2D Graphics Primer

8/3/2019 2D Graphics Primer

http://slidepdf.com/reader/full/2d-graphics-primer 1/12

2D Graphics Primer

Drawing Abstraction

Every computer graphics system (i.e., operating system) offers some notion of a

canvas to draw onto.

Usually, the canvases are separated into windows that are distinct from each other,

which provide a relative coordinate system and isolation. This abstraction allows the

same binary-level application to render onto different kinds of surfaces such as

screens, off-screen buffers (i.e., clipboards), and printers, plotters, and direct neural

implants...

Most current systems offer a resolution-independent (or device-independent) API. This

is crucial. If all rendering was done in pixels, a 100 pixel rectangle would be about an

inch big on most displays, but would be less than 1/10th of an inch big on some

printers.

Java offers the Graphics/Graphics2D classes that present this abstraction, and C#

offers the System.Drawing.Graphics class.

Coordinates

Device coordinatesCoordinates of the physical device:

• Usually has origin (0, 0) at upper left

• X increases to the right

• Y increases down

• Integral coordinates always has X in [0, width] and Y in [0, height]

• Coordinates correspond to pixels

• Need to know pixel density (DPI - dots per inch) to create a specific-sized object

Window coordinates

Coordinates within an operating system window

• Similar to device coordinates

• Can have out of bounds coordinates which are clipped by the OS

• If there is a frame around the window ("window dressing"), that is

outside the dimensions of the window



Physical coordinates("device-independent coordinates") - correspond to physical measurements

• Defined in universal measurements - inches, millimeters, points (1/72 of

an inch)

• Necessary to make graphics the same size independent of device

Model ("local") coordinatescorrespond to the object you are defining

• You may want to define an application-specific coordinate system

(i.e. Origin at the center of screen with one "unit" per inch)

• Need to convert between coordinate systems

Transformations

• A "transform" object encapsulates a coordinate system

• A sequence of transforms can efficiently switch coordinate systems

Graphics Description Models

Stroke Model

describes images with strokes of specified color and thickness. There are also

several other parameters, such as how the line segments are connected, and

whether the line is drawn solid, or with dashes. In addition, strokes can be anti-aliased.

Line from (4,5) to (9,7) in red with a thickness of 5

Circle centered at (19,8) in blue with a radius of 8 and a thickness

of 3

...

Dash StylesEnd Cap Styles



Miter Join

Round Join Bevel Join

Unantialiased Line Antialiased Line

Region Model

describes images with filled areas such as arcs, text, splines, and other shapes -

often other stroke objects. The area may be filled with a color, or a more

complex fill such as a gradient or texture (known as a paint in Java or a Brush

in C#).

Polygon filling (0, 0)-(10, 5)-(5, 10) with yellow

...

Solid Fill Gradient Fill



Pixel Model

describes images as a discrete number of pixels. Each pixel is specified by a

color from one of several possible color models.

NOTE : In practice, graphics are described with a combination of stroke, region, and

pixel descriptions.

Drawing Shapes

Java has a generic Shape class with g2.draw() and g2.fill() methods. See

java.awt.geom. Including GeneralPath which connects points with lines or

curves. It can be rendered resolution-independent, or can be flattened with

FlatteningPathIterator

C# has fixed shapes rather than a generic Shape class. See Graphics.DrawEllipse,

DrawBezier, DrawCurve, etc. C# also has a generic path called

Drawing2D.GraphicsPath, rendered with Graphics.DrawPath.

Rendering Damage/Redraw

In most graphics systems, you override a window's paint method and put all your

rendering code there, but you don't actually call that method directly. Instead you will

request a render by telling the OS that a portion of the screen is "damaged." It is out of

date and needs to be repainted. The OS will collect and merge all the damaged regions

and at some point later it will call the window's paint method passing it the full region

that needs to be repainted.

In java, you can request that a window be rendered with JComponent.repaint(). In C#,you would use Control.Invalidate. Or, you could just damage a portion of the canvas

with repaint(x, y, w, h) in Java or Invalidate(Rectangle) in C#. Remember that these

methods only request repaints to happen in the future, they do not happen

immediately.



