Curves

1
Curves

• Path - trajectory - motion
• Parameterization
• Designing polyloops
• Velocity, tangent, curvature, jerk
• Smoothing polyloops
• B-spline, Bezier and 4-points subdivisions
• The Jarek subdivision schemes for polyloops
• Arc-length parameterization
• Tangent and curvature estimation
• Curvature-driven adaptive parameterization

Updated January 25, 2014
2
How to specify a curve?

• Implicit (equation) x2y2r2
• Not convenient for walking on the curve
• Parametric (function) P(s) ( r cos(s) , r
  sin(s) )
• Polyloop cycles of edges joining consecutive
  points (vertices)
• Not smooth,unless you use lots of vertices
• Subdivision result to which a subdivision
  process converges
• Procedural snowflake, fractal, subdivision

V2
V3
V1
V5
V4
V0
3
It is all about curves
Points, vectors
curves
Shapes
Motions
Animated shapes
4
Outline

• We will talk about polygonal loops (polyloops) in
  the plane
• We will discuss algorithms to transform them into
  smooth curves
• We want to support an intuitive design paradigm
• A designer adjust the vertices of the control
  polygon (polyloop)
• The corresponding smooth curve is regenerated
  automatically
• The effect of an adjustment should be local and
  predictible
• We will look at several subdivision schemes for
  converting a control polyloop into a smooth curve
• We will study their properties and look at
  several applications
• Smooth rendering, culling and clipping
• Computing intersections, areas
• We will extend these results to
• 3D curves
• Surfaces
• Animations

5
Geometry review

• The constructions and algorithms that follow are
  formulated in terms of simple geometric
  constructions that involve points, vectors, and
  operators.
• We will first
• provide an overview of these fundamental concepts
• propose a few convenient operations
• define our notation

6
Points

• A point is a location in space (zero dimensional
  entity)
• Initially space will the plane (later we will go
  3D)
• How to represent a point P?
• Assume a known origin O and two orthogonal
  directions I and J (unit basis vectors)
• Represent P by its two coordinates P(x,y) where
  POxIyJ
• To get to P
• start at O,
• walk x units along the direction I,
• then y units along the direction J
• REMEMBER THIS EXPLANATION!
• It is the key to understanding
• Coordinate systems
• Graphic transformations
• Animations

7
Vectors

• A vector is either a direction or a displacement
  between two points (locations).
• We will also use vectors to represent forces and
  velocities.
• A vector is often represented by its coordinates
  V(x,y)
• Please do not confuse points and vectors!
• A direction is a unit vector. It may represent
• the direction of a ray from the eye through a
  pixel
• the direction towards the light from a point on a
  surface
• the tangent to a curve at a specific point
• a basis vector for expressing the coordinates of
  points
• Displacement vectors may be computed as point
  subtractions
• Let V be the displacement vector from A(xA,yA)
  to B(xB,yB)
• V may be denoted BA, or for short AB
• Its coordinates may be computed as V (xBxA,
  yByA)
• Hence, given O, a point P corresponds to the
  vector OP
• Assuming that the coordinates of the origin O are
  (0,0)

8
Operations on vectors

• Consider two vectors V(x,y) and V(x,y)
• Scaling vectors sV (sx,sy)
• Reversing a vector V(x,y)
• Adding vectors VV (xx,yy)
• Subtracting vectors VV (xx,yy)
• What do the coordinate mean?
• Consider a basis of two orthogonal unit vectors I
  and J
• V(x,y) means that VxIyJ
• Walk x units along I and y units along J

9
Advanced vector operators and notation

• Let V(x,y) and U(a,b)
• Length of V is computed by V.normsqrt(x2y2)
• Unit length vector obtained by normalizing V
  V.unit(1/V.norm)V,
• Orthogonal vector (rotating V ccw by 90 degrees)
  V.left(y,x)
• Cascading . notation (left-to-right)
  V.left.norm(V.left).norm
• Dot product U?Vaxby (a scalar!)
• When V.normU.norm1, then U?V is the projection
  of U onto V
• Proof write UcVdV.left(cxdy,dxcy), solving
  for c with x2y21 yields caxby
• Otherwise U?V is the projection of U onto V
  scaled by the length of V
• U?V U.normV.normcos(angle(U,V))
• Properties of dot product
• U?V V?U
• (U)?V (V?U)
• U?V 0 implies that they are orthogonal or one
  has zero length
• Tangential and orthogonal decomposition U with
  respect to a vector V
• along V U.along(V)(U?V.unit)V.unit
• away from V U.away(V)UU.along(V)

10
Walking along an edge

• Consider an edge from point A to point B denoted
  Edge(A,B)
• You can use a parameter, s, in 0,1 to identify
  any point P(s) on it P(s)AsAB
• This is the proper notation start at A and move
  by s along vector AB
• For convenience, we often use another equivalent
  notation
• P As(BA)
• 
  (1s)AsB
• Note that P(0)A and P(1)B

1s
B
s
P(0.6)
A
11
Closest point on an edge

• Consider Edge(A,B) and a point P in the plane
• Compute the point C of Edge(A,B) that is the
  closest to P
• First compute the parameter s of the orthogonal
  projection H of P on Edge(A,B)
• Then test it against 0,1 and decide
• Procedure project (A,B,P)
• sAP?AB/AB?AB assuming A?B
• IF 0ltslt1 THEN RETURN AsAB
• ELSE IF slt0
• THEN RETURN A
• ELSE RETURN B
• To compute closest point to P on polyloop
• Compute it for all edges and take closest

12
Arclength of an edge

• Write the procedure L(A,B) that returns the
  arclength of edge (A,B)

