International Journal of Computer Applications (0975 – 8887) Volume 50– No.17, July 2012 7 , c c Cx y r Centre A New Dynamic Programming Approach for Scan Conversion of a Circle Sushil Chandra Dimri Professor Graphic Era University Dehradun- India Sushil Kumar Chamoli Assistant Professor Graphic Era University Dehradun- India Anuj Dimri Research Scholar Graphic Era University Dehradun- India ABSTRACT Generally curve may be generated as a sequence of many small lines; some curves like circle, parabola, ellipse, in particular can be generated with help of D.D.A. algorithm and other special algorithms. There are two main recursive algorithms for scan conversion of the circle on computer screen, the Bresenhams and the Midpoint circle generating algorithm both are pixel based; in this paper we are presenting a recursive new approach for scan conversion of the circle. The pixel in one octant has been determined with help of this algorithm and rest of the parts of the circle will be generated with help of symmetry. General Terms Dynamic Programming, Symmetry, Euclidean geometry, Octant Keywords Scan conversion, decision parameter, real theoretical point, mid point algorithm 1. INTRODUCTION Scan conversion of continuous figure is a fundamental process in any graphics display. It is required since the raster display consists of a rectangular matrix of pixels. Scan conversion is an approximation of a given continuous figure by integer valued points (pixels) [2, 3, 9 and 10]. Circle is a very important geometrical figure. A circle is a simple shape of Euclidean geometry consisting of those points in planes that are equidistant from a given point, the centre and the distance between any of the points and the centre is called the radius. Circles are simple closed curves which divide the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk [9, 10, and 15].Scan conversion of circle received considerable research attention since 1960 [2, 12, 13, 18, 21, 22 and 23].Scan conversion of a circle means to identify the pixels (grid points) which are closest to the corresponding real theoretical points on circle [ 12 and 13 ].There are mainly to such criteria used to identify the pixels i) Minimizing the value of function 2 2 2 , f xy x y r ii) Minimizing the Euclidean distance to the circle Circle is a symmetrical geometrical figure-1. The circle 2 2 2 x y r is symmetrical about x axis, y axis, line , y x and line , y x to generate a circle on computer screen it is sufficient to find pixels on one octant and rest parts can be generated with help of reflection about x axis, y axis, line , y x and line , y x [3, 9, 10, 13 and 14]. Cartesian coordinates: Equation of circle having centre , c c Cx y and radius r is given by 2 2 2 . c c x x y y r The parametric equation of the circle is c x x r cos c y y r sin The equation of a circle whose center is origin and radius r is given by 2 2 2 x y r Its parametric equation is given by x r cos , , y r sin Where is a parameter which varies from 0 0 to 0 360 [1 and 9]. Y P(x, y) r sin c y y O c x x r cos X Fig 1: The circle with centre , c c x y There are mainly two popular algorithms for scan conversion of circle these are Bresenham‟s and Midpoint circle algorithms, both the algorithms use dynamic programming approach to draw the circle. Both determine the pixel position in the octant 0 45 to 0 90 , and remaining part will be generated by reflection. These algorithms use the function 2 2 2 , f xy x y r to locate the pixels [2, 3 and 12].