Text

Text is a special kind of graphics for both performance and human reasons.

Characters in any font can be represented either as bitmaps or curves.

In some systems, characters are defined as the set of pixels (bitmap) that form each

character. This approach is efficient since we need the set of pixels that make up thecharacters in order to draw them. But, for large font sizes the space required to store

all the bitmaps becomes problematic.

Another approach is to store only the outlines of the characters as closed shapes.These character definitions can easily be scaled to any size and converted to bitmaps

when necessary. Plus, drawing packages can treat characters as geometric shapes that

can be manipulated like other graphical elements.

Typically you will allocate fonts up front, and then use them on demand. See the Font

classes in Java and C#. Fonts get measured with Ascent, Descent, Height, Leading,

Width, and Kerning. Higher quality text can be drawn with anti-aliasing or morerecently, sub-pixel anti-aliasing (e.g., Microsoft Cleartype)

Color

There are various color models, which are not reflected in the API. Rather the API

needs to support the color representation of the hardware. There are two basic kinds,

Indexed Color (8 bit) and True Color (16, 24, 32 bit).

Indexed color uses an array of color values as a palette. An index into the color table

is assigned to each pixel. Using 8 bits per pixel allows only 256 colors.

Models that can represent a large number of colors are called true color. Examples are

RGB, HSV, and CMYK. The common 8-bit RGB model assigns a 3 byte value to

each pixel, one byte per channel (red, green and blue). This model can represent up to

28

× 28

× 28

or 16,777,216 colors.

Clipping

Clipping, or limiting drawing to a particular area of the screen, is crucial for a window

manager. Displayed objects need to be clipped to the bounds of the window in which

they are displayed. But, clipping is also necessary for application efficiency and high-end graphics.

Regions, analytical descriptions of shape, are used for the basis of clipping. Regions

include algebra for adding/subtracting, manipulating shape and there are various types

(rectangular, rectilinear, rectilinear with holes).



Java has the Area class (made of Shapes) and C# has the Region class.

Efficiency

There are various efficiency considerations that must be made when working with 2D

computer graphics. Some common mechanisms/considerations include region

management (partial redraws), high-level descriptions for networked displays, displaylists for complex objects that are redrawn many times, space-time trade-offs (i.e. pre-

allocate thumb image and store it, or render it each time).

Geometric Equations

In a drawing program, there are some common problems such as determining what

object a user clicked on, finding the nearest object to where the user clicked, or

snapping a line to the nearest object. It turns out we can use a few simple geometric

equations to aid with these tasks. And, there are different forms of these equations for

different uses.

Implicit Forms: F(x, y) = 0

i.e.: Ax + By + c = 0

This type of equation defines a half-space. And, intersecting multiple half-spaces

defines polygons. So, this kind of equation can be used to determining if a point is

within a region or finding the distance from a line.

Explicit Forms: y = F(x)

i.e. y = Mx + B

Explicit functions are not good for graphics operations, since they don't plug a point

in.

Parametric Forms: x = G(t), y = H(t)

i.e.: x = cos(t), y = sin(t) or,

x = x1 + t * (x2 - x1)y = y1 + t * (y2 - y1)



Here, both x and y are functions of an independent variable. If you plug in some value

for t, you can plot a point on the line. The functions x(t) and y(t) vary along the line,

as t varies so you can interpolate along a line.

More generally, this is known in the computer graphics world as LERP (linear

interpolation):

p = p1 + t * (p2 - p1)

The LERP equation bounds t between 0 and 1. It directly supports animation,

computing intersections, scales to multiple dimensions and can be applied to many

domains (points, colors, etc.).

Control Points

Used to specify geometry - often accessed through "handles" in an interface

Cubic Bezier Curve Quad Bezier Curve

Geometric Transformations

Often an application will need to translate (move), scale, rotate, or shear it's graphical

elements. This is necessary for almost any animation and many interactions. One way

to achieve this would be to modify all of the points in the original object. For

example, in a game of asteroids, to move the triangular ship down ten units, you could

add 10 to each point in the triangle. The problem with this approach is that it forces

you to modify your original object. You will no longer have a copy of the original to

revert back to. And for some transformations, it's worse. What if you scale all the

points by zero? Then you've lost your data altogether. A better approach would be tosave your original object and somehow transform it when you render. This is

commonly done in computer graphics with matrices.

