computer graphics implementation i
DESCRIPTION
Computer Graphics Implementation I. Algorithms for Drawing 2D Primitives. Line Drawing Algorithms DDA algorithm Midpoint algorithm Bresenham’s line algorithm Circle Generating Algorithms Bresenham’s circle algorithm Extension to Ellipse drawing. Line Drawing. Line Drawing - PowerPoint PPT PresentationTRANSCRIPT
C O M P U T E R G R A P H I C S
Guoying Zhao 1 / 51
C O M P U T E R G R A P H I C S
Guoying Zhao 1 / 51
Computer Graphics
Implementation I
C O M P U T E R G R A P H I C S
Guoying Zhao 2 / 51
C O M P U T E R G R A P H I C S
Guoying Zhao 2 / 51
Algorithms for Drawing 2D Primitives
• Line Drawing Algorithms– DDA algorithm– Midpoint algorithm– Bresenham’s line algorithm
• Circle Generating Algorithms– Bresenham’s circle algorithm– Extension to Ellipse drawing
C O M P U T E R G R A P H I C S
Guoying Zhao 3 / 51
Line Drawing
C O M P U T E R G R A P H I C S
Guoying Zhao 4 / 51
Line Drawing
• Draw a line on a raster screen between two points
• What’s wrong with statement of problem?– doesn’t say anything about which points are allowed as endpoints
– doesn’t give a clear meaning of “draw”
– doesn’t say what constitutes a “line” in raster world
– doesn’t say how to measure success of proposed algorithms
Problem Statement
• Given two points P and Q in XY plane, both with integer coordinates, determine which pixels on raster screen should be on in order to make picture of a unit-width line segment starting at P and ending at Q
Scan Converting Lines
C O M P U T E R G R A P H I C S
Guoying Zhao 5 / 51
Special case:
• Horizontal Line:
Draw pixel P and increment x coordinate value by 1 to get next pixel.
• Vertical Line:
Draw pixel P and increment y coordinate value by 1 to get next pixel.
• Diagonal Line:
Draw pixel P and increment both x and y coordinate by 1 to get next pixel.
• What should we do in general case? => Fast
- Digital differential analyzer (DDA) algorithm
- Mid-point algorithm
- Bresenham algorithm
Finding next pixel:
C O M P U T E R G R A P H I C S
Guoying Zhao 6 / 51
C O M P U T E R G R A P H I C S
Guoying Zhao 6 / 516
Scan Conversion of Line Segments
• Start with line segment in window coordinates with integer values for endpoints
• Assume implementation has a write_pixel function
y = mx + h
x
ym
C O M P U T E R G R A P H I C S
Guoying Zhao 7 / 51
Basic Algorithm• Find equation of line that connects two points P(x1,y1) and
Q(x2,y2)• Starting with leftmost point P, increment xi by 1 to calculate yi =
m*xi + hwhere m = slope, h = y intercept
• Draw pixel at (xi, Round(yi)) where
Round (yi) = Floor (0.5 + yi)
C O M P U T E R G R A P H I C S
Guoying Zhao 8 / 51
C O M P U T E R G R A P H I C S
Guoying Zhao 8 / 518
Strategy 1 – DDA Algorithm
• Digital Differential Analyzer
– DDA was a mechanical device for numerical solution of differential equations
– Line y=mx+ h satisfies differential equation dy/dx = m = Dy/Dx = (y2-y1)/(x2-x1)
• Along scan line Dx = 1
for(x=x1; x<=x2;x++) { y+=m; write_pixel(x, round(y), line_color);}
C O M P U T E R G R A P H I C S
Guoying Zhao 9 / 51
Example Code// Incremental Line Algorithm
// Assume x0 < x1
void Line(int x0, int y0,
int x1, int y1) {
int x, y;
float dy = y1 – y0;
float dx = x1 – x0;
float m = dy / dx;
y = y0;
for (x = x0; x < x1; x++) {
WritePixel(x, Round(y));
y = y + m;
}
}
C O M P U T E R G R A P H I C S
Guoying Zhao 10 / 51
),( ii yx ))(,1( myRoundx ii
))(,( ii yRoundx ),1( myx ii
C O M P U T E R G R A P H I C S
Guoying Zhao 11 / 51
C O M P U T E R G R A P H I C S
Guoying Zhao 11 / 5111
Problem
• DDA = for each x plot pixel at closest y– Problems for steep lines
C O M P U T E R G R A P H I C S
Guoying Zhao 12 / 51
C O M P U T E R G R A P H I C S
Guoying Zhao 12 / 5112
Using Symmetry
• Use for 1 m 0
• For m > 1, swap role of x and y– For each y, plot closest x
C O M P U T E R G R A P H I C S
Guoying Zhao 13 / 51
Problem
void Line(int x0, int y0,
int x1, int y1) {
