1. No category

Why Lines?
Lines:
Design of Line Algorithms
Line Requirements
– most common 2D primitive - done 100s or
1000s of times each frame
– even 3D wireframes are eventually 2D lines!
– optimized algorithms contain numerous
tricks/techniques that help in designing
more advanced algorithms
Basic Math Review
Must compute integer coordinates of pixels which lie on or near a
line or circle.
Pixel level algorithms are invoked hundreds or thousands of times
when an image is created or modified – must be fast!
Lines must create visually satisfactory images.
Point-slope formula for a line
6
– Lines should appear straight
– Lines should terminate accurately
– Lines should have constant density
Line algorithm should always be defined.
P = (X,Y)
Given two points (X1,Y1), (X2, Y2)
Consider a third point on the line: P = (X,Y)
Length of line segment between P1 and P2:
L=
( x2 − x1 ) 2 + ( y2 − y1 ) 2
Midpoint of a line segment between P1 and P3:
P2 = ( (X1+X3)/2 , (Y1+Y3)/2 )
Two lines are perpendicular if
1) M1 = -1/M2
2) Cosine of the angle between them is 0.
4
P2 = (X 2,Y2)
3
M = Slope = (Y2 - Y1)/(X2 - X1)
= (Y - Y1)/(X - X1)
2
1
Solving For Y
Y = [(Y2-Y1)/(X2-X1)]*(X-X1)+ Y1
P1 = (X 1,Y1)
1
can be arranged as Y=M*X+B
with B=(X2Y1-X1Y2)/(X2-X1)
Other Helpful Formulas
5
2
SLOPE =
3
RISE
RUN
4
5
=
Y2-Y 1
X2-X 1
Parametric Form
Given points P1 = (X1, Y1)
and P2 = (X2, Y2)
X = X1 + t(X2-X1)
Y = Y1 + t(Y2-Y1)
t is called the parameter. When
t = 0 we get (X1,Y1)
t = 1 we get (X2,Y2)
As 0 < t < 1 we get all the other points on the line segment
between (X1,Y1) and (X2,Y2).
6
Simple DDA* Line Algorithm
void DDA(int X1,Y1,X2,Y2)
{
int Length, I;
float X,Y,Xinc,Yinc;
Algorithm ideas based on parametric
form?
X = X1;
Y = Y1;
while(X<X2){
Plot(Round(X),Round(Y));
X = X + Xinc;
Length = ABS(X2 - X1);
if (ABS(Y2 - Y1) > Length)
Length = ABS(Y2-Y1);
Xinc = (X2 - X1)/Length;
Yinc = (Y2 - Y1)/Length;
Y = Y + Yinc;
}
}
DDA creates good lines but it is too time consuming due to the round
function and long operations on real values.
*DDA: Digital Differential Analyzer
DDA Example
DDA Example
Line from (6,9) to (11,12).
Length := Max of (ABS(11-6), ABS(12-9)) = 5
Xinc := 1
Yinc := 0.6
13
12
Values computed are:
(6,9), (7,9.6),
(8,10.2), (9,10.8),
(10,11.4), (11,12)
11
10
9
6
7
8
9
10
11
12
13
13
12
11
10
9
6
!
!
8
9
10
11
12
13
Fast Lines (cont.)
Fast Lines – Midpoint Method
!
7
NE = (xi + 1, yi + 1)
Simplifying assumptions: Assume we
wish to draw a line between points (0,0)
and (a,b) with slope m between 0 and 1
(i.e. line lies in first octant).
The general formula for a line is
y = mx + B
where m is the slope of the line and B is
the y-intercept. From our assumptions
m = b/a and B = 0.
y = (b/a)x + 0 => f(x,y) = bx - ay = 0 is an
equation for the line.
+y
-x
(a,b)
For lines in the first octant, given
one pixel on the line, the next
pixel is to the right (E) or to the right
and up (NE).
(xi +1, yi + ½ + e)
e
(xi +1, yi + ½)
+x
P = (xi ,yi )
-y
E = (xi + 1, yi)
Having turned on pixel P at (xi, yi ), the next pixel is NE at (xi+1, yi+1) or
E at (xi+1, yi). Choose the pixel closer to the line f(x, y) = bx - ay = 0.
Fast Lines (cont.)
