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Computer Graphics: Graphics Output Primitives

Circle Drawing, and Fill Area Polygons

By: A. H. Abdul Hafez

[email protected],

March 8, 2017 1 CG, by Dr. A.H. Abdul Hafez, CE Dept. HKU

mailto:[email protected]

Outlines

March 8, 2017 CG, by Dr. A.H. Abdul Hafez, CE Dept. HKU 2

1. Circle drawing algorithm

2. Frame buffer values

3. Fill area polygon

4. Plane equation and inside-outside test

5. Character primitives

6. End

Circle generation algorithm


A circle is defined as the set of points that are all at a given

distance r from a center position (xc, yc).

Symmetry of a circle.

Calculation of a circle point

(x, y) in one octant yields

the circle points shown for

the other seven octants.

Midpoint circle algorithm


To apply the midpoint method, we define a

circle residual function as

Our decision parameter is the circle function

evaluated at the midpoint between these two

pixels

Midpoint circle algorithm


The initial decision parameter is obtained by evaluating the circle

function at the start position (x0,y0)=(0,r):

If pk < 0, this midpoint is inside the circle and the pixel on scan line

yk is closer to the circle boundary. Otherwise, the mid position is

outside or on the circle boundary, and we select the pixel on scan

line yk - 1.


Example on Midpoint circle algorithm






Fill-area


Fill area is an area that is filled with some solid color or pattern.

Fill areas are used to describe surfaces of solid objects, but they are

also useful in a variety of other applications.

Fill regions are usually planar surfaces, mainly polygons. But, in

general, there are many possible shapes for a region in a picture

that we might wish to fill with some color option.

Solid-color fill areas specified with various boundaries

(a) A circular fill region

(b) A fill area bounded by a closed plotlines

(c) A filled area specified with an irregular curved boundary.

Classification of Polygons


A polygon is a plane figure specified by a set of three or more

coordinate positions, called vertices, that are connected in sequence

by straight-line segments, called the edges or sides of the polygon.

An interior angle of a polygon is an angle inside the polygon

boundary that is formed by two adjacent edges. If all interior angles

of a polygon are less than or equal to 180°, the polygon is convex.

A polygon that is not convex is called a concave polygon.

Convex and concave polygons


A concave polygon has at least one interior angle greater than 180.

Also, the extension of some edges of a concave polygon will

intersect other edges.

Some pair of interior points will produce a line segment that

intersects the polygon boundary.

Convex and concave polygons


Algorithmic:

1. If we set up a vector for each

polygon edge, then we can use

the cross product of adjacent

edges to test for concavity.

2. All such vector products will be

of the same sign (positive or

negative) for a convex polygon.

3. Therefore, if some cross products

yield a positive value and some a

negative value, we have a

concave polygon. Qi = (Vi – Vi-1)x(Vi+1 - Vi)

Qi. Qi+1 >0 for all i, then polygon is convex

Splitting a

concave polygon


Splitting a polygon


Polygons tables


As information for each polygon is input, the data are placed into

tables that are to be used in the subsequent processing, display, and

manipulation of the objects in the scene.

These polygon data tables can be organized into two groups:

Geometric tables and attribute tables.

1. Geometric data tables contain vertex coordinates and parameters to identify

the spatial orientation of the polygon surfaces.

2. Attribute information for an object includes parameters specifying the

degree of transparency of the object, its surface reflectivity, and texture

characteristics.

Polygons tables


Geometric data for the objects

in a scene are arranged

conveniently in three lists:

1. a vertex table,

2. an edge table, and

3. a surface-facet table.

The edge table contains

pointers back into the vertex

table to identify the vertices for

each polygon edge.

And the surface-facet table

contains pointers back into the

edge table to identify the edges

for each polygon.

Plane Equations


Each polygon in a scene is contained within a plane of infinite

extent. The general equation of a plane is

Solution with 3 points


Front and back faces


The normal vector N for a plane described with the equation Ax + By +Cz + D = 0 is perpendicular to the plane and has Cartesian components (A, B, C) .

The end of the Lecture


Thanks for your time

Questions are welcome

Documents

Computer Graphics - Bilgisayar Mühendisliği Bölümü | …compe.hku.edu.tr/Dosyalar/COME-306--Bilgisayar...Computer Graphics: Graphics Output Primitives Circle Drawing, and Fill