curve modeling dr. s.m. malaek assistant: m. younesi
TRANSCRIPT
Curve Modeling
Dr. S.M. MalaekAssistant: M. Younesi
Parametric Polynomials
Parametric Polynomials For interpolating n point we need a
polynomial of degree n-1.
Parametric PolynomialsPolynomial interpolation has several disadvantages: If the polynomial degree is high, unwanted wiggles
are introduced. If the polynomial degree is law, it gives too little
flexibility. No local operations.
Solution:Polynomial Splines
Spline Curve
Spline Curve In drafting terminology, a spline is a
flexible stripflexible strip used to produce a smooth curve through a designated set of points.
Flexible Strip
Spline Curve Several small weightsweights are distributed
along the length of the strip to hold it in position on the drafting table as the curve is drawn.
Weights
Spline Curve The term spline curvespline curve originally
referred to a curve drawn in this manner.
Spline Curve In modeling, the term spline curvespline curve refers
to a any composite curve formed with polynomial section satisfying specified continuity conditions at the boundary of the pieces.
Control Points A spline curve is specified by given a set of
coordinate positions, called control points.
Control points indicates the general shape of the curve.
Control Points A spline curve is defined,
modified, and manipulated with operations on the control points.
Convex Hull The convex polygon boundary
that enclose a set control points is called the convex hull.
Convex Hull The convex hull is a rubber band stretched
around the positions of the control points so that each control point is either on the perimeter of the hull or inside it.
ConvexRemind: A region is convex if the line segment
joining any two points in the region is also within the region.
Shape
of
the Spline Curve
Shape of the Curve Control points are fitted with piecewise
continuous parametric polynomial function in one of two way:
1.1. Interpolation SplinesInterpolation Splines
2.2. Approximation SplinesApproximation Splines
Interpolation Splines When polynomial sections are fitted so that
the curve passespasses through each control point,
the resulting curve is said to interpolateinterpolate the set of control points.
Interpolation Splines Interpolation curves are commonly used to
digitize drawing or to specify animation paths.
Approximation Splines When polynomial sections are fitted to the
general control point path withoutwithout necessarily passing through any control point, the resulting
curve is said to approximateapproximate the set if control points.
Approximation Splines Approximation curves are uses as design tools
to structure object surfaces.
ParametricContinuity Conditions
Parametric Continuity Conditions To represent a curve as a series of
piecewise parametric curves, these
curves to fit together reasonably …Continuity!
Parametric Continuity Conditions Each section of a spline is described
with a set of parametric coordinate functions of the form:
21),(),(),( uuuuzzuyyuxx
Parametric Continuity Conditions Let C1(u) and C2(u) , be two
parametric Curves.
We set parametric continuity by matching the parametric derivation of adjoining curve sections at their common boundary.
10 u
Parametric Continuity Conditions Zero order parametric (C0): Simply the
curves meet, C1(1) = C2(0) .
First order parametric (C1): The first parametric derivations for two successive curve sections are equal at their joining point, C´1(1)= C´2(0)
Parametric Continuity Conditions Second order parametric (C2): Both the first
and second parametric derivatives of the two curve sections are the same at the intersection, C˝1(1)= C˝2(0)
Parametric Continuity ConditionsSimple Example: The curve consists of two parabolas, domains [-1,0] and [0,1],
respectively
f(u) = ( u, -u2, 0 )
g(v) = ( v, v2, 0 ) f'(u) = ( 1, -2u, 0 ) ,f''(u) = ( 0, -2, 0 )
g'(u) = ( 1, 2v, 0 ) ,g''(v) = ( 0, 2, 0 ) Two curves are C1 continuous at the origin. f''(0) = ( 0, -2, 0 ) is not equal to g''(v) = ( 0, 2, 0 ), they are not
C2 continuous at the origin.