! "# $
%
%& '
$ )
#
*
) ' ' -
%& '
. "/ )
f(xi+1, yi+ ½ + e) = 0 (point on line)
= b(xi + 1) - a(yi+ ½ + e)
= b(xi + 1) - a(yi + ½) – a*e
= f(xi + 1, yi + ½) – a*e
NE = (xi + 1, yi + 1)
e
(xi +1, yi + ½)
* '
0
) +,
P = (xi ,yi )
7
8 9
: 6
7
;
<
=
> ? @
Therefore, we only need to know the sign of di to choose
between pixels E and NE. If di ≥ 0 choose NE, else choose E.
But, calculating di directly each time requires at least two adds,
a subtract, and two multiplies => too slow!
E = (xi + 1, yi)
' 1
Fast Line Algorithm
Decision Variable calculation
B C D
Algorithm:
Calculate d0 directly, then for each i >= 0:
if di ≥ 0 Then
Choose NE = (xi + 1, yi + 1) as next point
di+1 = f(xi+1 + 1, yi+1 + ½) = f(xi + 1 + 1, yi + 1 + ½)
= b(xi + 1 + 1) - a(yi + 1 + ½) = f(x i + 1, yi + ½) + b - a
= di + b - a
else
Choose E = (xi + 1, yi) as next point
di+1 = f(xi+1 + 1, yi+1 + ½) = f(xi + 1 + 1, yi + ½)
= b(xi + 1 + 1) - a(yi + ½) = f(xi + 1, yi + ½) + b = d i + b
A
3 45 6
Let di = f(xi + 1, yi + ½) = a*e; di is known as the decision
variable.
Since a ≥ 0, di has the same sign as e.
") %
2
(xi +1, yi + ½ + e)
((' # % !
The Decision Variable
Knowing di, we need only add a constant term to find di+1 !
{Bresenham for lines with slope between 0 and 1}
a = ABS(xend - xstart);
b = ABS(yend - ystart);
d = 2*b - a;
for (i = 0; i<a; i++){
Incr1 = 2*(b-a);
Plot(x,y);
Incr2 = 2*b;
x = x + 1;
if (xstart > xend) {
if (d ≥ 0) {
y = y + 1;
x = xend;
d = d + incr1;
y = yend
}
}
else
else {
d = d + incr2;
x = xstart;
}
y = ystart
}
}
G C D
O T
LXY
\
LM N G C D
L M N U K IH
]
B C D ND L M N D S G C D
^S Y
]
O D P EQ EM F
H G
^_ ` Z
H I b M N EG C U
\
R X ^Z a
LM N H
IEF D
H X ^_ ` Z
LNM U
\
R D
P H I P K IH G D O
O E N D P G IV
R a H _ `
XY SY Z GM
XH SR Z
EF
G CD
L E N Q G M P G H F G EQ c
else {
x = x + 1;
d = d + b
i++) {
}
}
+ 1;
+ 1;
+ b - a;
d e f g f h i f f h g e j k l j e j m nj f g o g p q i k r g ns i t u v
i p nf h z g f n v ‚ h g i ko e p nf h z
J H N EH R ID S O T S U H V
W M EF G X Y S Y Z [
x = 0;
y = 0;
d = b - a/2;
for(i = 0; i < a;
Plot(x,y);
if (d ≥ 0) {
x = x
y = y
d = d
}
Bresenham’s Line Algorithm
We can also generalize the algorithm to work for lines beginning at points other than
(0,0) by giving x and y the proper initial values. This results in Bresenham's Line Algorithm.
E F EG EH I J H IK D
LNM U
w x y g f h g j z r kf n{ k l | l u f e o g f } ~  u } € y g  i j } e i k k nj f g o g p
s f n k k y e p ƒ s s nj  g y g e j kl  i p g i | e r f f h g s no j € j e f f h g q i kr g e x } v
Optimizations
„
Speed can be increased even more by detecting cycles in the decision
variable. These cycles correspond to a repeated pattern of pixel choices.
„
The pattern is saved and if a cycle is detected it is repeated without
recalculating.
16
15
14
13
12
11
10
9
6
di=
7
2
8
-6
9
6
10
-2
11
10
12
13
2 -6
14
15
16
6 -2 10
17
The aliasing problem
Antialiasing - solutions
Aliasing can be smoothed out by using higher addressability.
Aliasing is cause by finite addressability of the display.
„
If addressability is fixed but intensity is variable, use it to control the address of
a "virtual pixel". Two adjacent pixels can be used to give the impression of a
point between them. The perceived location of the point depends upon the ratio
of the intensities used at each. The impression of a pixel located halfway
between two addressable points can be given by having two adjacent pixels at
half intensity.
„
Approximation of lines and circles with discrete points
often gives a staircase appearance or "Jaggies".