int x, y;
float dy = y1 – y0;
float dx = x1 – x0;
float m = dy / dx;
y = y0;
for (x = x0; x < x1; x++) {
WritePixel(x, Round(y));
y = y + m;
}
}
Rounding takes time
C O M P U T E R G R A P H I C S
Guoying Zhao 14 / 51
• Assume that line’s slope is shallow and positive (0 < slope < 1); other slopes can be handled by suitable reflections about principle axes
• Call lower left endpoint (x0, y0) and upper right endpoint (x1, y1)
• Assume that we have just selected pixel P at (xp, yp)
• Next, we must choose between pixel to right (E pixel) (xp+1, yp), or one right and one up (NE pixel) (xp+1, yp+1)
• Let Q be intersection point of line being scan-converted and vertical line x=xp+1
Strategy 2 – Midpoint Line Algorithm (1/3)
C O M P U T E R G R A P H I C S
Guoying Zhao 15 / 51
Strategy 2 – Midpoint Line Algorithm (2/3)
),( pp yxP 1 pxx
Previous pixel Choices for current pixel
Choices for next pixel
E pixel
NE pixel
Midpoint MQ
C O M P U T E R G R A P H I C S
Guoying Zhao 16 / 51
• Line passes between E and NE
• Point that is closer to intersection point Q must be chosen
• Observe on which side of line midpoint M lies:– E is closer to line if midpoint M lies above line, i.e., line crosses
bottom half
– NE is closer to line if midpoint M lies below line, i.e., line crosses top half
• Error (vertical distance between chosen pixel and actual line) is always <= ½
Strategy 2 – Midpoint Line Algorithm (3/3)
• Algorithm chooses NE as next pixel for line shown
• Now, need to find a way to calculate on which side of line midpoint lies E
pixel
NE pixel
MQ
C O M P U T E R G R A P H I C S
Guoying Zhao 17 / 51
Line equation as function f(x):
Line equation as implicit function:
for coefficients a, b, c, where a, b ≠ 0
from above,
Properties (proof by case analysis):• f(xm, ym) = 0 when any point M is on line• f(xm, ym) < 0 when any point M is above line• f(xm, ym) > 0 when any point M is below line• Our decision will be based on value of function at midpoint M at (xp +
1, yp + ½)
Line
Bxdx
dyy
Bmxy
0),( cbyaxyxf
dxBcdxbdya
dxBdxyxdy
dxBxdydxy
,,
0
C O M P U T E R G R A P H I C S
Guoying Zhao 18 / 51
void MidpointLine(int x0, int y0, int x1, int y1) {
int dx = x1 - x0;
int dy = y1 - y0;
int x = x0;
int y = y0;
float c = y0 * dx – dy * x0;
writePixel(x, y);
x++;
while (x < x1) {
d = dy * x – (y + 0.5) * dx + c;
if (d > 0) { // Northeast Case
y++;
}
writePixel(x, y);
x++;
} /* while */
}
C O M P U T E R G R A P H I C S
Guoying Zhao 19 / 51
Decision Variable d:
• We only need sign of f(xp + 1, yp + ½) to see where line lies,
and then pick nearest pixel
• d = f(xp + 1, yp + ½)
- if d > 0 choose pixel NE
- if d < 0 choose pixel E
- if d = 0 choose either one consistently
How do we incrementally update d?
• On basis of picking E or NE, figure out location of M for that
pixel, and corresponding value d for next grid line
• We can derive d for the next pixel based on our current
decision
Decision Variable
C O M P U T E R G R A P H I C S
Guoying Zhao 20 / 51
Increment M by one in x direction
dnew = f(xp + 2, yp + ½)
= a(xp + 2) + b(yp + ½) + c
dold = a(xp + 1) + b(yp + ½) + c
• dnew - dold is the incremental difference E
dnew = dold + a E = a = dy
• We can compute value of decision variable at next step incrementally without computing F(M) directly
dnew = dold + E = dold + dy
E can be thought of as correction or update factor to take dold to dnew • It is referred to as forward difference
If E was chosen:Strategy 3 – Bresenham algorithm
C O M P U T E R G R A P H I C S
Guoying Zhao 21 / 51
Increment M by one in both x and y directions
dnew = F(xp + 2, yp + 3/2)
= a(xp + 2) + b(yp + 3/2) + c
NE = dnew – dold
dnew = dold + a + b
NE = a + b = dy – dx
• Thus, incrementally,
dnew = dold + NE = dold + dy – dx
If NE was chosen:
C O M P U T E R G R A P H I C S
Guoying Zhao 22 / 51
• At each step, algorithm chooses between 2 pixels based on sign of decision variable calculated in previous iteration.