Curvature of f(u) = 2/(1 + 4u2)1.5 Curvature of g(v) = 2/(1 + 4v2)1.5
Two curve segments may be C1 continuous and even curvature continuous, but not C2
continuous
Problems withParametric Continuity Conditions
Representation
Problems with Parametric ContinuityTwo line segments:
f'(u) = B - A
g'(v) = C - B Thus, f'(u) = B - A is, in general, not equal to g'(v) = C - B and consequently
these two line segments are not C1 continuous at the joining point B!
Is it strange?Is it strange?Re-parameterizing the curve segments may overcome the problem
F(u) = A + u(B - A)/ | B - A |
G(v) = B + v(C - B)/ | C - B |
f(u) = A + u(B - A)
g(v) = B + v(C - B)
u is in the range of 0 and | B - A | and v is in the range of 0 and | C – B|
Problems with Parametric ContinuitySemi circle:
At (0,1,0) = f(1) = g(0). ) C0)f'(u) = ( PI u sin(u2 PI/2), PI u cos(u2 PI/2), 0 )
f''(u) = ( PI2 u2 cos(u2 PI/2), -PI2 u2 sin(u2 PI/2), 0 )
f'(u) × f''(u) = ( 0, 0, -PI3 u3 )
| f'(u) | = PI u
| f'(u) × f''(u) | = PI3 u3
k(u) = 1
g'(v) = ( PI v cos(v2 PI/2), -PI v sin(v2 PI/2), 0 )
g''(v) = ( -PI2 v2 cos(v2 PI/2), -PI2 v2 cos(v2 PI/2), 0 )
g'(v) × g''(v) = ( 0, 0, -PI3 u3 )
| g'(v) | = PI v
| g'(v) × g''(v) | = PI3 v3
k(v) = 1
f(u) = ( -cos(u2 PI/2), sin(u2 PI/2), 0 )
g(v) = ( sin(v2 PI/2), cos(v2 PI/2), 0 )
Problems with Parametric ContinuitySemi circle:
Let u2 = p in f(u) and let v2 = q in g(v). The new equations are:
f(p) = ( -cos(p PI/2), sin(p PI/2), 0 )
g(q) = ( sin(q PI/2), cos(q PI/2), 0)
Their derivatives are
f'(p) = ( (PI/2) sin(p PI/2), (PI/2) cos(p PI/2), 0 )
f''(p) = ( (PI/2)2 cos(p PI/2), -(PI/2)2 sin(p PI/2), 0 )
g'(q) = ( (PI/2) cos(q PI/2), -(PI/2) sin(q PI/2), 0 )
g''(q) = ( -(PI/2)2 sin(q PI/2), -(PI/2)2 cos(q PI/2), 0 )
Therefore, after changing variables, both f'(1) and g'(0) are equal to ( PI/2, 0, 0 ) and hence they are C1 continuous. Moreover, both f''(1) and g''(0) are equal to ( 0, -(PI/2)2, 0 ) and hence they are C2
continuous! They are also curvature continuous because they have curvature 1 everywhere (remember they are circles).
GeometricContinuity Conditions
Geometric Continuity Conditions Two curve segments are said Gk
geometric continuous at the joining point if and only if there exists two parameterizations, one for each curve segment, such that all i-th derivatives, i less than or equal to k, computed with these new parameterizations agree at the joining point.
Geometric Continuity Conditions Two C0 curve segments are said G1 geometric
continuous at the joining point if and only if vectors f'(u) and g'(v) are in the same direction at the joining point. Note that f'(u) and g'(v) are evaluated at the joining point.
The tangent vectors point to the opposite directions, and, as a result, they are not G1 at the joining point
Geometric Continuity Conditions
Two C1 curve segments are said G2 geometric continuous at the joining point if and only if vector f''(u) - g''(v) is parallel to the tangent vector at the joining point. Note that f''(u) and g''(v) are evaluated at the joining point.