An antialiased line has a series of virtual pixels each located at the proper
address.
Aliased rendering of the line
„
Desired line
Antialiased Bresenham Lines
Aliasing / Antialiasing Examples
Line drawing algorithms such as Bresenham's can easily be
modified to implement virtual pixels. We use the distance (e = di/a)
value to determine pixel intensities.
Three possible cases which occur during the Bresenham algorithm:
e>0
0 > e > -0.5
A
e < -0.5
A
A
B
B
e
e
e
B
C
A = 0.5 + e
B = 1 - abs(e+0.5)
C=0
C
C
A = 0.5 + e
B = 1 - abs(e+0.5)
C=0
A=0
B = 1 - abs(e+0.5)
C = -0.5 - e
Polygons
Processing Polygons
A polygon is a many-sided planar figure composed of vertices and
edges.
Vertices are represented by points (x,y).
Edges are represented as line segments which connect two points,
(x1,y1) and (x2,y2).
P = { (xi , yi ) } i=1,n
E1
(x1,y1)
(x2,y2)
E2
E3
(x3,y3)
Convex and Concave Polygons
Convex Polygon - For any two points P1, P2 inside the polygon, all
points on the line segment which connects P1 and P2 are inside the
polygon.
– All points P = uP1 + (1-u)P2, u in [0,1] are inside the polygon
provided that P1 and P2 are inside the polygon.
Concave Polygon - A polygon which is not convex.
Inside test: A point P is inside a polygon if and only if a scanline intersects the
polygon edges an odd number of times moving from P in either direction.
2
trivial
simple sequence of line renderings
requires proper termination of lines at
endpoints
Max-Min Test
Inside Polygon Test
1
Rendering unfilled polygons
When crossing a vertex, if the vertex is a local maximum or minimum then count
it twice, else count it once.
3
1
or
or
Count twice
Count once
Does the vertex
count as two
points?
Or should it
count as one
point?
Filling Polygons
Fill the polygon 1 scanline at a time
Scan-Line Algorithm
For each scan-line:
1. Find the intersections of the scan line
with all edges of the polygon.
2. Sort the intersections by increasing xcoordinate.
3. Fill in all pixels between pairs of
intersections.
For scan-line number 7 the sorted list
of x-coordinates is (1,3,7,9)
Therefore fill pixels with x-coordinates
1-3 and 7-9.
8
Determine which pixels on each scanline are inside the polygon
and set those pixels to the appropriate value.
Key idea: Don’t check each pixel for “inside-ness”. Instead, look
only for those pixels at which changes occur.
Problem:
Calculating intersections is slow.
Solution:
Incremental computation / coherence
6
4
2
1
3
7
9
Edge Coherence
Processing Polygons
Observation: Not all edges intersect each scanline.
Many edges intersected by scanline i will also be intersected
by scanline i+1
Polygon edges are sorted according to their minimum Y. Scan lines are
processed in increasing (upward) Y order. When the current scan line
reaches the lower endpoint of an edge it becomes active. When the
current scan line moves above the upper endpoint, the edge becomes
inactive.
Active Edges
Formula for scanline s is y = s, for an edge is y = mx + b
Their intersection is
s = mxs + b => xs = (s-b)/m
For scanline s + 1,
xs+1 = (s+1 - b)/m = x s + 1/m
Not yet active edges
Finished edge
Ignored horizontal edge
Incremental calculation: xs+1 = xs + 1/m
Polygon fill rules (to ensure consistency)
1. Horizontal edges: Do not include in edge table
2. Horizontal edges: Drawn on the bottom, not on the top.
3. Vertices: If local max or min, then count twice, else count
once.
4. Vertices at local minima are drawn, vertices at local maxima
are not.
5. Only turn on pixels whose centers are interior to the
polygon: round up values on the left edge of a span, round
down on the right edge
Polygon fill example
D (16,18)
F (2,15)
E (8,12)
C (16,6)
A (2,3)
B (8,1)
Antialiasing Polygons
!
"
#
'
!
"
#
$%
$%
$%
$%
#
"
Xi Xi+1
1 02 3
+ , - 0 )
( - 0 2 3 ) 7 8
4 02 3 5 6
4 ) ) , 6 . , . 9 1 1 92 3
( - 9
! &"
&
' ' $"
$%
Fill patterns can be used to put a noticeable texture inside a polygon. A fill
pattern can be defined in a 0-based, m x n array. A pixel (x,y) is assigned
the value found in:
pattern((x mod m), (y mod n))
Xj Xj+1
0 1 + , , - . ) ) / + , - $%
Fill Patterns
( ) * / + $#
) 7 8 9 2 3 4 5 96
Pattern
Pattern filled polygon
Halftoning
For bitmapped displays, fill patterns with different fill densities can be used to
vary the range of intensities of a polygon. The result is a tradeoff of resolution
(addressability) for a greater range of intensities and is called halftoning. The
pattern in this case should be designed to avoid being noticed.