• It then updates decision variable by adding either E or NE to old value depending on choice of pixel. Simple additions only!
• First pixel is first endpoint (x0, y0), so we can directly calculate initial value of d for choosing between E and NE.
Summary (1/2)
C O M P U T E R G R A P H I C S
Guoying Zhao 23 / 51
• First midpoint for first d = dstart is at (x0 + 1, y0 + ½)
• f(x0 + 1, y0 + ½) = a(x0 + 1) + b(y0 + ½) + c = a * x0 + b * y0 + c + a + b/2 = f(x0, y0) + a + b/2
• But (x0, y0) is point on line and f(x0, y0) = 0
• Therefore, dstart = a + b/2 = dy – dx/2 – use dstart to choose second pixel, etc.
• To eliminate fraction in dstart : – redefine f by multiplying it by 2; f(x,y) = 2(ax + by + c)– this multiplies each constant and decision variable by 2, but does not
change sign
Summary (2/2)
C O M P U T E R G R A P H I C S
Guoying Zhao 24 / 51
Example Codevoid Bresenham(int x0, int y0, int x1, int y1) {
int dx = x1 - x0;
int dy = y1 - y0;
int d = 2 * dy - dx;
int incrE = 2 * dy;
int incrNE = 2 * (dy - dx);
int x = x0;
int y = y0;
writePixel(x, y);
while (x < x1) {
if (d <= 0) { // East Case
d = d + incrE;
} else { // Northeast Case
d = d + incrNE;
y++;
}
x++;
writePixel(x, y);
} /* while */
} /* MidpointLine */
C O M P U T E R G R A P H I C S
Guoying Zhao 25 / 51
Drawing Circles
C O M P U T E R G R A P H I C S
Guoying Zhao 26 / 51
C O M P U T E R G R A P H I C S
Guoying Zhao 26 / 51
Bresenham’s Circle Algorithm
• Circle Equation– (x-xc)2 + (y- yc)2 = R2
• Problem simplifying– Making the circle
origin (xc, yc) in the coordinate origin
– Symmetry points(y,x), (y,-x),(x,-y),(-x,-y),(-y,-x),(-y,x),(-x,y)
C O M P U T E R G R A P H I C S
Guoying Zhao 27 / 51
Version 1: really bad
For x = – R to R
y = sqrt(R * R – x * x);
Pixel (round(x), round(y));
Pixel (round(x), round(-y));
Version 2: slightly less bad
For x = 0 to 360
Pixel (round (R • cos(x)), round(R • sin(x)));
(17, 0)
(0, 17)
(17, 0)
(0, 17)
Scan Converting Circles
C O M P U T E R G R A P H I C S
Guoying Zhao 28 / 51
Suppose circle origin is (xc,yc),
radius is R.
Let D(x, y) = (x-xc)2+(y-yc)
2–R2
• If point(x,y) is outside the circle, D(x,y)>0 。
• If point(x,y) is inside the circle, D(x,y)<0 。
(xc, yc)
C O M P U T E R G R A P H I C S
Guoying Zhao 29 / 51
Bresenham’s Circle Algorithm
Two choice: (xi+1, yi) or (xi+1, yi-1)
(xi, yi) (xi+1, yi)
(xi+1, yi-1)
(xi, yi-1)
(xi+1, yi)
(xi+1, yi-1)
(xi, yi)
(xi, yi-1)
C O M P U T E R G R A P H I C S
Guoying Zhao 30 / 51
• Symmetry: If (x0 + a, y0 + b) is on circle
– also (x0 ± a, y0 ± b) and (x0 ± b, y0 ± a); hence 8-way symmetry.