Geometric Continuity Conditions Example:two parabola segments with joining point at ( 0, 1, 0): f(u) = ( -1 + u2, 2u - u2, 0 )
g(v) = ( 2u - u2, 1 - u2, 0 )
f'(u) = ( 2u, 2 - 2u, 0 ) f''(u) = ( 2, -2, 0 ) f'(u) × f''(u) = ( 0, 0, -4 ) | f'(u) | = 2SQRT(1 - 2u + 2u2 ) | f'(u) × f''(u) | = 4 k(u) = 1/(2(1 - 2u + 2u2)1.5)
g'(v) = ( 2 - 2v, -2v, 0 ) g''(v) = ( -2, -2, 0 ) g'(v) × g''(v) = ( 0, 0, -4 ) | g'(v) | = 2SQRT(1 - 2v + 2v2 ) | g'(v) × g''(v) | = 4 k(v) = 1/(2(1 - 2v + 2v2)1.5)
Geometric Continuity Conditions From f'(1) = g'(0) = ( 2, 0, 0 ), we know that these two curve
segments are C1 at the joining point. Since f''(1) = ( 2, -2, 0 ) is not equal to g''(0) = ( -2, -2, 0 ), they are not C2.
Since the curves are C1, checking for G2 is meaningful. Since f''(1) - g''(0) = ( 4, 0, 0 ) is parallel to the tangent vector ( 2, 0, 0 ) at the joining point, these two curve segments are G2 continuous at the joining point ( 0, 1, 0 ).
Geometric Continuity Conditions
Geometric Continuity Conditions: Only require parametric derivatives of the two sections to be proportional to each other at their common boundary instead of equal to each other.
Geometric Continuity Conditions Zero order geometric (G0): Same as C0
C1(1) = C2(0) First order geometric (G1): The parametric first
derivatives are proportional at the intersection of two successive sections.
C´1(1)= αC´2(0) Second order geometric (G2): Both the first and
second parametric derivatives of the two curve sections are proportional at their boundary,
C˝1(1)= αC˝2(0)
Parametric & Geometric Continuity A curve generated with geometric
continuity condition is similar to one generated with parametric continuity , but with slight differences in curve shape.
With geometric continuity, the curve is pulled toward the section with the greater tangent vector.
Spline Interpolation
LinearLinearSpline
Interpolation
Linear Spline Interpolation Given a set of control points, linear
interpolation spline is obtained by fitting the input points with a piecewise linear polynomial curve that passes through every control points.
Linear Spline Interpolation A linear polynomial:
)10(
)(
,)(
)(
u
buauz
buauy
buaux
zz
yy
xx
10,)( uuu bap
Linear Spline Interpolation We need to determine the values of a and b in the
linear polynomial for each of the n curve section.
Substituting endpoint values 0 and 1 for parameter u:
b
au
uu
1
)( bap
b
a
p
p
11
10
1
0
Linear Spline Interpolation Solving this equation for the polynomial
coefficients:
P(u) can be written in terms of the boundary conditions:
1
0
1
0
1
10
1111
10
p
pp
p
b
a
1
0
10
111)(
p
puup
b
auu 1)(p
Mspline Mgeometry
U
Linear Spline Interpolation The matrix representation for linear spline
representation:
Mgeometry is the matrix containing the geometry constraint values (boundary condition) on the spline.
Mspline is the matrix that transform the geometric constraint values to the polynomial coefficients and provides a characterization for the spline curve (Basis Matrix).
geometrysplineu MMUP )(
Linear Spline Interpolation We can expand the matrix representation:
1
0
10),()(k
kkuubu PP
1
0
1
0
1
10
111)(
p
puu
p
puup
Blending Functions
1100
10
)()(
)()1()(
PP
PPP
ubub
uuu
Linear Spline Interpolation
The Linear spline blending functionThe Linear spline blending function::
1
0
10),()(k
kkuubu PP
1100
10
)()(
)()1()(
PP
PPP
ubub
uuu
Linear Spline Interpolation The curve is linear combination of the blending
functions. The piecewise linear interpolation is C0
continuous. For a linear polynomial, 2 constraints are
required for each segment.
