Polygon DrawingPolygon Drawing

CG SeminarBy

Ali Abdul_Zahraa

polygon

0 A polygon is a bounded region of a plane. 0 It is bounded by a series of line segments.0 A line segment is a piece of a line that runs from one

point to another point.0 The line segment end point is called a vertex.0 Each vertex is shared by exactly two line segments

polygon

0 A polygon in a computer graphics system is a two-dimensional shape that is modeled and stored within its database (memory) .

0 A polygon can be colored, shaded and textured, and its position in the database (memory) is defined by the coordinates of its vertices (corners(

Polygons typesMain Types Are :0 Convex polygons0 Concave polygons0 Complex polygons

Convex polygons

0A convex polygon is a simple polygon whose interior is a convex set.

0 In a convex polygon, all interior angles are less than 180 degrees

Convex polygons

0 Every line segment between two vertices remains inside or on the boundary of the polygon.

Convex polygons0The intersection of two convex polygons is a

convex polygon

Convex polygons0 Helly's theorem: For every collection of at least 3

convex polygons: if the intersection of every 3 of them is nonempty, then the whole collection has a nonempty intersection.

Convex polygons

0 Every polygon inscribed in a circle (such that all vertices of the polygon touch the circle), if not self-intersecting, is convex. However, not every convex polygon can be inscribed in a circle

Concave or non-convex polygons

0 A simple (non-self-intersecting) polygon that is not convex is called concave, non-convex or reentrant. A simple concave polygon will always have an interior angle with a measure that is greater than 180 degrees.

0 It is always possible to partition a concave polygon into a set of convex polygons

Complex polygons

0 Complex polygons are a real mess. The edges of complex polygons cross each other.

Regular & irregular polygon

0 A polygon is equilateral if all of its sides are of the same length.

0 A polygon is equiangular if all of its angles are of equal measure.

0 A regular polygon is a polygon that is both equilateral and equiangular.

Named Polygons

0Triangle 3 sides0Quadrilateral 4 sides

0Tetragon 4 sides 0Pentagon 5 sides0Hexagon 6 sides0Heptagon 7 sides0Octagon 8 sides0Decagon 10 sides

0Dodecagon 12 sides

process of drawing

0 There are at least three different names for the process of drawing :

1 . it called "rendering" which means to make a copy of something. when you draw a polygon you are making a visible copy of its representation in memory.

process of drawing

2. Another term "scan conversion." This means drawing it as a set of scan lines. It also implies that you want to draw it in scan line order, from the top of the screen to the bottom of the screen, to match the way the electron beam scans out the picture on the tube. The term comes from the days before frame buffers when the picture had to be scan converted on the fly for each frame.

process of drawing

3. The other term used to describe drawing into a frame buffer is "rasterization.". That makes a raster the picture part of a CRT and rasterization is the process of drawing something on a CRT.

make a good polygon rendering algorithm

0 Speed is important. Faster is better as long as the picture is good enough for your purpose.

0 Memory isn't cheap. Don't use more than you need. But, if you can trade memory for speed, go for it.

0 It is not OK to leave gaps between polygons that share edges.

0 What you see on the screen has to be an accurate representation of the mathematical model.

0 The algorithm must be extendible. Vertices contain more than just X and Y coordinates. You need a Z coordinate for hidden surface removal. You might also have colors at each coordinate to support color blending across the polygon face. And, you might want to carry texture coordinates along.

Vertex & edge

0 A vertex is a point. vertex can fall in a lot of different places on a pixel.

0 An edge is a line that runs from one vertex to another. That means that an edge starts somewhere inside a pixel and ends inside another pixel

2D polygon drawing

0 The simplest way to represent a polygon is as a list of its vertices. You just assume that there is a line

segment connecting the first vertex on the list with the second, the second with the third, and so on until

you reach the last vertex. The last vertex is connected to the first vertex to close the polygon boundary.

0 bresenham's , DDA (useful).

Parity test

0 The well defined ways to decide what part of a plane is inside a polygon. The idea of the parity test is to trace a line from outside the polygon ("from infinity") to the other side of the polygon (negative infinity) and count how many times the line crosses an edge of the polygon.

0 If the line has crossed an odd number of edges it's inside the polygon and if it's crossed an even number of edges it's outside the polygon .

0A frame buffer is a two dimensional array of pixels. 0Each row of pixels in the frame buffer tells the

video hardware what color to draw along a scan line of the display screen.

0This means that we can draw a polygon by coloring the parts of the scan line that are inside the polygon and the parity test is just perfect for telling when a scan line is inside a polygon.

0 For any scan line find all the places that it crosses a polygon edge, sort the crossings by their X coordinate, and fill in the part of the scan line between each pair of crossings.

The edge problem

0 It’s happen when the drawing algorithm entered in infinite loop of edge (lines) drawing that when the first and last edge not determined, the drawer may draw the edge between vertices many time

0 Or when the edge fall between two polygons will be filled in by both polygons. Filling the pixel twice is wasteful

The edge problem0 So to solve this problem must adding another vector to save

how is vertices was visited.0 Also it’s a solving to a problem that happen In parity test

when the vertex fall on scan line, each vertex belong to two edges so that will give not accurate number of scan line hit , so it will solved by this method.

3D polygon Drawing

0 An easy way to determine a path along a polygonal surface mesh is to determine a plane that contains the two points and is generally perpendicular to the surface.

0 The intersection of the plane with the faces making up the surface mesh will define a path between the two points

0 a greedy-type algorithm can be used to construct a path of edges along a mesh surface from a given start vertex to a given destination vertex

0 For each edge emanating from the current vertex (initially the start vertex), calculate the projection of the edge onto the straight line between the current vertex and the destination vertex

0 Divide this distance by the length of the edge to get the cosine of the angle between the edge and the straight line.

0The edge with the largest cosine is the edge most in the direction of the straight line; choose this edge to add to the path

0Keep applying this until the destination edge is reached.

0 Improvements to this approach can be made by allowing the path to cut across polygons to arrive at points along opposite edges rather than by going vertex to vertex

0Candidate edges can be generated by projecting the straight line onto polygons containing the vertex and then computing their cosine values for consideration.

0 If a path downhill from an initial point on the surface is desired, then the surface normal and global up vector can be used to determine the downhill vector.

0The cross product of the normal and global up vector defines a vector that lies on the surface perpendicular to the downhill direction. So the cross product of this vector and the normal vector defines the downhill (and uphill) vector on a plane

polyhedron

0 A polyhedron is a 3-dimensional figure whose surfaces are polygons.

Which of these is polyhedron???

References

0 http://www.mathopenref.com/polygon.html0 Computer Animation Algorithms and Techniques,

Rick Parent 0 http://en.wikipedia.org/wiki/Polygon

Thank U

☺ ☺ ☺

0 Hyperplane separation theorem: Any two convex polygons have a separator line. If the polygons are closed and at least one of them is compact, then there are even two parallel separator lines (with a gap between them)

0Krein–Milman theorem: A convex polygon is the convex hull of its vertices. ) one only needs the corners of the polygon to recover the entire polygon shape.(

0 Inscribed triangle property: Of all triangles contained in a convex polygon, there exists a triangle with a maximal area whose vertices are all polygon vertices