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• How do we draw curves and surfaces?
• We need smooth curves and surfaces in many applications:
– model real world objects
– computer-aided design (CAD)
– high quality fonts
– data plots
– artists sketches
– Draw fractal lines, curves and surfaces
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Curve Generation I
• True curve  DDA
• Approximated  Interpolation
- Circular Arc Generation using DDA Algorithm
- Uses differential equation of the curve
x  R cos(  )  x 0
y  R sin(  )  y 0
where ( x 0 , y 0 ) is the center of curvature , and R is the radius of arc
x 2  x1  dx  x1  ( y 1  y 0 ) d 
y 2  y 1  dy  y 1  ( x 2  x 0 ) d 

R
d   Min ( 0 . 01 , 1 /( 3 . 2  ( x  x 0  y  y 0 )))
( x0 , y0 )
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Curve Generation II
- Drawbacks:
1.
2.
3.
4.
Need more information than endpoints.
Ability to scale a picture is limited.
New clipping algo is required.
Complex for airplane wings, cars and human faces.
- Advantage:
Very smooth
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Interpolation
- Drawing curves using approximation methods
Find suitable mathematical expression for the known curve
(Polynomial, trigonometric and exponential) to approximate the curve
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Spline Representation
-
To produce a smooth curve through a designated set of points (control
points)  flexible strip is used (Spline).
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Parametric Equation
•
•
•
A (2-D) parametric curve is expressed as:
– A pair of (mathematical) functions: P(t) = ( x(t), y(t) ).
• In 3-D, we add a third function for z.
– And an interval of legal values for t: [a,b].
t is called the parameter.
Example: x(t) = t2–2t, y(t) = t–1, t in [0,3].
y
t = 3 (end)
(t2–2t, t–1)
x
t = 0 (start)
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Bezier Curves I
• French mathematician
• Used to construct curves and surfaces
• Determined by defining polygon
• Useful for curve and surface design
• Easy to implement
• Available in CAD system and various
graphic packages
• Cubic Bezier curve is used
to avoid no. of calculations
• Always passes through the first
and last control points
• Designed curve follows the shape
of the defining polygon
• Invariant under an affine transformation
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Bezier Curves II
P ( u )  (1  u ) P1  3 u (1  u ) P2  3 u (1  u ) P3  u P4
3
2
2
3
Example: construct the Bezier curve of order 3 and with 4
polygon vertices A(1,1), B(2,3), C(4,3) and D(6,4).
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2D Bezier Curves
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3D Bezier Curves
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Bezier Surfaces
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Bézier Surface
Properties
boundary curves
lie on surface
boundary curves
defined by
boundary polygons
Bézier Surface
Properties
Nice, intuitive method for creating surfaces
Variable display resolution
Minimal storage
Bézier Surface
Multiple patches
connected smoothly
Conditions on control net
similar to curves …
difficult to do manually