Computer Graphics

1. Computer Graphics Scan Conversion Polygon Faculty of Physical and Basic Education Computer Science Dep. 2012-2013 Lecturer: Azhee W. MD. E-mail: azhee.muhamad@univsul.net azheewria07@gmail.com

2. Scan Conversion polygon • Polygon • Polygon Classifications • Identifying Concave Polygons • Splitting Concave Polygons • Inside-Outside Tests • 2 University of sulaimanyiah - Faculty of Physical and Basic Education - Computer Dep. 2012-2013

3. Polygon • A polyline is a chain of connected line segments. When starting point and terminal point of any polyline is same, i.e. when polyline is closed then it is called polygon. • 3 University of sulaimanyiah - Faculty of Physical and Basic Education - Computer Dep. 2012-2013

4. Polygon Classifications • interior angle: is an angle inside the polygon boundary • that is formed by two adjacent edges. • convex polygon: interior angles of a polygon are less than or equal to 180° • OrConvex : is a polygon in which the line segment joining any two points within the polygon lies completely inside the polygon • 4 University of sulaimanyiah - Faculty of Physical and Basic Education - Computer Dep. 2012-2013

5. Polygon Classifications • Concave polygon: the polygon that is not convex. • or Concave: is a polygon in which the line segment joining any two points within the polygon may not lie completely inside the polygon. • 5 University of sulaimanyiah - Faculty of Physical and Basic Education - Computer Dep. 2012-2013

6. Identifying Concave Polygons • At least one interior angle greater than 180° • 2. The extension of some edges of a concave polygon will • intersect other edges • 3. Some pair of interior points will produce a line segment • that intersects the polygon boundary • 4. Find a cross product for adjacent edges • For a convex polygon all cross products will be of the • same sign (positive or negative) • If some cross products yield a positive value and some a negative value, we have a concave polygon • 6 University of sulaimanyiah - Faculty of Physical and Basic Education - Computer Dep. 2012-2013

7. Convex or concave? • Convex if no line that contains a side of the polygon contains a point in the interior of the polygon. • Concave or non-convex if a line does contain a side of the polygon containing a point on the interior of the polygon. • 7 University of sulaimanyiah - Faculty of Physical and Basic Education - Computer Dep. 2012-2013

8. Identifying Concave Polygons(2) • 8 University of sulaimanyiah - Faculty of Physical and Basic Education - Computer Dep. 2012-2013

9. Splitting Concave Polygons: Cross Product • 9 University of sulaimanyiah - Faculty of Physical and Basic Education - Computer Dep. 2012-2013

10. example • Q)Use cross product to find normal vector of a polygon with the following vertices: (0.2, -0.4, 0.2), (0.6, 0.7, 0.5), (-0.3, 0.4, -0.3), • (-0.4, -0.3, -0.4) • Answer: • Ek = Vk+1 - Vk • E1 = (a1, a2, a3), E2 = (b1, b2, b3) • E1 x E2 = (a2b3 − a3b2, a3b1 − a1b3, a1b2 − a2b1) • V1 = (0.2, -0.4, 0.2), V2 = (0.6, 0.7, 0.5), V3 = (-0.3, 0.4, -0.3) • E1 = V2 – V1 • = (0.6, 0.7, 0.5) - (0.2, -0.4, 0.2) • = (0.4, 1.1, 0.3) • E2 = V3 – V2 • = (-0.3, 0.4, -0.3) - (0.6, 0.7, 0.5) • = (-0.9, -0.3, -0.8) • E1 x E2 = (1.1*-0.8-0.3*-0.3, 0.3*-0.9-0.4*-0.8, 0.4*-0.3-1.1*-0.9) = (-0.79, 0.05, 0.87) • 10 University of sulaimanyiah - Faculty of Physical and Basic Education - Computer Dep. 2012-2013

11. Splitting Concave Polygons: Set of Triangles • 11 University of sulaimanyiah - Faculty of Physical and Basic Education - Computer Dep. 2012-2013

12. Splitting Concave Polygons: Set of Triangles (2) • 12 University of sulaimanyiah - Faculty of Physical and Basic Education - Computer Dep. 2012-2013

13. Inside-Outside Tests • It is not clear which regions of the xyplane we • should call interior and which regions we should • designate as exterior for a complex polygon with • intersecting regions. • algorithms: • odd-even rule • nonzero winding-number rule • 13 University of sulaimanyiah - Faculty of Physical and Basic Education - Computer Dep. 2012-2013

14. Inside-Outside Tests (2) • odd-even rule • Draw a reference line from any position to a • distant point outside a closed polyline • The line must not pass through any endpoints • Count the number of line segments crossed along this line • If the number is odd then the • region considered to be • interior • Otherwise, the region is • exterior • 14 University of sulaimanyiah - Faculty of Physical and Basic Education - Computer Dep. 2012-2013

15. Inside-Outside Tests (2) • nonzero winding-number rule • Counts the number of times the boundary of an object winds • around a particular point in the counterclockwise direction • The count is called the winding number • Initialize winding number to zero • Draw a reference line from any position to a distant point • The line must not pass through any endpoints • Add one to the winding number if the • intersected segment crosses the reference • line from right to left (counterclockwise) • Subtract one from the winding number if the • segment crosses from left to right (clockwise) • If winding number is nonzero, the region is • considered to be interior • Otherwise, the region is exterior • 15 University of sulaimanyiah - Faculty of Physical and Basic Education - Computer Dep. 2012-2013

16. Example • Q) Use nonzero winding number rule to determine the interior and exterior regions of the following polygon ? • 16 University of sulaimanyiah - Faculty of Physical and Basic Education - Computer Dep. 2012-2013

17. Example • 17 University of sulaimanyiah - Faculty of Physical and Basic Education - Computer Dep. 2012-2013

18. References •  Donald Hearn, M. Pauline Baker, • Computer Graphics with OpenGL, 3rd • edition, Prentice Hall, 2004 •  Chapter 3, 4 • 18 University of sulaimanyiah - Faculty of Physical and Basic Education - Computer Dep. 2012-2013