03: Basic Principles of 2D Graphics Slide 2 Lecturer Details Dr.
Walid Khedr, Ph.D. Email: [email protected] Web:
www.staff.zu.edu.eg/wkhedr Department of Information Technology
Slide 3 Main Topics Introduction Basic principles of
two-dimensional graphics Drawing lines and curves Areas, text and
colors Basic principles of three-dimensional graphics Modeling
three-dimensional objects Visible surface determination
Illumination and shading Special effects and virtual reality Slide
4 Basic Principles of 2D Graphics 1. Objectives 2. Representing an
Image Vector and Raster Graphics 3. Getting Started with Java 2D 4.
Basic Geometric Objects 5. Geometric Transformations scaling,
rotation, shearing and translation. 6. Homogeneous Coordinates 7.
Moving Objects 8. Interpolators Slide 5 Objectives Learn basic
concepts that are required for the understanding of two-dimensional
graphics. Distinguish between the representation of images on
output devices and the model of the image itself. Learn the basic
geometric objects and transformations Slide 6 Representing an Image
Original ImageVector graphics: Representation by basic geometric
objects (lines, circles, ellipses, cubic curves,..). Raster
graphics: Representation in the form of a pixel matrix. Slide 7
Raster Graphics Raster graphics: is the representation of images as
an array of pixels. Colors are assigned to each pixel. Used for
computer monitors, printers, and file formats like bitmap or jpg.
Slide 8 Storing Bitmap Images Frequently large files Example: 600
rows of 800 pixels with 1 byte for each of 3 colors ~1.5MB file
File size affected by Resolution (the number of pixels per inch)
Amount of detail affecting clarity and sharpness of an image
Levels: number of bits for displaying shades of gray or multiple
colors Palette: color translation table that uses a code for each
pixel rather than actual color value Data compression Slide 9
Vector Graphics Vector graphics is the use of geometrical
primitives such as points, lines, curves, and shapes or polygon(s),
which are all based on mathematical equations, to represent images.
Efficient storage Scalable Slide 10 Slide 11 Raster or Vector ?
Computer monitors, printers and also various formats for storing
images are based on raster-oriented graphics, also called
pixel-oriented graphics. In order to display vector-oriented
graphics in the form of raster graphics, all geometrical shapes
must be converted into pixels. Scan Conversion Lead to high
computational efforts Possible aliasing effects Slide 12 Aliasing
Effect (Jagged Edges) Slide 13 Getting Started with Java 2D Before
modelling of two-dimensional objects is discussed in more detail, a
short introduction into how Java 2D can be used to generate images
in general is provided. Slide 14 AWT AWT is a java package that we
will be using in order to create graphical user interfaces Some
important classes within the AWT package Containers: Frame: has an
title bar, can contain many things Panel Each of the above
containers has a paint method that we will inherit but will usually
override when we want to customize the containers graphics. Slide
15 AWT Example Slide 16 Paint Method paint called automagically
when needed applet/application starts/resizes/unhidden/etc Paint
method takes a Graphic object as an argument. This argument is
passed to the paint method when drawing operation is required Slide
17 Java 2D & Paint Method The class Graphics2D within Java 2D
extends the class Graphics. In order to exploit the options of Java
2D, the Graphics object must be casted into a Graphics2D object
within the paint() method. In order to keep the example programs as
simple and understandable as possible, the computations required
for defining and positioning the geometric objects are carried out
directly in the paint method. Slide 18 Java 2D & Paint Method,
Cont. Slide 19 Window Coordinates Slide 20 Basic Geometric Objects
The basic geometric objects in computer graphics are usually called
primitives or graphics output primitives. The basic primitives are
the following ones: Points that are uniquely defined by their x-
and y-coordinate. Their main function is the description of other
objects like lines Lines, polylines or curves can be defined by two
or more points. For a line two points are needed curves require
additional control points. Polylines are connected sequences of
lines. Areas are usually bounded by closed polylines or polygons.
Areas can be filled with a color or a texture. Slide 21 Polygons
Polygon, closed polyline: The last line segment of a polyline ends
where the first line segment started. Non-self-overlapping
Convexity: Whenever two points are within the polygon the
connecting line between these two points lies completely inside the
polygon as well. Slide 22 Parametric Curves Parametric curves are
defined as parametric polynomials Quadratic Curves: Two endpoints
and one control point. Cubic Curves: Two endpoints and two control
points The curve begins and ends in the two specified endpoints. In
general, it will not pass through control points. The control
points define the direction of the curve in the two endpoints.