Modeling Curves and Surfaces
These fill patterns are chosen to minimize banding.
Motivation
Naïve approach
Need representations of smooth real
world objects
Art / drawings using CG need smooth
curves
Animation: camera paths
– curve: piecewise linear function
– lots of storage if high accuracy desired
– hard to interactively manipulate the shape
Instead, we’ll use higher-order functions
Note: difference between stored model
and rendered shape
Parametric Curves
Approaches
Explicit Functions:
Simply use lines & polygons to
approximate curves & surfaces
y = f(x) [e.g. y = 2x2]
Linear:
1. Only one value of y for each x
y = ayt + by
2. Difficult to represent a slope of infinity
Implicit Equations:
f(x,y) = 0 [e.g. x2 + y2 - r2 = 0]
x = axt + bx
z = azt + bz
Quadratic:
1. Need constraints to model just one part of a curve
x = axt2 + bxt + cx
y = ayt2 + byt + cy
2. Joining curves together smoothly is difficult
z = azt2 + bzt + cz
Parametric Equations:
x = f(t), y = f(t) [e.g. x = t2+3, y = 3t2+2t+1]
1. Slopes represented as parametric tangent vectors
2. Easy to join curve segments smoothly
Cubic:
x = axt3 + bxt2+ cxt + dx
y = ayt3 + byt2+ cyt + dy
z = azt3 + bzt2+ czt + dz
≤ ≤ Joining Curve Segments Together
G0
G1
geometric continuity: The directions of the two
segments’ tangent vectors are equal at the join point
Joining Examples
geometric continuity: Two curve segments join together
S
R0
TV3
C1
continuity: Tangent vectors of the two segments are equal
in magnitude and direction
Q3
TV2
– (C1 ➜ G1 unless tangent vector = [0,0,0])
P
Cn continuity: Direction and magnitude through the nth
derivative are equal at the join point
R2
R1
Q2
Q1
Alternate Notation
Notation
Q(t) = T M G
x = axt3 + bx t2+ cxt + dx
y = ayt3 + by t2+ cyt + dy
[t
Q(t) = T C
z = azt3 + bzt2+ czt + dz
[
T=t
t
2
ax
bx
C=
cx
dx
ay
by
cy
dy
]
t 1
az
bz
cz
dz
ax
Q (t ) = [t 1]
bx
m13
m21
m22
m23
m24
m31
m32
m33
m34
m41
m42
m43
m44
Basis Matrix
m
Q(t ) = [t 1] 11
m21
!
"
#
m12 ! G1 x
"
m22 G2 x
G1 = P1
ay
by
az
bz
G2 y
G2 z
We know that at t = 0,
t=0
&)
)
Q′(t ) = [1 0]
G1
G2
G3 G4
Geometry Matrix
G2 = P2.
&
m11
m12
G1 y
G1z
m21
m22 G2 x G2 y
G2 z
'
% $(
G1x
t<0
P1
%$
P2
Q(0) = P1
Q'(0) = P2 - P1
at t = 1,
'
(
What is the basis matrix?
t=1
G1 z
Since two points, P1 & P2 define a straight line, then the
geometry matrix should be:
G1 y
The basis matrix contains information about the general family of curve
that will be produced.
t>1
#
m14
Example (cont.)
ax
Q′(t ) = [1 0 ]
bx
az
bz
m12
Idea: Different curves can be specified by changing the geometric
information in the geometry matrix.
ay
by
t 1]
m11
T Matrix
x'(t) = ax
y'(t) = ay
z'(t) = az
2
Example: Parametric Line
x(t) = axt + bx
y(t) = ayt + by
z(t) = azt + bz
t
3
3
Q(1) = P2
Q'(1) = P2 - P1
Example (cont.)
Example (cont.)
