Page 88 - DCAP313_LAB_ON_COMPUTER_GRAPHICS

Lab on Computer Graphics

Notes Equation (12) are found by substituting the expressions for the point (x, y) into a standard ellipse
equation, yielding the following ellipse equation:
[cos(j - j 1 ) (x TR - (h 1 - h 2 )) - sin(j - j 1 )(y TR - (k 1 - k 2 ))] 2
◊
◊
2
2
A 2 2
◊
◊
+ [sin(j - j 1 ) (x TR - (h 2 - h 2 )) - cos(j - j 1 )(y TR - (k 1 - k 2 ))] 2 = 1 (16)
2
2
B 2 2
Where (x TR , y TR ) are defined in Equation (15). Expanding the terms of Equation (16), and then
simplifying yields expressions for the implicit polynomial coefficients in Equation (12) for a
general ellipse with parameters (A 2 , B 2 , h 2 TR, k 2 TR, φ 2 – φ 1 ):
2
2
cos(j - j ) sin (j - j )
AA = 2 1 + 2 1 (17a)
A 2 2 B 2 2
BB = 2 ◊sin(j 2 - j 1 ◊ )cos(j 2 - j 1 ) - 2 ◊sin(j 2 - j 1 ◊ )cos(j 2 - j 1 ) (17b)
A 2 2 B 2 2
2
2
CC = sin(j - j 1 ) + cos (j - j 1 ) (17c)
2
2
A 2 2 B 2 2
-◊ cos(j - j ◊ )[h ◊ cos(j - j ) + k ◊sin(j - j )]
2
DD = 2 1 2 TR 2 1 2TR 2 1 +
A 2 2
-◊sin(j - j ◊ )[k ◊ cos(j - j ) - h ◊sin(j - j )]
2
2 1 2 TR 2 1 2 TR 2 1 (17d)
B 2 2
2
-◊sin(j - j ◊ )[h ◊ cos(j - j ) - k ◊sin(j - j )]
EE = 2 1 2 TR 2 1 2TR 2 1 +
A 2
2
2◊cos(j - j ◊ )[h ◊ sin(j - j ) - k ◊cos(j - j )]
2 1 2 TR 2 1 2 TR 2 1 (17e)
B 2 2
[h ◊ cos(j - j ) - k ◊ sin(j - j )] 2 [h ◊ sin(j - j ) - k ◊cos(j - j )] 2
FF = 2 TR 2 1 2 TR 2 1 + 2 TR 2 1 2 TR R 2 1 – 1 (17f)
A 2 2 B 2 2
Intersection points are found by solving simultaneously the two implicit polynomials denoted
above as Equation (11) and Equation (12). Solving for x in Equation (11):
x 2 y 2 2 Ê y 2 ˆ
+ = 1  x = ± A 1 ◊ 1 - ˜ (18)
Á
A 2 B 2 Ë B 2 ¯
1 1 1
Substituting the expressions for x of Equation (18) into Equation (12), where the coefficients are
defined in Equation (17) yields a quadratic polynomial in y. substituting either the positive or
the negative root gives the same quadratic polynomial coefficients, which are:
cy[4]  y + cy[3]  y + cy[2]  y + cy[1]  y + cy[0] = 0 (19)
3
2
1
4
where:
cy[]
4
)
2
◊
2
2
2
2
◊
2
= A ◊ AA + B ◊ È A ◊( BB -◊ AACC + B CC ˘
4
2
1 Î
˚
B 1 1 1 1
cy[]
3
˘
(
◊ ÈB CC EE
◊
◊
= 2 ◊ B 1 Î 2 1 ◊ ◊ + A 2 1 ◊ BBDD - AAEE)
˚
B 1
cy[] 2 2 2 2 2 2 2
2
2
◊
2
)
◊
= A ◊ È { Î B ◊( 2 ◊ AA CC - BB + DD -◊ AA FF˘ -◊ A ◊ AA } + B 1
˚
1
1
1
B 1
2
 (2  CC  FF + EE )
cy[]
1
◊ ÈA
˘
◊
◊
(
◊
= 2 ◊ B 1 Î 1 2 ◊ AA EE - BB DD) + EE FF
˚
B 1
0
cy[]
◊
)
◊
)
˚
= Î È A ◊( A AA - DD + FF˘◊[ AAA + DD + FF]
1
1
1
B 1
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