Slide 23 Quadratic Curve Example Slide 24 Cubic Curve Example Slide
25 Parametric curves When fitting quadratic or cubic curves
together it is not sufficient to simply use the endpoint of the
previous curve as a starting point for the next curve. In order to
avoid sharp bends, the first control point of a succeeding curve
must be on the line defined by the last control and endpoint of the
previous curve. Other important curves in computer graphics are
circles and ellipses Slide 26 Definition of Areas Polygons, circles
and ellipses define areas. Areas can be bounded by a closed curve.
Areas can be filled by colors and textures. Slide 27 Set-Theoretic
Operations Modifying the shape of elementary geometric objects by
geometric transformations. Application of set-theoretic operations
like union, intersection, difference and symmetric difference to
previously defined areas. Union, intersection, difference and
symmetric difference of a circle and a rectangle. Slide 28 Basic
Geometric Objects in Java 2D The abstract class Shape with its
various subclasses allows the construction of various
two-dimensional geometric objects. Shapes will not be drawn until
the draw or the fill method is called with the corresponding Shape
as argument in the form graphics2d.draw(shape) or
graphics2d.fill(shape), respectively. Slide 29 Basic Geometric
Objects in Java 2D The abstract class Point2D is NOT a subclass of
Shape. Points cannot be drawn in Java. They are only supposed to be
used for the description of other objects. Subclasses of Point2D:
Point2D.Float and Point2D.Double. Slide 30 Basic Geometric Objects
in Java 2D Line (segment): Two endpoints are needed to define
aline. Line2D.Double line = new Line2D.Double(x1,y1,x2,y2);
Quadratic curve: Two endpoints and a control point are needed to
define a quadratic curve. QuadCurve2D.Double qc = new
QuadCurve2D.Double(x1,y1, ctrlx,ctrly, x2,y2); Cubic curve: Two
endpoints and two control points are needed to define a cubic
curve. CubicCurve2D.Double cc = new CubicCurve2D.Double(x1,y1,
ctrlx1,ctrly1, ctrlx2,ctrly2, x2,y2); Slide 31 Basic Geometric
Objects in Java 2D General path: A GeneralPath is sequences of
lines, quadratic and cubic curves. GeneralPath gp = new
GeneralPath(); gp.moveTo(50,50); gp.lineTo(50,200);
gp.quadTo(150,500,250,200); gp.curveTo(350,-100,150,150,100,100);
gp.lineTo(50,50); Slide 32 General Path Car Slide 33 Slide 34 Slide
35 Slide 36 Slide 37 Slide 38 Slide 39 Slide 40 Slide 41 Slide 42
Slide 43 Slide 44 Rectangles Rectangle: By Rectangle2D.Double r2d =
new Rectangle2D.Double( x,y,width,height); A rectangle is defined
by left corner has the coordinates (x,y) width width and height
height. Slide 45 Circles and Ellipses Circles and ellipses: by
Ellipse2D.Double elli = new Ellipse2D.Double(x,y,width,height); An
axes-parallel ellipse is defined by a bounding Rectangle with the
parameters x,y,width,height. Ellipse2D.Double elli = new
Ellipse2D.Double(x-r,y- r,2*r,2*r); defines a circle with centre
(x,y) and radius r. Slide 46 Example: Rectangle and Ellipse Slide
47 Ellipse Arcs Arcs (of ellipses) are defined by Arc2D.Double arc
= new Arc2D.Double( rect,start,extend,type); rect2D: specifies the
bounding rectangle of the corresponding ellipse in the form of a
Rectangle2D. Start: is the angle. The angle corresponds to the
angle with the x-axis Extend: is the opening angle of the arc,
i.e., the arc extends from the start angle start to the angle start
+ extend. Type: can take one of the three values Arc2D.OPEN,
Arc2D.PIE and Arc2D.CHORD Slide 48 Ellipse Arcs Slide 49 Area
Objects Slide 50 Geometric Transformations Geometric
transformations play a crucial role in computer graphics. Geometric
transformations can be used to position objects: To scale them To
rotate them To change the shape of objects To move them The most
important geometric transformations are scaling, rotation, shearing
and translation. Slide 51 Representation of Points in Matrix Form
In two-dimensional coordinate system, any point is represented in
terms of x and y coordinates. The point (x, y) can be converted
into matrix in the following two ways: Slide 52 Representation of
2D Objects in Matrix Form Slide 53 Transformation of Points The
result of the multiplication of a matrix (x y) containing the
coordinate of a point P and a general 2x2 transformation matrix:
Slide 54 Transformation The term transform' means 'to change'. This
change can either be of shape, size or position of the object. To
perform transformation on any object, object matrix X is multiplied
by the transformation matrix T. Slide 55 Scaling Scaling means to
change the size of object. This change can either be positive or
negative. After applying scaling factor or matrix on any object,
coordinate is either increased or decreased depending on the value
of scaling factors. Scaling matrix is given by: where, S x and S y
are scaling factors in x-direction and y- direction respectively.