Solve these simultaneous equations to find the
basis matrix Mline:
m21x1 + m22x2 = x1
(m11 + m21)x1 + (m12 + m22)x2= x2
m21y1 + m22y2 = y1
(m11 + m21)y1 + (m12 + m22)y2 = y2
m21z1 + m22z2 = z1
(m11 + m21)z1 + (m12 + m22)z2= z2
m11x1 + m12x2 = x2 - x1
m11 = -1
m12 = 1
m11y1 + m12y2 = y2 - y1
m21 = 1
m22 = 0
Q(0) =
(0, 1) Mline G = P1
Q'(0) =
(1, 0) Mline G = P2 - P1
Q(1) =
Q'(1) =
(1, 1) Mline G = P2
m11z1 + m12z2 = z2 - z1
M line
(1, 0) Mline G = P2 - P1
Multiplying the T and M matrices gives you a set of functions in t; these are
called blending functions.
There is one blending function for each of the pieces of geometric information
in the G matrix.
The value of a blending function at a certain value of t determines the effect of
the corresponding piece of geometric information at that point along the
curve.
[t 1]
Hermite Curves
Blending functions
Line blending functions:
−1 1
=
1 0
−1 1 t]
= [1 − t
1 0
R1
P4
R4
f(t)
x
1 P1 fn.
P2 fn.
0
1
= (1 − t ) x1 + tx2
P1
y = (1 − t ) y1 + ty2
t
Hermite Curve Examples (cont.)
Hermite Curve Examples
Tangent vector direction
at point P 1. Magnitude
varies for each curve
P1
Tangent vector direction
at point P 4. Magnitude
varies for each curve
P4
Family of Hermite parametric cubic curves. Only the tangent vector
at P1 varies for each curve, increasing for the higher curves
Family of Hermite parametric cubic curves. Only the direction of the tangent vector at the
left starting point varies; all tangent vectors have the same magnitude. A smaller
magnitude would eliminate the loops in some of the curves
Hermite Matrices
Q(t) = T MH GH
GH =
P1
T = [t
3
t
2
Q(t) = T MH GH
t 1]
Q(t) =
(2t 3 − 3t 2 + 1)P1 +
P4
R1
R4
Hermite Blending Functions
−2
2
−3
MH =
0
1
1
1
0
0
0
(−2t 3 + 3t 2 )P4 +
− 2 −1
1
0
3
0
(t 3 − 2t 2 + t)R1 +
(t 3 − t 2 )R4
Bézier Curves
Bézier Matrices
GB =
P3
P2
Q(t) = T MB GB
P4
• R1 = 3(P2-P1), R4 = 3(P4-P3)
T = [t 3 t 2
P3
P2
• constant velocity curves if
control points are equally
spaced
MB =
Two Bezier curves and their control points. Notice that the convex
hulls shown as dashed lines need not involve all four control points
Bézier Blending Functions
• P1 & P4: endpoints
P4
P1
P1
P1
P2
P3
P4
t 1]
−1
3
3
−6
3
0
−3
3
0
0
1
0
−3 1
0
General Bézier Curves
Let P1 through Pn+1 be points that define the curve.
Then:
Q (t ) =
QB (t) =
(−t + 3t − 3t + 1) P1 +
3
The Bernstein polynomials, which
are weighting functions for Bezier
curves. At t=0, only BB1 is nonzero,
so the curve interpolates P1;
similarly, at t=1, only BB4 is nonzero,
and the curve interpolates P4
2
(3t 3 − 6t 2 + 3t ) P2 +
(−3t 3 + 3t 2 ) P3 + t 3 P4
n
i =0
Pi +1 Bi ,n (t),t ∈[0,1]
where
Bi,n(t) = C(n,i) ti (1-t)n-i
{Blending functions}
and
C(n,i) = n!/(i!(n-i)!)
{Binomial coefficient}
0
General Bézier Curves
Bézier Curve Characteristics
The functions interpolate the first and last vertex points.
For two points, n = 1:
The tangent at P0 is given by P1 - P0, and the tangent at Pn by Pn - Pn-1
QB(t) = (1-t)P1 + tP2
The blending functions are symmetric with respect to t and (1-t). This
means that we can reverse the sequence of vertex points defining the
curve without changing the shape of the curve.
For three points, n = 2:
The curve, Q(t), lies within the convex hull defined by the control
points.
QB(t) = (1-t)2P1 + 2t(1-t)P2 + t2P3
If the first and last vertices coincide, then we produce a closed curve.
If complicated curves are to be generated, they can be formed by
piecing together several Bézier sections.
For four points, n = 3:
QB(t) = (1-t)3P1 + 3t(1-t)2P2 + 3t2(1-t)P3 + t3P4
Modeling surfaces
By specifying multiple coincident points at a vertex, we pull the curve
in closer and closer to that vertex.
