CG OpenGL line & area-course 3

Line & Area
Chen Jing-Fung (2006/10/24)
Assistant Research Fellow,
Digital Media Center,
National Taiwan Normal University

Video Processing Lab
臺灣師範大學數位媒體中心視訊處理研究室


Coordinate
representations
Viewing & Projection
Coordinates

1 1
Modeling 1 Video Monitor
(local or master)
Coordinates
Plotter
World Normalized
(Cartesian) Coordinates
Coordinates
Other Output

Device
Coordinates
Video Processing Lab 2

Reshape function


Monitor

monitor

http://groups.csail.mit.edu/graphics/classes/6.837/F04/calendar.html

2D graph
• Plot figure in 2D coordinate
– X-coordinate indicated horizontal
– Y-coordinate indicated vertical
– Z-coordinate indicated depth

simple

nonsimple


Coordinate Reference
• A set of coordinate extents y
– Bounding box (3D), bounding rectangle (2D)
5
– Max & Min (x, y, z), (x, y) 4
• Screen coordinates 3
2
– Scan line number: y 1
– Column number: x 0
x
01 2345
• A low-level procedure of the pixel position
– setPixel(x, y);
• Retrieve the current frame-buffer setting for a
pixel location
– getPixel(x, y, color);
Combine RGB bit codes


Coordinate specifications
• Absolute coordinate & Relative
coordinate
– Last position (current position): (3, 8)
– => Some graph packages is used relative
coordinate: (2, -1) -> Absolute position:
(5, 7)


2D world-coordinate in
OpenGL
• An orthogonal projection function
– gluOrtho2D
glMatrixMode (GL_PROJECTION);
glLoadIdentity (); Assign the identity matrix
gluOrtho2D (xmin, xmax, ymin, ymax); World coordinate range

ymax

Display window

ymin
xmin
xmax

How to a parameter
drawing line?



Bresenham’s line algorithm
• Bresenham’s line is an accurate and
efficient raster line-generating algorithm
• Use only incremental integer calculations.
• Determine at next sample point whether
to plot.
– The positive-slope line:
if start point (50, 50), next point?
• (51, 50) or (51, 51)


Parallel Bresenham line algorithm
• Drawing parallel processing
– Two different line on the same plane and
along a line path simultaneously.

yk+1

yk

xk+1

11
P. 99-101

How to buffer value by
the structure?



Setting frame-buffer values
• Since scan-conversion algorithms generate
pixel positions which are labeled from (0, 0)
to (xmax, ymax), the frame-buffer bit
address is calculated as
– (x, y) Screen
• addr(x, y) = addr(0, 0) + y(xmax + 1)+x yend
– (x+1, y)
• addr(x+1, y) = addr(x, y) + 1
(x, y)
– (x+1, y+1) y0
• addr(x+1, y+1) = addr(x, y) + xmax + 2
x0 xend
Frame buffer:
(0, 0) (1, 0) (2, 0) (0, 1) (x, y)
(xmax, 0) (xmax, ymax)
addr(0, 0) addr(x, y)

Another drawing line
method
Use Midpoint



Circle-generating algorithms

• A circle is defined as the set of points
– Radius = r, center position (xc, yc)
(x-xc)2+(y-yc)2=r2
– X axis in unit steps from xc- r to xc+ r and
calculating the corresponding y values at each
position as
y  y c  r 2  ( xc  x )2
– using polar coordinates r and θ
p.103-104 臺灣師範大學數位媒體中心視訊處理研究室

Symmetry of a circle
• Eight octants
– Calculation if a circle point (x, y) in one octant
yields the circle points shown for the other
seven octants
• More efficient circle algorithm are based
on incremental calculation of decision
parameters, which involves only simple
integer operations. (as in the Bresenham
line algorithm)


Midpoint circle algorithm (1)
• The basic idea is to test the halfway
position between two pixels
– Determine inside or outside the circle
boundary.
• Midpoint method = Bresenham line
algorithm
– Integer circle radius
– A straight-line segment