Matrix Transformations

We can represent some transformations as 2x2 matrices of the following form.



We can then multiply the matrix by a column vector to apply the transformation to a

point.

x′ = Ax + By

y′ = Cx + Dy

Matrices also allow as to represent a sequence of transformations.

Multiplying two matrices of the same size yields another matrix of the same size. So,

the three transformations above can actually be represented as a single matrix. As

long as we can represent transformations as 2x2 matrices, we can multiple them

together to create one representative matrix.

Scaling

One common transformation involves scaling around the origin (0, 0). Here, wemultiply all the x-coordinates by some scale factor Sx and we multiply the y-

coordinates by some scale factor Sy.

x′ = x × Sx

y′ = y × Sy

If we define P as a point [x, y], we can combine the equations above to get the

following representation.

P′ = S · P or,



We can then represent the scale vector as 2x2 matrix.

Rotation

Another common type of transformation is rotation, where we rotate the points by

some angle θ.

x′ = x × cos(θ) - y × sin(θ)

y′ = x × sin(θ) + y × cos(θ)

Again, we can represent rotation as a 2x2 matrix.

P′ = R · P or,

Translation

A simple transformation involves adding some offset Tx to all the x-coordinates, and

adding some offset Ty to all the y-coordinates. This is known as a translation.



x′ = x + Tx

y′= y + Ty

If we define P as a point [x, y], we can combine the equations above to get the

following representation:

P′ = P + T or,

But, we cannot represent translation as a 2x2 matrix! This means we won't be able to

combine it with rotation and scaling transformations through matrix multiplication.

The solution is to use homogeneous coordinates.

Homogeneous Coordinates

We can take a 2-dimensional point and represent it as a 3-vector.

We add a third coordinate h to every 2D point, where (x, y, h) represents the point at

location (x/h, y/h). We can now represent translation as a 3x3 matrix of the following

form.

P′ = T · P or,

We can then change our scale and rotation matrices into 3x3 matrices as well.



P′ = S · P or,

P′ = R · P or,

This may not seem intuitive or exciting. But, it is quite useful for graphics operations

because it allows us to combine translation, scale and rotation transformations, simplyby using matrix multiplication.

Affine Transformations

The 3x3 matrix that we derived above is called an Affine Transform. It can

encapsulate translate, rotate scale, shear and flip transformations. AffineTransformations have various properties.

• Origin may not map to the (0,0)

• Lines map to lines

• Parallell lines remain parallel

• Ratios are preserved

• Closed under composition

Matrix Composition

As we discussed above, we can multiply various matrices together, each of which

represent a transformation, in order get one general representation. For example, if wescale, rotate and then translate, we will have done the following.

P′ = ( T · (R · (S · P)))

We can then separate out our matrix.

P′ = ( T · R · S) · P

M = TRS

However, matrix multiplication is not commutative.



M1 · M2 != M2 · M1

To apply a transformation after the current one, we post-multiply the matrix.

P′ = Mnew · Mcurrent P

To apply a transformation first, we pre-multiply.

P′ = Mcurrent · Mnew P

Coordinate Systems

Transforms can manipulate objects or views. If we transform, draw an object, and

then transform back, we are manipulating objects. But, if we set a transform once atthe beginning and then draw the whole model we are manipulating the view. Actually,

we are defining a new coordinate system.

It turns out an affine transform actually defines a coordinate system. Imagine weapply a rotation, followed by a translation. We can think of this as creating a rotated,

translated coordinate system, with a new origin.

We can now draw objects as normal using the local coordinates of this new coordinate

system. So, if we draw an object a (0, 0), it will appear at the new origin of the newcoordinate system. Thinking of things this way is often simpler than thinking about

transforming individual objects.

2D Graphics Primer

Documents

Transcript of 2D Graphics Primer