Continuity at Join Points
(from Lecture 2)
• Discontinuous: physical separation
• Parametric Continuity
• Positional (C0 ): no physical separation
• C1 : C0 and matching first derivatives
• C2 : C1 and matching second
derivatives
• Geometric Continuity
• Positional (G0 ) = C0
• Tangential (G1) : G0 and tangents are
proportional, point in same direction,
but magnitudes may differ
• Curvature (G2) : G1 and tangent lengths
are the same and rate of length change
is the same
source: Mortenson, Angel (Ch 9), Wiki
Continuity at Join Points
• Hermite curves provide C1 continuity at curve
segment join points.
– matching parametric 1st derivatives
• Bezier curves provide C0 continuity at curve
segment join points.
– Can provide G1 continuity given collinearity of some
control points (see next slide)
• Cubic B-splines can provide C2 continuity at
curve segment join points.
– matching parametric 2nd derivatives
Composite Bezier Curves
Joining adjacent curve segments is
an alternative to degree elevation.
Collinearity of cubic Bezier control
points produces G1 continuity at join
point:
Evaluate at u=0 and u=1 to show tangents related to first and last control polygon line segment.
pu (0) 3(p1 p0 ) pu (1) 3(p3 p 2 )
For G2 continuity at join point in cubic case, 5 vertices must be coplanar.
(this needs further explanation – see later slide)
Composite Bezier Surface
• Bezier surface patches can
provide G1 continuity at patch
boundary curves.
• For common boundary curve
defined by control points p14,
p24, p34, p44, need collinearity
of: {p , p , p }, i [1 : 4]
i ,3 i,4 i ,5
• Two adjacent patches are Cr
across their common boundary
iff all rows of control net
vertices are interpretable as
polygons of Cr piecewise
Bezier curves.
•Cubic B-splines can provide C2 continuity at surface patch boundary curves.
source: Mortenson, Farin
Continuity within a
(Single) Curve Segment
• Parametric Ck Continuity:
– Refers to the parametric curve representation and parametric
derivatives
– Smoothness of motion along the parametric curve
– “A curve P(t) has kth-order parametric continuity everywhere in the
t-interval [a,b] if all derivatives of the curve, up to the kth, exist and
are continuous at all points inside [a,b].”
– A curve with continuous parametric velocity and acceleration has
2nd-order parametric continuity.
b
Note that Ck continuity implies Ci
x( ) Ke cos y ( ) Ke sinb
continuity for i < k.
apply product rule
x' ( ) ( Ke b )( sin ) (cos )(Ke b )(beb )
y ' ( ) ( Ke b )(cos ) (sin )(Ke b )(beb )
Example
1stderivatives of parametric expression are
continuous, so spiral has 1st-order (C1) parametric
continuity.
source: Hill, Ch 10
Continuity within a
(Single) Curve Segment (continued)
• Geometric Gk Continuity in interval [a,b] (assume P is curve):
– “Geometric continuity requires that various derivative vectors have
a continuous direction even though they might have discontinuity in
speed.”
– G0 = C0
– G1: P’(c-) = k P’(c+) for some constant k for every c in [a,b] .
• Velocity vector may jump in size, but its direction is continuous.
– G2: P’(c-) = k P’(c+) for some constant k and P’’(c-) = m
P’’(c+) for some constants k and m for every c in [a,b] .
• Both 1st and 2nd derivative directions are continuous.
Note that, for these definitions, Gk continuity implies Gi continuity for i < k.
These definitions suffice for that textbook’s treatment, but there is more to the story…
source: Hill, Ch 10
Reparameterization Relationship
• Curve has Gr continuity if an arc-length
reparameterization exists after which it has Cr
continuity. source: Farin, Ch 10
• “Two curve segments are Gk geometric
continuous at the joining point if and only if there
exist two parameterizations, one for each curve
segment, such that all ith derivatives, i k ,
computed with these new parameterizations
agree at the joining point.” source: cs.mtu.edu
Additional Perspective
• “Parametric continuity of order n implies
geometric continuity of order n, but not
vice-versa.”
Continuity at Join Point
Parametric Continuity Geometric Continuity
• Defined using parametric • Defined using intrinsic differential
differential properties of properties of curve or surface (e.g.
curve or surface unit tangent vector, curvature),
• Ck more restrictive than Gk independent of parameterization.