Midpoint circle algorithm (2)
• To apply the midpoint method, we define a
circle function as:
– fcirc(x, y) = x2+y2-r2 x 2 + y2 - r 2 = 0
yk
<0, if (x,y) is inside the circle boundary yk-1 Midpoint
fcirc(x,y) =0, if (x,y) is on the circle boundary yk-2
>0, if (x,y) is outside the circle boundary xk xk+1 xk+2
– Decision parameter (each xk, k=0 & pk<0)
• pk= fcirc(xk+1, yk-1/2)
• pk+1=pk+2(xk+1) + (yk+12-yk2) - (yk+1-yk)+1
– Increment?
What is different between midpoint and line-
based? Video Processing Lab 18
p.105-109 臺灣師範大學數位媒體中心視訊處理研究室

Other curves
• Various curve functions are useful in object
modeling, animation path specification and other
graphics applications.

• Commonly encountered curves include conics,
general polynomials and spline functions.
– Conic sections: Ax2+ By2+Cxy+Dx+Ey+F=0 v0
g
– Parabolic y0
• trajectory: y=y0+a(x-x0)2+b(x-x0)
• Projection point: x=x0+vx0t; y=y0+vy0t-1/2gt2
– Hyperbola: (x/rx)2-(y/ry)2=1 x0

Spline curve formed with individual ry
cubic polynomial sections between
specified coordinate positions rx


Fill-Area



Fill-Area Primitives
• Polygon area (solid-color, patterned)
– scan-line to fill polygon
• Intersection points located and sorted
• Consecutive pairs define interior span
• Attention with vertex intersections
• Exploit coherence (incremental calculations)
– flood fill algorithm


Fill-area primitives
• Describing components of a picture is
an area that is filled with some solid
color
– Polygon is a plane figure specified by
• Vertices: three or more coordinate positions
• Edges or sides: straight-line segments

concave convey
<1800
>1800
• Area….

Concave polygons
• The direction of surface is “z’’ axle
• Given two consecutive vertex (V)
positions, Vk and Vk+1, we define the
edge vector (E) as Ek = Vk+1 - Vk (Cross product)
y
V6 V5 (E1 x E2)z > 0
E4 (E2 x E3)z > 0
V4
Where is “z” (E3 x E4)z < 0 convex
E3
E2 V3
V1 V2
E1 x concave
Test!!

Splitting concave polygons
• Vector method
– All vector cross products have the same
sign => convex
• Rotational method
– rotate polygon-edges onto x-axis, always
same direction => convex
– Each vertex vk Area 1
V1
• Origin: translate Vk V3
V2
• Rotate clockwise Vk+1 on x-axis
• If vertex below x-axis => split Area 2
V4


Inside-Outside Tests
• Testing interior vs. exterior object
boundaries
– Odd-even (parity) rule or nonzero winding
parity rule
• Odd-even rule
– number of segments crossed by the line drawn
from P to distant point outside the coordinate
extents
• bounding rectangle or box is odd it is interior point
• else it is exterior point
– use the rule for color filling of rings, 2
concentric polygons


Nonzero winding parity
rule
• Nonzero winding parity rule may yield
different interior and exterior regions
– Setup vectors along object edges and the
reference line.
– Compute cross product of vector u along the
line from P to a distant point, with object
vector E for each edge that crosses the line.
– With 2 D object in XY plane direction of cross
product will be + z direction or –z direction.
• if z component for a crossing is +ve that segment
crosses from left to right, add 1 to winding number
• else if it is left to right subtract 1 from winding
number simpler


3D inside-outside
• Consider a sphere, x2 + y2 + z2 = 1.
– Interior: the set of all points that
satisfy x2 + y2 + z2 < 1
– The solid's surface: x2 + y2 + z2 <= 1.
– Exterior: x2 + y2 + z2 > 1.
Interior point A boundary

Boundary point C

Exterior point B
http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/model/closure.html

Polygon Surfaces (1)
• Set of surface polygons enclose
object interior = Boundary
Representation (“B-rep”)

wireframe representation
of a cylinder with back
(hidden) lines removed


Polygon Surfaces (2)
• Polygon tables (B-rep lists)
– Geometric and attribute tables
– Vertex, edge, polygon tables
– Consistency, completeness checks