A polynomial of degreeA polynomial of degree kk has has k+1 k+1 coefficientscoefficients and thus requiresand thus requires k+1k+1 independent independent constraintsconstraints to uniquely to uniquely determine it.determine it.
CubicCubicSpline
Interpolation
Cubic Spline InterpolationCubic splines are used to: Set up paths for object motion A representation for an existing object
or drawing Design object shape
Cubic splines are more flexible for modeling arbitrary curve shapes.
Cubic Spline Interpolation Given a set of control points:
Cubic interpolation spline is obtained by fitting the input points with a piecewise cubic polynomial curve that passes through every control points.
nkzyxP kkkk ,,2,1,0),,,(
Cubic Spline Interpolation A cubic polynomial:
)10(
)(
,)(
)(
23
23
23
u
ducubuauz
ducubuauy
ducubuaux
zzzz
yyyy
xxxx
10,)( 23 uuuuu dcbap
Linear Spline Interpolation
We need to determine the values of a , b, c and d in the linear polynomial for each of the n curve section.
By setting enough boundary conditions at the “joints” between curve sections we can obtain numerical values for all the coefficients.
10,)( 23 uuuuu dcbap
Cubic Spline InterpolationMethods for setting the boundary conditions
for cubic interpolation splines:
1. Natural Cubic Splines
2. Hermit Splines
3. Cardinal Splines
4. Kochanek-Bartels Splines
Natural
Cubic Splines
Natural Cubic Splines Natural cubic splines are developed
for graphics application.
Natural cubic splines have C2
continuity.
Natural Cubic SplinesIf we have n+1 control points to fit, then: We have n curve sections 4n polynomial coefficients to be
determined.3 control points (n=2)
2 curve sections
4×2 polynomial coefficients to be determined.
Natural Cubic SplinesAt each n-1 interior control points, we have 4
boundary condition: The two curve sections on either side of a control
point must first and second parametric derivatives at that control point, and each curve must pass through that control point.
At P1 interior control:
1. C´1(1)=C'2(0)
2. C˝1(1)=C˝2(0)
3. C1(1)=P1
4. C2(0)=P1
Natural Cubic SplinesAdditional equation: First control point P0
End control points Pn
5. C1(0)=P0
6. C2(1)=P2
Natural Cubic SplinesWe still need twotwo more conditions to
be able to determine values for all coefficients.
Natural Cubic SplinesMethod 1:Set the second derivatives at p0 and pn to 0
7. C˝1(0)=0
8. C˝2(1)=0
Natural Cubic SplinesMethod 2:Add a control point P-1 and a control point Pn+1.
7. P-1
8. P3
PP-1-1 PP33
Natural Cubic Splines Natural cubic splines are a mathematical
model for the drafting spline.
Disadvantage: NoNo “Local Control”: we cannot
reconstruct part of the curve without specifying an entirely new set of control points.
Hermit Splines
Hermit SplinesGiven a set of control points:
Hermit splines is an interpolating piecewise cubic polynomial with a
specified tangentspecified tangent at each control point.
nkzyxP kkkk ,,2,1,0),,,(
Hermit Splines Hermit splines can be adjusted
locally because each curve section is only dependent on its endpoint constraints (unlike the natural cubic splines).
Hermit Splines Example: Change in MagnitudeMagnitude of T0
Hermit Splines Example: Change in DirectionDirection of T0
Hermit Splines
10,)( 23 uuuuu dcbap
1
1
)1(
)0(
)1(
)0(
k
k
k
k
Dpp
Dpp
pp
pp
P(u) represents a parametric cubic point function for the section between pk and pk+1.