• G1: common tangent line
• G2: same curvature, requiring
conditions from Hill (Ch 10) & (see
differential geometry slides)
– Osculating planes coincide or
– Binormals are collinear.
source: Mortenson Ch 3, p. 100-102
Parametric Cross-Plot
For Farin’s discussion of C1 continuity at join point, cross-plot notion is useful.
source: Farin, Ch 6
Composite Cubic Bezier Curves
(continued) source: Farin, Ch 5
Domain
violates
(5.30) for y
component.
curves are
identical in x,y
space
Domain
satisfies
(5.30) for y
component.
Parametric C1 continuity, with
parametric domains considered,
3
b3 b2 3
b4 b3 (5.30)
requires (for x and y components): (b a) (c b )
Composite Bezier Curves
For G2 continuity at join point in cubic case, 5 vertices
p m2 , p m1 , p m q 0 , q1 , q 2
must be coplanar.
(follow-up from prior slide)
Achieving this might require adding control points (degree elevation).
2 p1 p 0 p 2 p1 2 p 2 p1 p3 p 2
0 1
3 p1 p 0
3
3 p3 p 2
3
curvature at endpoints of curve segment
p ui p uu
consistent with: i
i
3
p ui
source: Mortenson, Ch 4, p. 142-143
C2 Continuity at Curve Join Point
• “Full” C2 continuity at join point requires:
– Same radius of curvature*
– Same osculating plane*
– These conditions hold for curves p(u) and r(u) if:
p i ri
p ui riu
p r
uu
i i
uu
* see later slides on topics in differential geometry
source: Mortenson, Ch 12
Piecewise Cubic B-Spline Curve
Smoothness at Joint
familiar situation
looks incorrect
looks incorrect
looks incorrect
familiar situation
curvature discontinuity
source: Mortenson, Ch 5
Control Point Multiplicity Effect on
Uniform Cubic B-Spline Joint
C2 and G2 C2 and G2
control point One control point multiplicity = 2
multiplicities = 1
C0 and G0
One control point multiplicity = 4
C2 and G2 One curve segment degenerates into a
One control point multiplicity = 3 single point. Other curve segment is a
straight line. First derivatives at join
point are equal but vanish. Second
derivatives at join point are equal but
vanish.
1 1 1 1 2
p (u ) (u 2u 1)p 0 (3u 4u )p1 (3u 2u 1)p 2 u p 3
u 2 2 2
2 2 2 2
puu (u) (u 1)p0 (3u 2)p1 (3u 1)p2 up3
Knot Multiplicity Effect on Non-
uniform B-Spline
• If a knot has multiplicity r, then the B-
spline curve of degree n has smoothness
Cn-r at that knot.
source: Farin, Ch 8
A Few Differential Geometry
Topics Related to Continuity
Local Curve Topics
• Principal Vectors
– Tangent
– Normal
– Binormal
• Osculating Plane and Circle
• Frenet Frame
• Curvature
• Torsion
• Revisiting the Definition of Geometric Continuity
source: Ch 12 Mortenson
Intrinsic Definition
(adapted from earlier lecture)
• No reliance on external frame of reference
• Requires 2 equations as functions of arc
length* s: 1
*length measured along the curve
1) Curvature: f ( s)
2) Torsion: g (s )
Torsion (in 3D) measures how much
curve deviates from a plane curve.
• For plane curves, alternatively:
1 d
ds
Treated in more detail in Chapter 12 of Mortenson and Chapter 10 of Farin.
source: Mortenson
Calculating Arc Length
• Approximation: For parametric
interval u1 to u2, subdivide curve
segment into n equal pieces.
li
n
L li where li p i p i 1 p i p i 1
i 1
pp p
2
using
u2
L
u1
pu pu du is more accurate.
source: Mortenson, p. 401
Tangent
pui
unit tangent vector: ti u
pi
source: Mortenson, p. 388
Normal Plane
• Plane through pi perpendicular to ti
q ( x, y , z )
xiu x yiu y ziu z ( xi xiu yi yiu zi ziu ) 0
source: Mortenson, p. 388-389
Principal Normal Vector and Line
Moving slightly
along curve in
neighborhood of pi
causes tangent Use dot product
vector to move in to find projection
direction specified of puu onto p ui
i
by: puu
i
Principal normal
vector is on
intersection of Binormal vector
normal plane with bi t i ni
(osculating) plane
shown in (a). lies in normal
plane.
source: Mortenson, p. 389-391
Osculating Plane
Limiting position
of plane defined
by pi and two
neighboring i
points pj and ph
on the curve as Tangent
these neighboring vector lies in
points osculating
independently i plane.
approach pi . Normal vector lies in osculating plane.
