Digital Media Dr. Jim Rowan ITEC 2110

Vector Graphics Elegant way to construct digital images that –have a compact representation –are scalable –are easy to edit Mandatory for 3-D modeling

Coordinate systems Bitmapped pixel coordinates (integer) 4,04,14,24,9 3,03,13,23,9 0,00,10,20,9 x y

Coordinate systems bitmapped A E CD x y B A= (, ) B= (, ) C= (, ) D= (, ) E= (, )

Coordinate systems Vector graphics coordinates (real values) A point is defined by its x and y coordinate x y (1.41, 1.74) (x,y) Can be fractional Can be negative

Vector graphics coordinates (real values) A displacement (distance between points or movement from one point to another) can be defined by a pair of points Coordinate systems x y (1.41, 1.74) point 1 (3.12, 0.95) displacement from point 1 to point 2... = [(x2 - x1), (y2 - y1)] ( , ) (1.71,-0.79) point 2

Vectors have magnitude (length) and direction This vector goes down 0.79 and to the right 1.71 It is a two dimensional (x&y) vector that goes down and to the right (1.71,-0.79) AKA vector x y (1.41, 1.74) (3.12, 0.95)

Another grandfathered term From the time when displays were directly “steered” by their programs Give the display two points and it would move the beam from one point to the other drawing a line... This movement could be described as a vector since it has magnitude (length) and direction (from one point to the other) Vector graphics?

Absolute vs Window coordinates (top paragraph of page 88) rendering software renders images in a window windows can be moved at will so... the rendering program only knows where the object is relative to the window it is in rendering software does not know where the window is on the screen a coordinate transformation must be performed

Bounding Box what point is used to place an object? images can be contained inside a “bounding box” which is the smallest box that contains all the points found in an object bounding box is placed in relation to its (0,0) position

Window coordinates x y (0,0) window 1 desktop screen bounding box (150,200) window (410,290) (760,570) (100,103)

Vector graphics Store shapes (like the line) very economically in the form of formulas or equations But which formula? A line is defined by y = mx + b (but that is for an infinitely long line) In computer graphics we want a line segment, i.e. it has endpoints Since a line segment can also be completely described by its endpoints…

Here’s the vector we saw earlier defined by it’s two endpoints x (1.41, 1.74) (3.12, 0.95)

Now let’s convert this vector graphic to a bitmapped graphic

Convert Vector to Bitmapped Visually... “completely cover the vector with pixels” x

Vector to Bitmapped x

The artifact created: The “Jaggies” This can result in “aliasing” x

Aliasing When converting a vector graphic to a bitmapped graphic we can’t create a perfect representation What we end up with is a bitmapped version of the vector It’s not really the original but rather an alias of the original version

Aliasing Even we view a bitmapped digital photograph on a display, the reconstruction is performed by a display device, and by our eyes and by our brain. If the reconstructed image differs from the original image, we are seeing an alias.

To combat aliasing: anti-aliasing Techniques to mitigate the affect of aliases Smoothes the edges of jagged lines

Using anti-aliasing techniques on our original line x our pixellated line to be displayed (the external model) is in black

combating aliases: anti-aliasing techniques x the original vector graphic line stored in the internal model is in red our pixellated line to be displayed (the external model) is in black

combating aliases: anti-aliasing techniques x

combating aliases: anti-aliasing techniques “average the grayness” x

So far we have just looked at a line –The line was stored as its endpoints What about other shapes?

vector graphics: shapes connected lines are stored as a polyline –internally the polyline is stored as a series of points closed polylines form a shape open polyline is a jagged line

vector graphics: rectangles and squares rectangles can be described by two corners squares are special cases of the rectangle

vector graphics: ellipses and circles ellipses can be described by two points circles are special cases of the ellipses –they are stored as a single point

vector graphics: curves Question: How would you draw a curve using a computer with a mouse? You can’t draw smooth lines very easily A line tool with handles (based on the Bezier curve) can be manipulated by those handles

Lines and curves Bezier curves can be smoothly joined together An anchor point is the point where one joins the other When a curve closes on itself it is considered a closed curve When it doesn’t it’s an open curve

Lines and curves Closed (and open for that matter) lines can be filled –This is how drawn shapes become objects like the cowboy on Toy Story (except that the characters in toy story were created in 3D, not 2D) –solid color, pattern or gradient (linear or radial) –Patterns are built of tiles that match across x & y Lines have ends –ends can be messy when joined –mitre, rounded, square, bevel

Manipulating objects AKA closed curves Translation is a simple up/down side-to- side movement Scaling: make bigger or smaller Rotation about a point Reflection about a line

3-D Way more complex than 2-D 3-D shapes (objects) are defined by their surfaces Made even more complicated by the fact that a 3-D object inside the computer must be translated into 2-D to be rendered on a computer screen... –This results in the need to specify the viewpoint

Structural hierarchy Things in the real world are compositions of smaller things Things in the 3-D graphics world are also compositions of smaller things Hierarchical structure is an excellent way of coping with complexity Also seen in object-oriented programming like Java and Squeak!

Structural hierarchy Car –Wheels (4) tire wheel hubcap –Doors (2) handles –inside »lever –outside »button »handle window(s) –Lights headlights (2) tail lights (2+) stop lights (2+)

3-D: additional complexity rendering is extremely computationally expensive (demanding) –lighting natural artificial reflections off other objects in the scene shadows cast by other objects in the scene –surface texture affects appearance

Questions?