7 a curves and surfaces
TRANSCRIPT
-
8/2/2019 7 a Curves and Surfaces
1/19
-
8/2/2019 7 a Curves and Surfaces
2/19
Agenda
Parametric curves in space
Parametric surfaces in space
-
8/2/2019 7 a Curves and Surfaces
3/19
Parametric Curves
The idea is to be able to define complexcurves (or surfaces in 3D) by controlling a
set of points. This can be used for:
Designing objects
Designing trajectories of objects or cameras inanimations
Design surfaces
-
8/2/2019 7 a Curves and Surfaces
4/19
Parametric Curves
See examples in Paint:
You define a starting and ending point.
Then you define two additional controlpoints The curve passes through the first and lastcontrol points
The curve gets close to the intermediatecontrol points
-
8/2/2019 7 a Curves and Surfaces
5/19
Bzier Curves
Invented by French engineer Pierre Bzier
Are simple to implement
Have several features that are useful indesigning objects
-
8/2/2019 7 a Curves and Surfaces
6/19
Bzier Curves
Desirable features Order-0 continuity: curve n+1 starts where
curve n ends. Order-1 continuity: the derivative of curve n+1where it starts is the same as the derivative ofcurve n where it ends.
The curve is confined within the convex-hull formed by the control points
-
8/2/2019 7 a Curves and Surfaces
7/19
Bzier Curves
Some examples:
-
8/2/2019 7 a Curves and Surfaces
8/19
Bzier Curves
How to create a closed curve:
-
8/2/2019 7 a Curves and Surfaces
9/19
Bzier Curves
How to make the curve pass closer to apoint:
-
8/2/2019 7 a Curves and Surfaces
10/19
Bzier Curves the Math
For n+1 controlpoints:
BEZ, the blendingfunctions, are theBernsteinpolynomials:
C(n,k) are thebinomialcoefficients:
n
k
nkk uuBEZu0
,10),()( pP
1
, )1(),()(
kk
nk uuknCuBEZ
)!(!
!),(
knk
nknC
-
8/2/2019 7 a Curves and Surfaces
11/19
Bzier Curves the Math
The vectorialequationrepresents three
parametricequations:
n
k
nkk uBEZxux0
, )()(
n
k
nkk uBEZyuy0
, )()(
n
k
nkk uBEZzuz0
,)()(
-
8/2/2019 7 a Curves and Surfaces
12/19
Bzier Curves the Math
For instance, for n = 3
3
3,0 )1()( uuBEZ 2
3,1 )1(3)( uuuBEZ
)1(3)(2
3,2 uuuBEZ
3
3,3 )( uuBEZ
-
8/2/2019 7 a Curves and Surfaces
13/19
Bzier Curves the Math
Blending functions
from: www.cs.virginia.edu/~gfx/Courses/2001/Intro.fall.01/Lectures/lecture21.ppt
-
8/2/2019 7 a Curves and Surfaces
14/19
Bzier Curves
Zero and one continuity The curve connects the first and last control
points:
Values for the parametric first derivatives of aBzier curve can be calculated as:
np
p
)1(
)0( 0
P
P
nn nn
nn
ppP
ppP
1
10
)1('
)0('
-
8/2/2019 7 a Curves and Surfaces
15/19
Bzier Curves
Therefore, it is easy to connect two curveswith 0 and 1 continuity:
-
8/2/2019 7 a Curves and Surfaces
16/19
Parametric Surfaces
Parametric Bzier surfaces:
Note that two (u, v) parameters areneeded.
n
k
nkmjkj
m
j
uBEZvBEZpvuP0
,,,
0
,
-
8/2/2019 7 a Curves and Surfaces
17/19
Parametric Surfaces
Examples:
-
8/2/2019 7 a Curves and Surfaces
18/19
Parametric Surfaces
Surfaces can also be connected with 0-and 1-continuity:
-
8/2/2019 7 a Curves and Surfaces
19/19
Other types of curves (andsurfaces
Hermite Interpolation
Cardinal Splines
Kochanek-Bartels Splines B-splines
Non-uniform rational b-spline curves
(NURBS)