Note: pi, pj and
ph cannot be
collinear.
x xi xiu xiuu
y yi yiu yiuu 0
z zi ziu ziuu
source: Mortenson, p. 392-393
Frenet Frame
Rectifying plane
at pi is the plane
through pi and
perpendicular to
the principal
normal ni:
(q p i ) n i 0
i i
Note changes to Mortenson’s figure 12.5.
source: Mortenson, p. 393-394
Curvature
• Radius of curvature is
i and curvature at
point pi on a curve is:
1 p ui p uu
i
i
i p ui
3
Recall that vector puu
i
lies in the
osculating plane.
Curvature of a planar curve
in x, y plane:
Curvature is intrinsic and does not change
1 d 2 y / dx 2
1 (dy / dx)
with a change of parameterization.
2 3/ 2
source: Mortenson, p. 394-397
Torsion
• Torsion at pi is limit of ratio of
angle between binormal at pi and
binormal at neighboring point ph to
arc-length of curve between ph
and pi, as ph approaches pi along
the curve.
i
p u
i p uu
i p uuu
i p p
u
i
uu
i p uuu
i
uu 2 uu 2
p p
u
i i p p
u
i i
Torsion is intrinsic and does not change
with a change of parameterization.
source: Mortenson, p. 394-397
Reparameterization Relationship
• Curve has Gr continuity if an arc-length
reparameterization exists after which it has
Cr continuity.
• This is equivalent to these 2 conditions:
– Cr-2 continuity of curvature
– Cr-3 continuity of torsion
Local properties torsion and curvature are
intrinsic and uniquely determine a curve.
source: Farin, Ch 10, p.189 & Ch 11, p. 200
Local Surface Topics
• Fundamental Forms
• Tangent Plane
• Principal Curvature
• Osculating Paraboloid
source: Ch 12 Mortenson
Local Properties of a Surface
Fundamental Forms
• Given parametric surface p(u,w)
• Form I: dp dp Edu 2 2 Fdudw Gdw2
E pu pu F pu p w G pw pw
• Form II: dp(u, w) dn(u, w) Ldu 2 2Mdudw Ndw2
pu p w
L p uu n M p uw n N p ww n n u
p pw
• Useful for calculating arc length of a curve on a
surface, surface area, curvature, etc.
Local properties first and second fundamental forms
are intrinsic and uniquely determine a surface.
source: Mortenson, p. 404-405
Local Properties of a Surface
Tangent Plane
p u p(u, w) / u
p w p(u, w) / w
q p pu p w 0
x xi xiu xiw
y yi yiu yiw 0
z zi ziu
ziw
q p(ui,wi) components of parametric tangent
vectors pu(ui,wi) and pw(ui,wi) source: Mortenson, p. 406
Local Properties of a Surface
Principal Curvature
• Derive curvature of all parametric curves C on parametric surface S
passing through point p with same tangent line l at p.
contains l
normal curvature vector kn =
projection of curvature vector k
onto n at p
k n (k n)n
in tangent plane with
normal curvature: n k n parametric direction
dw/du
L(du / dt) 2 2M (du / dt)(dw / dt) N (dw / dt) 2
n
E (du / dt) 2 2 F (du / dt)(dw / dt) G (dw / dt) 2
source: Mortenson, p. 407-410
Local Properties of a Surface
Principal Curvature (continued)
curvature extrema:
Rotating a plane
principal normal
around the normal
curvatures
changes the
curvature n.
typographical
error?
source: Mortenson, p. 407-410
Local Properties of a Surface
Osculating Paraboloid
Second
fundamental form
helps to measure
distance of surface
from tangent
plane.
| d | (q p) n
d f Ldu 2 2Mdudw Ndw2
As q approaches p:
1
2
Osculating Paraboloid
source: Mortenson, p. 412
Local Properties of a Surface
Local Surface Characterization
source: Mortenson, p. 412-413
a) LN M 2 0
Elliptic Point:
locally convex
b) LN M 2 0
Hyperbolic Point:
“saddle point”
c) LN M 2 0
L2 M 2 N 2 0
LM N 0 typographical
error?
Planar Point
(not shown) Parabolic Point:
single line in
tangent plane along
which d =0