• Reduce the problem to finding the pixels for 1/8 of the circle
R
(x0 + a, y0 + b)
(x-x0)2 + (y-y0)2 = R2
(x0, y0)
Use Symmetry
C O M P U T E R G R A P H I C S
Guoying Zhao 31 / 51
• Scan top right 1/8 of circle of radius R
• Circle starts at (x0, y0 + R)
• Let’s use another incremental algorithm with decision variable evaluated at midpoint
(x0, y0)
Using the Symmetry
C O M P U T E R G R A P H I C S
Guoying Zhao 32 / 51
E
SE
x = x0; y = y0 + R; Pixel(x, y);for (x = x0+1; (x – x0) < (y – y0); x++) {
if (decision_var < 0) {/* move east */update decision_var;
}else {
/* move south east */update decision_var;y--;
}Pixel(x, y);
}
Sketch of Incremental Algorithm
C O M P U T E R G R A P H I C S
Guoying Zhao 33 / 51
• Decision variable– negative if we move E, positive if we move SE (or vice versa).
• Follow line strategy: Use implicit equation of circle
f(x,y) = x2 + y2 – R2 = 0
f(x,y) is zero on circle, negative inside, positive outside
• If we are at pixel (x, y)– examine (x + 1, y) and (x + 1, y – 1)
• Compute f at the midpoint
What we need for Incremental Algorithm
C O M P U T E R G R A P H I C S
Guoying Zhao 34 / 51
P = (xp, yp)M ME
MSE
SE
Decision VariableE
• Evaluate f(x,y) = x2 + y2 – R2
at the point
• We are asking: “Is
positive or negative?” (it is zero on circle)
• If negative, midpoint inside circle, choose E– vertical distance to the circle is less at
(x + 1, y) than at (x + 1, y–1).• If positive, opposite is true, choose SE
22
2
21)1(
21,1 Ryxyxf
21,1 yx
C O M P U T E R G R A P H I C S
Guoying Zhao 35 / 51
• Decision based on vertical distance
• Ok for lines, since d and dvert are proportional
• For circles, not true:
• Which d is closer to zero? (i.e. which of the two values below is closer to R):
RyxCircyxd
RyxCircyxd
22
22
)1()1()),1,1((
)1()),,1((
2222 )1()1()1( yxoryx
The right decision variable?
C O M P U T E R G R A P H I C S
Guoying Zhao 36 / 51
• We could ask instead:“Is (x + 1)2 + y2 or (x + 1)2 + (y – 1)2 closer to R2?”
• The two values in equation above differ by
12])1()1[(])1[( 2222 yyxyx
fE – fSE = 290 – 257 = 33
(0, 17) (1, 17)
(1, 16)
fE = 12 + 172 = 290E
SE
fSE = 12 + 162 = 257
2y – 1 = 2(17) – 1 = 33
Alternate Phrasing (1/3)
C O M P U T E R G R A P H I C S
Guoying Zhao 37 / 51
• The second value, which is always less,
is closer if its difference from R2 is less than
i.e., if
then
so
so
so
2 )12(1 y])1()1[( 222 yxR
)12(21
y
21)1(0 222 Ryyx
2112)1(0 222 Ryyyx
)1()1(210 222 Ryxy
22
2
41
21)1(0 Ryx
Alternate Phrasing (2/3)
C O M P U T E R G R A P H I C S
Guoying Zhao 38 / 51
• The radial distance decision is whether
is positive or negative
• And the vertical distance decision is whether
•
is positive or negative; d1 and d2 are apart.
• The integer d1 is positive only if d2 + is
positive (except special case where d2 = 0).4
1
4
1
22
2
41
21)1(1 Ryxd
22
2
21)1(2 Ryxd
Alternate Phrasing (3/3)
C O M P U T E R G R A P H I C S
Guoying Zhao 39 / 51
• How to compute the value of
at successive points?• Answer: Note that
is just
and that
is just
(1/2)
22
2
21)1(),( Ryxyxf
2232),(
),()1,1(
32),(
),(),1(
SE
E
yxyx
yxfyxf
xyx
yxfyxf
Incremental Computation, Again
C O M P U T E R G R A P H I C S
Guoying Zhao 40 / 51
F F’
F’’
Incremental Computation (2/2)• If we move E, update by adding 2x + 3
• If we move SE, update by adding 2x + 3 – 2y + 2.
