Line Drawing - vda.univie.ac.atvda.univie.ac.at/Teaching/Graphics/19s/LectureNotes/14_lines_18w.pdf · lies on or near an ideal, infinitely thin line imposed on a 2D raster grid Assumptions

© Machiraju/Zhang/Möller© Machiraju/Zhang/Möller

Line Drawing

Foundations of Computer Graphics

Torsten Möller


Rendering Pipeline

Hardware

Modelling Transform Visibility

Illumination +Shading

ColorPerception,Interaction

Texture/Realism


Reading

Angel – Chapter 8.8-8.10


Today

Recap - the graphics pipeline Rasterizing lines

brute-force DDA Bresenham

Rasterizing quadrics Anti-aliasing


From the modeling stage to image formation Pipelined approach helps increase system

throughput Throughput: rate at which data flows through

the system In a pipeline, data processing in subsequent

blocks can be done in parallel Especially effective when the same sequence of

operations are to be performed on large quantity of data – true in graphics

The graphics pipeline


Vertex processor Per-vertex operations (vertices define

objects/primitives) Two main functionalities:

Coordinate transformations Color computation at each vertex (shading models)

Objects/geometry first defined in their own coordinate systems, then transformed into a world space — modeling transformation

Then objects are transformed from world space into the camera coordinate system — viewing transformation and projection transformation


Clipping and primitive assembly

Model a finite field of vision Remove objects outside a

finite clipping volume Need to be done on a primitive by primitive basis,

not on vertices Output is a set of primitives

whose projections can appear in the image


Rasterizer The rasterizer converts a primitive into a set

of fragments A fragment stores per pixel information for its

associated primitive, later used to determine whether the fragment contributes to pixel color and to compute the pixel color Raster/pixel location (in the frame buffer) Depth, e.g., to determine whether this fragment

“survives” Interpolated attributes, e.g., color and texture

coordinate, from the previous stages Alpha value (for blending) Window ID, etc.


Fragment processing Performs per-fragment operations Main function is to compute the color at a pixel,

using information stored in the fragments, e.g., depth, alpha, texture coordinates; can also add in fog and other effects

A (programmable) fragment shader is a program that performs the processing which replaces the OpenGL fixed functionality, e.g., simply using interpolated attributes

Similar to a vertex shader (per-vertex operations) Shader programs typically have limited

instruction set


Drawing or rasterization of primitives first. Consider lines and polygons.

Where are we at now?


Line drawing in OpenGL

glColor3f(1.0, 1.0, 1.0);

glLineWidth(2.0);

glBegin(GL_LINES);

glVertex2f(0.5, 0.5);




glEnd();

glBegin(GL_LINE_STRIP);

glBegin(GL_LINE_LOOP);

There is no GL_CIRCLE


Assumptions Transformation, clipping, projection

already done Primitives to rasterize are actually inside

the screen Work with 2D screen coordinates with

square pixels

(0, 0)

(N, M)


Primitives

Polygons explicit curves

lines, quadrics (conic sections) parametric curves/surfaces

Curves, Surfaces implicit description, e.g. x2+y2-1=0 Misc

particle systems/points, fractals


Explicit Descriptions

Given - an explicit equationy = f(x)

cannot get multiple values of y no infinite slope (vertical line) axis dependent (y depends on x) specify one variable and resolve for

the other


Lines

Given - an explicit line equationy = mx+n

Which pixels to set?http://bert.stuy.edu/pbrooks/graphics/demos/BresenhamDemo.htm


Scan Converting Lines Compute the coordinates of pixels that

lies on or near an ideal, infinitely thin line imposed on a 2D raster grid

Assumptions line will be 1 pixel thick and will approximate

an infinitely fine line pixels represented as disjoint circles, centred

on a grid (unless specified otherwise) integer coordinates of endpoints pixel on or off (2 states)


Scan Converting Lines (2) Desirable properties:

constant brightness (irrespective of length or orientation)

1 pixel per column (-1 <= slope <= 1), 1 pixel per row (slope > +/- 1)

as close to the ideal/as straight as possible allow control of pen, line and endpoint

styles drawn as smoothly as possible

(anti-aliasing) drawn as rapidly as possible

http://bert.stuy.edu/pbrooks/graphics/demos/BresenhamDemo.htm


Lines - solutions

• Brute force– for each column xi compute the row

yi=round(mxi+n)

– very costly

• incremental! (Newtons idea)– we know yi - find yi+1

– yi+1= round(yi +m)


Basic – Brute force Find slope m = rise/run or y = mx + b increment x by 1 (xi); calculate yi = mxi + b pixel (xi, round(yi)) turned on simple, but inefficient:

floating point multiplication addition round


Incremental - DDA multiplication can

be eliminated:

called digital differential analyzer (DDA) - after mechanical device that solves differential equations by numerical methods

Drawbacks: floating point values (m,y) round operation special cases could be done more quickly


Lines - Bresenham Special case of Midpoint Line Algorithm uses only integer arithmetic & no

rounding idea is to provide the best-fit

approximation to a true line by minimizing the error (distance) to the true line

slope (rest is done with reflection)

© Machiraju/Zhang/Möller

Explicit line equation:

Implicit line equation:

Main idea - call on decision variable:

Lines - Bresenham (2)


if di < 0, pick O, and:

if di > 0, pick NO, and:

if di = 0, pick either one, but consistently


© Machiraju/Zhang/Möller


Bresenham [Wikipedia]

dx=6dy=3



Additional issues: - covers 0 <= m <= 1 case, other 7 octants: exchange x/y, mirror x and y Endpoint order: we want P0 to P1 to

look exactly the same as P1 to P0

Starting at edge of a clip rectangle: we must use the midpoint test result rather than computing intersections analytically if we are to ensure the correct sequence of pixels




Quadrics - Circles

Circles, ellipsoids, etc.

divide in 8 quadrants What is special

about 8-foldsymmetry?


Quadrics - Circles (2) Midpoint Circle Algorithm

trace out arc in 2nd octant and draw the rest by symmetry

choose between E and SE pixel at each stage

https://en.wikipedia.org/wiki/Bresenham%27s_line_algorithm


Quadrics - Circles (3)

incremental - now have higher order

higher order difference system


Quadrics - Ellipses

Can a midpoint algorithm be developed for ellipses? What are the important differences from circles?

The result is Da Silva's algorithm (the idea is illustrated below):


Lines - Anti-Aliasing

A continuous (zero-width) line is “invisible” in a discrete grid.

Give line thickness - How? Signal Processing Approach:

line is high-frequency want to avoid aliasing what to do first?


Lines - Anti-Aliasing (2)

Bandlimit signal - how? Convolve with smoothing filter -

which?



Box filter not all that good convolve it again - why? Then resample!



Other alternatives? Can we improve this algorithm?


General Anti-Aliasing: super-sampling

simple but expensive simply sample n times within a

pixel and average!


remember …

© Machiraju/Möller© Machiraju/Möller

Aliasing example

© Machiraju/Möller© Machiraju/Möller

Point vs. supersampling


Xiaolin Wu's line algorithm

Multi-sample AA: https://www.youtube.com/watch?v=hqi0114mwtY

Documents

Line Drawing - vda.univie.ac.atvda.univie.ac.at/Teaching/Graphics/19s/LectureNotes/14_lines_18w.pdf · lies on or near an ideal, infinitely thin line imposed on a 2D raster grid Assumptions