- University of Sulaimani
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1
• 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
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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