Slide 56 Scaling, Cont. Slide 57 Scaling Example Slide 58 General
Fixed-Point Scaling Applying a scaling to an object that is not
centered around the origin of the coordinate system will lead to a
translation in addition to the scaling. But sometimes it may be
required to perform scaling without altering the object's position.
Translate Object so that it is centered around the origin, perform
scaling then reverse the translation. Slide 59 Reflection If S x is
negative, in addition to scaling in the x- direction, a reflection
with respect to the y-axis is applied. If S Y is negative, in
addition to scaling in the y- direction, a reflection with respect
to the x-axis is applied. If S x or S y equal -1, only reflection
is applied. Slide 60 Rotation Slide 61 Derivation of the Rotation
Matrix Slide 62 Rotation Example 1 Slide 63 Rotation Example 2
Slide 64 Shearing The shear transformation is an elementary
geometric transformation that causes a certain deformation of
objects. Shearing can be done either in x-direction or y-
direction. The off-diagonal terms in transformation matrix is
responsible for shearing. Hence the matrix of shearing is: Slide 65
Shearing Example Slide 66 Translation A translation T(d x, d y )
causes a shift by the vector d = (d x, d y ) T. This means the
translation maps the point (x, y) to the point: Slide 67
Translation, Cont. Slide 68 Slide 69 Homogeneous Coordinates To
produce a sequence of transformation with above equations, we must
calculate the transformed coordinates one step at a time. A more
efficient approach is to combine sequence of transformations into
one transformation In order to combine sequence of transformation
we use Homogeneous Coordinates Slide 70 Homogeneous Coordinates In
order to compute transformations based on matrix multiplications,
homogeneous coordinates are introduced. The homogeneous coordinates
of non-homogeneous vector [x y] are [x' y' h]. where x = x'/h and y
= y/h and where h is any real number. One set of homogeneous
coordinates is always of the form [x y 1]. We select this form to
represent the position vector [x y] in physical xy-plane systems.
There is no unique homogeneous coordinate representation i.e., [8 4
2]. [ 12 6 3], [16 8 4] all represent the physical point (4, 2),
Slide 71 Homogeneous Coordinates The general 2D-Transformation
matrix is now 3x3, i.e. Translation in homogeneous coordinates will
be Slide 72 Transformation Matrices Slide 73 Combined
Transformations The composition of geometric transformations can be
computed by matrix multiplication in homogeneous coordinates.
Matrix multiplication is noncommutative. The order in which
geometric transformations are applied matters. Changing the order
might lead to a different result. It should also be taken into
account that transformations in matrix notation or as compositions
of mappings are carried out from right to left Slide 74 Combined
Transformations Slide 75 Application of Transformation Slide 76
Application of Transformation, Cont. The required transformation is
Slide 77 Geometric Transformations in Java 2D Slide 78 Slide 79
Slide 80 Moving Objects Moving objects (animated graphics) can be
implemented based on geometric transformations. The transformations
must describe the movement of the object(s): The object is modeled
by suitable geometric transformations The object must be drawn The
objects position in the next frame is computed based on
transformations. The old object (or the whole window) is deleted.
The updated object and the background are drawn again. Slide 81
Moving Objects Example A moving clock with a single hand for the
seconds is considered sliding from the lower left to the upper
right of a display window. The clock itself consists of a quadratic
frame and the single rectangular hand for the seconds. A
translation is needed in order to move the quadratic frame of the
clock from the lower left to the upper right corner of the window.
This translation must also be applied to the hand for the seconds.
In addition to the translation, the hand must also rotate. Slide 82
Moving Objects Example, Cont. Slide 83 Slide 84 Method 1: Keep
track of the position, orientation and scaling of the object and
apply the transformations step by step to the already transformed
object. Method 2: Generate an object in the origin and keep track
of the accumulated transformations defining the movement. Always
apply the accumulated transformations to the object in the origin
of the coordinate system in a suitable order. Slide 85 Moving
Objects Example, Cont. Slide 86 Interpolators Is another ways to
model movements or changes of objects in animated graphics. Aim:
Continuous transition from an initial state to final state.
Simplest case: moving from position (point) p 0 to p 1 The points p
on the line between p 0 and p 1 are the convex combinations of
these two points given by: Slide 87 Interpolation of
Transformations Slide 88 Interpolation of Transformations, Cont.
Slide 89 Simple Object Interpolation Slide 90 Simple Object
Interpolation, Cont. Both Letters C and D are described by two
quadratic curves Slide 91 Reading Chapter 2 Slide 92 Next Lecture
Scan conversion Drawing lines and curves Slide 93 Thank You
