Page 81 - DCAP313_LAB_ON_COMPUTER_GRAPHICS

Unit 5: Implementing Ellipse Algorithm

int main (int argc, char ** argv) Notes
{
double A, B;double X 1 , Y 1 ;double X 2 , Y 2 ;
double area1, area2;int rtn;char msg[1024];
printf (“Calling ellipse_segment.c\n”); //-- case shown in Fig. 1A = 4.;
B = 2;
X 1 = 4./sqrt (5.);
Y 1 = 4./sqrt (5.);
X 2 = –3;
Y 2 = –sqrt (7.)/2;
area1 = ellipse_segment (A, B, X 1 , Y 1 , X 2 , Y 2 , &rtn);
sprintf (msg, “Figure 5.1: segment area = %15.8f, return_value = %d\n”,
area1, rtn);
printf (msg);
//-- case shown in Figure 5.2
A = 4;
B = 2;
X 1 = –3;
Y 1 = –sqrt (7.)/2;
X 2 = 4./sqrt (5.);
Y 2 = 4./sqrt (5.);
area2 = ellipse_segment (A, B, X 1 , Y 1 , X 2 , Y 2 , &rtn);
sprintf (msg, “Fig 5.2: segment area = %15.8f, return_value = %d\n”,
area2, rtn);
printf (msg);
sprintf (msg, “sum of ellipse segments = %15.8f\n”, area1 + area2);
printf (msg);
sprintf (msg, “total ellipse area by pi*a*b = %15.8f\n”, pi*A*B);
printf (msg);
return rtn;
}

5.4 Extending the Core Segment Algorithm to More General Cases

Core segment algorithm is given below:

5.4.1 Segment Area for a (Directional) Line through a General Ellipse
The ELLIPSE_SEGMENT algorithm is based on an ellipse that is centred at the origin with its axes
aligned to the coordinate axes. The core algorithm can be extended to rotated and/or translated
ellipse forms. Consider an ellipse described by Equation (1), with semi-major axis lengths of A
and B that is centred at the origin with its axes aligned with the coordinate axes. Suppose that
the ellipse is rotated counter-clockwise through an angle φ, and that the ellipse is then translated
so that its centre is at the point (h, k). The rotated + translated ellipse could be defined by the set
of parameters (A, B, h, k, φ), with the understanding that the rotation through φ is performed
before the translation through (h, k). Figure 3 illustrates this idea, showing the ellipse of Figure
1 rotated counter-clockwise through an angle φ = +3/8, then translated by (h, k) = (–6, +3).

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