The boundary conditionsboundary conditions are:
Dpk and Dpk+1 are the values for the parametric derivations (slope of the curve) at control points Pk and Pk+1
Hermit Splines10,)( 23 uuuuu dcbap
d
c
b
a
p 1)( 23 uuuu
d
c
b
a
p 0123)( 2 uuu
The derivative of the point function:
d
c
b
a
Dp
Dp
p
p
0123
0100
1111
1000
1
1
k
k
k
k
We can write the constraints
in a matrix form:
1
1
)1(
)0(
)1(
)0(
k
k
k
k
Dpp
Dpp
pp
pp
Boundary conditions
Hermit Splines
Solving this equation for the polynomial coefficients:
d
c
b
a
Dp
Dp
p
p
0123
0100
1111
1000
1
1
k
k
k
k
1
1
1
1
1
1
1
0001
0100
1233
11220123
0100
1111
1000
k
k
k
k
H
k
k
k
k
k
k
k
k
Dp
Dp
p
p
M
Dp
Dp
p
pDp
Dp
p
p
d
c
b
a
MH, the Hermit matrix, is the inverse
of the boundary constraint matrix.
d
c
b
a
p 1)( 23 uuuu
1
123232323 232132)(
k
k
k
k
uuuuuuuuuu
Dp
Dp
p
p
p
1
123 1)(
k
k
k
k
Huuuu
Dp
Dp
p
p
Mp
Blending Functions
)()()()()(312110
uHuHuHuHukkkk
DpDpppP
d
c
b
a
Dp
Dp
p
p
0123
0100
1111
1000
1
1
k
k
k
k
Hermit SplinesHermit Blending Functions
Hermit Splines Hermit polynomials can be useful for some
digitizing applications where it may not be too difficult to specify or approximate the curve slopes.
For most problems in modeling, it is more useful to generate spline curve without requiring input values for
curve slopes.
But…
Cardinal
Splines
Cardinal Splines Cardinal splines are interpolating piecewise
cubics with specified endpoint tangents at the boundary of each curve section.
The difference is that we do not have to give the values for the endpoint tangents.
For a cardinal spline, the value for the slope at a control point is calculated
from the coordinates of the two adjacent control points.
Cardinal Splines A cardinal spline section is completely
specified with four consecutive control points.
The middle two control points are the section endpoints, and the other two points are used in the calculation of the endpoint slopes.
Cardinal Splines The boundary conditions:
))(1(2
1)1(
))(1(2
1)0(
)1(
)0(
2
11
1
kk
kk
k
t
t
ppP
ppP
pP
pP k
Thus, the slopes at control points pk and pk+1 are taken to be proportional to the chords and11 kk pp 2kkpp
Cardinal Splines The boundary conditions:
))(1(2
1)1(
))(1(2
1)0(
)1(
)0(
2
11
1
kk
kk
k
t
t
ppP
ppP
pP
pP k
Cardinal Splines
Parameter t is called the tension parameter.
Tension parameter controls how loosely or tightly the cardinal spline fits the input control points.
))(1(2
1)1(
))(1(2
1)0(
)1(
)0(
2
11
1
kk
kk
k
t
t
ppP
ppP
pP
pP k
Cardinal Splines
When t=0 is called the Catmull-Rom splines, or Overhauser splines.
Cardinal Splines
))(1(2
1)1(
))(1(2
1)0(
)1(
)0(
2
11
1
kk
kk
k
t
t
ppP
ppP
pP
pP k
2
1
1
23 1)(
k
k
k
k
Cuuuu
p
p
p
p
MP
0010
00
2332
22
ss
ssss
ssss
CM
2)1( ts
Cardinal Splines
))(1(2
1)1(
))(1(2
1)0(
)1(
)0(
2
11
1
kk
kk
k
t
t
ppP
ppP
pP
pP k
)()()()(
)(])23()2[(
]1)3()2[()2()(
3221101
232
231
23231
uCARuCARuCARuCAR
sususuusus
usussususuu
kkkk
kk
kk
pppp
pp
ppP
The polynomials CARk(u) for k=0,1,2,3 are
the cardinal functions.
Cardinal SplinesCardinal functions