• Forward differences of a 1st degree polynomial are constants and those of a 2nd degree polynomial are 1st degree polynomials – this “first order forward difference,” like a partial derivative, is one
degree lower
C O M P U T E R G R A P H I C S
Guoying Zhao 41 / 51
• The function is linear, hence amenable to incremental computation, viz:
• Similarly
Second Differences (1/2)
32),(E xyx
2),()1,1( EE yxyx
2),(),1( EE yxyx
4),()1,1( SESE yxyx2),(),1( SESE yxyx
C O M P U T E R G R A P H I C S
Guoying Zhao 42 / 51
• For any step, can compute new ΔE(x, y) from old ΔE(x, y) by adding appropriate second constant increment – update delta terms as we move.
– This is also true of ΔSE(x, y)
• Having drawn pixel (a,b), decide location of new pixel at (a + 1, b) or (a + 1, b – 1), using previously computed d(a, b).
• Having drawn new pixel, must update d(a, b) for next iteration; need to find either d(a + 1, b) or d(a + 1, b – 1) depending on pixel choice
• Must add ΔE(a, b) or ΔSE(a, b) to d(a, b)
• So we…– Look at d(i) to decide which to draw next, update x and y– Update d using ΔE(a,b) or ΔSE(a,b) – Update each of ΔE(a,b) and ΔSE(a,b) for future use– Draw pixel
Second Differences (2/2)
C O M P U T E R G R A P H I C S
Guoying Zhao 43 / 51
Bresenham’s Eighth Circle Algorithmvoid MEC (int R) /* 1/8th of a circle w/ radius R */
{ int x = 0, y = R;
int delta_E, delta_SE;
float decision;
delta_E = 2*x + 3;
delta_SE = 2*(x-y) + 5;
decision = (x+1)*(x+1) + (y - 0.5)*(y - 0.5) –R*R;
Pixel(x, y);
while( y > x ) {
if (decision < 0) {/* Move east */
decision += delta_E;
delta_E += 2; delta_SE += 2; /*Update delta*/
}
else {/* Move SE */
y--;
decision += delta_SE;
delta_E += 2; delta_SE += 4; /*Update delta*/}
x++;
Pixel(x, y);
}
}
C O M P U T E R G R A P H I C S
Guoying Zhao 44 / 51
• Uses floats!
• 1 test, 3 or 4 additions per pixel
• Initialization can be improved
• Multiply everything by 4 No Floats!– Makes the components even, but sign of decision variable remains
same
Questions
• Are we getting all pixels whose distance from the circle is less than ½?
• Why is y > x the right stopping criterion?
• What if it were an ellipse?
Analysis
C O M P U T E R G R A P H I C S
Guoying Zhao 45 / 51
• Equation is
i.e,
• Computation of and is similar
• Only 4-fold symmetry
• When do we stop stepping horizontally and switch to vertical?
Aligned Ellipses
12
2
2
2
by
ax
222222bayaxb
E SE
C O M P U T E R G R A P H I C S
Guoying Zhao 46 / 51
• When absolute value of slope of ellipse is more than 1, viz:
• How do you check this? At a point (x,y) for which f(x,y) = 0, a vector perpendicular to the level set is f(x,y) which is
• This vector points more right than up when
Direction Changing Criterion (1/2)
),(),,( yxyfyx
xf
0),(),(
yx
yfyx
xf
C O M P U T E R G R A P H I C S
Guoying Zhao 47 / 51
• In our case,
and
so we check for
i.e.
• This, too, can be computed incrementally
Direction Changing Criterion (2/2)
xayxx
f 22),(
ybyxy
f 22),(
022 22 ybxa
022 ybxa
C O M P U T E R G R A P H I C S
Guoying Zhao 48 / 51
• Patterned primitives
• Non-integer primitives
• General conics
Other Scan Conversion Problems
C O M P U T E R G R A P H I C S
Guoying Zhao 49 / 51
• Patterned line from P to Q is not same as patterned line from Q to P.
• Patterns can be geometric or cosmetic– Cosmetic: Texture applied after transformations– Geometric: Pattern subject to transformations
Cosmetic patterned line
Geometric patterned line
Patterned Lines
P Q
P Q
C O M P U T E R G R A P H I C S
Guoying Zhao 50 / 51
Geometric Pattern vs. Cosmetic Pattern
+
Geometric (Perspectivized/Filtered) Cosmetic
C O M P U T E R G R A P H I C S
Guoying Zhao 51 / 51
C O M P U T E R G R A P H I C S
Guoying Zhao 51 / 51
Next Lecture
• Polygon Filling– Scan-line Conversion Approaches– Area Filling Approaches
• Antialiasing
• Clipping