Matrix representation
Extension of parametric cubic curves
called “parametric bicubic surfaces”
Idea: infinite # of curves stacked
together
– equations now have 2 parameters
– Q(s,t)
P1(t)
t=0.75
G1 (t )
G2 (t )
G=
G3 (t )
P4(t)
t=0.25
t
s
Matrix representation (cont.)
A single curve was expressed
Q(t) = TMG
Now, the geometric information varies
Q(s,t) = SMG
Since each Gi is a cubic curve, it can be
written:
Gi(t) = TMGi
Gi = g i1
gi4
Matrix representation (cont.)
Substituting, we obtain:
TMG1
G1
G2
= SMTM G3
G4
TMG3 TMG 4
Q(s,t) = SM TMG 2
gi2
gi 3
G4 (t )
each Gi(t) is itself
a cubic curve
This formulation does not work in
terms of matrix dimensions...
Matrix representation (cont.)
So, use the transpose rule:
Gi(t) = GiT MT TT
Q(s,t) = S M
g11
g12
g13
g14
g 21
g 31
g 22
g32
g 23
g33
g 24
g 34
g 41
g 42
g 43
g 44
=SM
T
G1
T
G2 MT TT
T
G3
T
G4
Finally, remember that the large geometry
matrix has 3 components (x, y, z) for each
gij, so that we get three parametric
equations:
x(s,t) = S M Gx MT TT
y(s,t) = S M Gy MT TT
z(s,t) = S M Gz MT TT
MT TT
Hermite surfaces
Matrix representation (cont.)
Hermite surface matrices
x(s,t) = S M GHx MT TT
y(s,t) = S M GHy MT TT
z(s,t) = S M GHz MT TT
extension of Hermite curves to
parametric bicubic surfaces
four elements of the geometry matrix are
now P1(t), P4(t), R1(t), R4(t)
can be thought of as interpolating the
curves Q(s,0) and Q(s,1) or
Q(0,t) and Q(1,t)
P1 x (t) = TMG1x
g11
g12
G1 x =
g13
g11
g 21
=
g
31
g 41
GH x
g12
g 22
g32
g 42
g13
g 23
g33
g 43
g14 g 24 g34 g 44
g14
Rendering curves
Iterative method based on parametric formula
evaluation
Rendering surfaces
– t=0
– plot x(t), y(t), z(t)
– increment t by a small amt. and repeat until t¡ 1
This is slow, depending on step size
Can increase performance using difference
methods (as with line drawing)
Can also use iterative methods in s and t
Grid representation:
– render curves of constant t and constant s
– use perspective and hidden lines to indicate
surface shape
Solid representation:
– subdivide surface into “flat” sections
– render quadrilateral for each section
– shape indicated by shading (need normal)
Examples
Examples
Closed surfaces can be formed by
setting the last control point equal to
the first. If the tangents also match
between the first two and last two
control points then the closed surface
will have first order continuity.
The corresponding properties of the Bézier curve apply to the Bézier surface
• The surface does not in general pass through the control points except
for the corners of the control point grid.
• The surface is contained within the convex hull of the control points.
Along the edges of the grid patch the Bézier surface matches that of a Bézier
curve through the control points along that edge.
While a cylinder/cone can be formed from a Bézier surface, it is not
possible to form a sphere.
•
Examples
Homework
Given 2 points in 3D: P1=(-4,4,12) and
P2=(5,7.5,7.5), center of projection O=(0,0,0),
image plane z=3, and the origin of the image
plane in XYZ coordinates Op=(-3,-2,3),
calculate:
•
•
(-4, 4, 12)
•
Y
Yp
(5,7.5,7.5)
•
Z
P3
•
Recalculate 1.b and 1.c when the center of
projection moves to O’=(1,2,0)
•
Recalculate 1.b and 1.c when the image plane
rotates with 45° around Y p (Op = (-3,-2,3) and
O=(0,0,0)) (the sense of the rotation is
indicated in the figure)
P4
(-3,-2,3)
image plane
(0,0,0)
Bezier surface patches rendered using different methods
Xp
X
the parametric equations for the line P1P2
determine the intersections P 3 and P 4 of the
OP 1 and OP2 lines with the image plane.
which are the coordinates of P3 and P4 in
image coordinates (XpYp)?
use Bresenham algorithm to determine which
pixels in the image plane are activated when
you draw the line P3P4.
Hint: compute the new image plane equation and in the
parametric line equations compute t that
constrains P3 and P4 to lay in the image plane