Two polygon surfaces
• Geometric data-table representation for
two adjacent polygon surface facets,
formed with six edges and five vertices
VERTEX TABLE EDGE TABLE
V1 E1: V1, V2
V1: x1, y1, z1
E2: V2, V3
V2: x2, y2, z2
E1 E3 E6 E3
V3
S1 E4
V4
S2 V5 E5
V5
E2 V E6
3 E
4 E5 SURFACE-FACET TABLE
V2
S1: E1, E2,E3
V4 S2: E3, E4, E5, E6


http://www.icg.tu-graz.ac.at/courses/bacgvis/slides

Plane equations

• Ax+By+Cz+D=0
– Plane parameter: A, B, C and D
• Normal vector (N) the plane equation:
x − 1 = 0.
The normal vector:
N=( A, B, C )
Plane equation
Ax + By + Cz + D = 0


3D cartesian coordinate

y axis
y axis
z axis

x axis x axis
z axis
Left-handed cartesian
Right-handed cartesian


Front and Back Polygon
Faces (1)
• Back face = polygon side that faces into
the object interior
• Front face = polygon side that faces
outward
• Behind a polygon plane = visible to the
polygon back face
• In front of a polygon plane = visible to the
polygon front face


Front and Back Polygon
Faces (2)
• Ax + By + Cz + D = 0 for points on the
surface
< 0 for points behind
> 0 for points in front
– right-handed coordinate system
– polygon points in counterclockwise order
• V1, V2, V3 counterclockwise => normal
vector N = (V2 - V1) x (V3 - V1)


What is Cross product (x) ?
• A mathematical joke asks,
– "What do you get when you cross a
mountain-climber with a mosquito? "

• On the television sitcom, Head of
the Class asks,
– "What do you get when you cross an
elephant and a grape?"

http://mathworld.wolfram.com/CrossProduct.html

Cross product (x)
• The orientation determined by the
right-hand rule

http://mathworld.wolfram.com/CrossProduct.html

OpenGL polygon fill-area
functions
• OpenGL provides a special rectangle function
– glRect*(x1, y1, x2, y2);
• *: i (integer), s (short), f (float), d (double) and v (vector)
• plRectiv(vertex1, vertex2);
• Fill rectangle use “glVertex”
• glBegin (GL_POLYGON);
– glVertex2iv (p1);
– glVertex2iv (p2); p6 p5
– glVertex2iv (p5); p1 p4
• glEnd (); p2 p3


How to design “font”?
A complete set of type of one size
and face.



South China Morning Post, 13
Sep., 2005


Texture

• Graphics displays often include
textural information :
– labels on graphs and charts
– signs on buildings or vehicles
– General identifying information for
simulation and visualization applications.


Typeface (1)
• Letters, numbers and other characters can
be displayed in a variety of sizes and
styles.
– The overall design style for a set of characters
– Routines for generating character primitives
• 20-point Courier Italic or 28-point Palatino
Bold
• font :
– 14 points, a total character height is about 0.5 cm. A

– 72 points is about 2.54 cm (1 inch) B

Typeface (2)

the letter B represented with an 8x8 bilevel
bitmap pattern and with an outline shape
defined with straight line and curve
segments

Homework (1)
• Select 6 points and try to draw primitive
constant
– (a) GL_POLYGON (b) GL_TRIANGLES (c)
GL_TRIANGLE_STRIP (d)
GL_TRIANGLE_FAN
• Select 8 points and try to draw primitive
constant
– (a) GL_QUADS (b) GL_QUAD_STRIP
• Draw
50 100
100

200


Homework (2)
• Draw switch
– (1) parabolic
• x = y2
(-10≦y ≦10)
– (2) 2D curve path
• 2D : Set (x0, y0) & step (relate to θ),
m:integer
– r=sin (mθ)
r=A+Bcos(mθ)
r=e-mθ
r=mθ


Reference
• http://www.icg.tu-
graz.ac.at/courses/bacgvis/slides
• South China Morning Post, 13 Sep.,
2005
• Computer Graphics with OpenGL 3/e,
Hearn Baker, Third edition, ISBN：
0131202383


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