13
Total arclength of a polyloop

• Write the procedure L(C) that returns the total
  arclength of polyloop with vertices Ci, for i
  0, 1, n

14
Terminology (for lack of a better one)

• A path is a parameterized curve (2D or 3D)
• A point (of an object) may follow a curve
• The path indicates where we are along the curve
  at each moment of the animation
• The orientation of the object is not defined
  (could be pure translation?)
• A motion defines the evolution of a pose
• Position and orientation of the local CS or
  object(s) defined in it
• A trajectory is a motion defined by a path
• The orientation is defined by the tangent to the
  curve
• and bi-tangent in 3D

15
Curve and path

• Given a path P(t), what is the curve traced by
  P(t)?
• Union of all points P(t)
• What if P(t) starts at A, moves a few times back
  and forth along the segment (A,B) then finishes
  at B?
• The path is complicated
• The curve is trivial

16
Motivation

• Assume that you have a polyloop P path defined by
  sampling positions Pi or produced by subdividing
  a control polygon L
• In general, the vertices of P are not uniformly
  spaced
• Their spacing may reflect the animators desire
  to slow down near turns or at specific places
• or it may be the undesired side effect of data
  acquisition or creation
• We may want to re-parameterize P to obtain a path
  P that moves at constant speed if you use
  s(Pi,t,Pi1) on each segment
• Then, you can warp time (as in project P1a) to
  impose your own speed profile, independently on
  the original sampling of P

17
Example where resampling is needed

• Subdivision of a bad control polygon

18
Arclength reparameterization

• Write the procedure for computing a point on the
  polyloop that corresponds to parameter t in 0,1
• Demonstrate it by animating a point at constant
  speed

19
Research challenge!!!

• Given a polyloop P, place an ordered set of k
  points along P so that each point is at the same
  Euclidean (shortest) distance to its two
  neighbors.
• Is is always possible?
• Is there a deterministic algorithm?
• Would an iterative algorithms always converge?

20
Polygon area calculation 2 methods

• Sum of signed areas of triangles, each joining an
  arbitrary origin o to a different edge(a,b)
• SUM (oa?ob) for each edge (a,b)
• u?v is the dot product of v with the the vector
  obtained by rotating u by 90
• Sum of signed areas of trapezoids between each
  oriented edge and its orthogonal projection
  (shadow) on the x-axis

y
by
b
a
ay
x
ax
bx
area(a,b)(ayby)(bxax)/2
21
Area implementation

• float area ()
• float A0
• for (int i0 iltvn i)
  AtrapezeArea(Pi,Pthis.in(i))
• return(A)
• float trapezeArea(pt A, pt B) return((B.x-A.x)(B
  .yA.y)/2.)

22
Centroid of polygonal region

• Weighted sum of centroids of trapezoids
• Weight signed area of trapezoid / total area

23
Centroid of trapezoid

• Any line through centroid cuts shape in 2 parts
  with same moment
• x 0?1 sy(s)ds / 0?1 y(s)ds, with y(s)as(ba)
• x 0?1 (ba)s2 as ds / 0?1 (ba)s a ds
• x(a2b)/(3(ab))
• Similar derivation
• y (a2abb2)/(3(ab))

y
b
a
(x,y)
x
0
1
24
Implementation of centroid

• pt barycenter () G.setTo(0,0)
• for (int i0 iltvn i)
• G.addScaledPt(
• trapezeArea(Pi,Pthis.in(i)),trapezeCente
  r(Pi,Pthis.in(i)))
• G.scaleBy(1./this.area())
• return(G)
• pt trapezeCenter(pt A, pt B)
• return(new pt(
• A.x(B.x-A.x)(A.y2B.y)/(A.yB.y)/3.,
  (A.yA.yA.yB.yB.yB.y)/(A.yB.y)/3.)
• )

25
Principal directions of polyloop

• For a set of points sampled along the curve
• Not the same as principal axes (second order
  inertia moments)

26
Implementation of principal direction

• void showAxis() pt Gthis.barycenter()
  stroke(black) G.show(10)
• float xx0, xy0, yy0, px0, py0, mx0,
  my0
• for (int i0 iltvn i) xx(Pi.x-G.x)(P
  i.x-G.x) xy(Pi.x-G.x)(Pi.y-G.y)
  yy(Pi.y-G.y)(Pi.y-G.y)
• float a atan(2xy/(xx-yy))/2.
• vec V new vec(50,0) V.rotateBy(a) vec U
  V.makeTurnedLeft()
• for (int i0 iltvn i) vec
  WG.makeVecTo(Pi) float vddot(W,V) float
  uddot(W,U) if(vdgt0) pxvd else mx-vd
  if(udgt0) pyud else my-ud
• if(mxgtmax(px,py,my)) V.rotateBy(PI)
  if(pygtmax(px,mx,my)) V.rotateBy(PI/2.)
  if(mygtmax(px,mx,py)) V.rotateBy(-PI/2.)
• strokeWeight(1) stroke(black)
  V.showArrowAt(G)
• stroke(black,60) V.turnLeft() V.showAt(G)
  V.turnLeft() V.showAt(G) V.turnLeft()
  V.showAt(G)

27
Questions

• Describe two techniques for testing whether a
  point q lies inside a polygon P
• Describe two techniques for computing the area of
  a polygon P
• Assume that you have a function fill_tri(a,b,c)
  that will visit all pixels whose center falls
  inside the triangle with vertices (a,b,c). Assume
  that each time the function visits a pixel, it
  toggles its status. Initially, all the status of
  each pixel is OFF. Explain how you could turn to
  ON the status of all pixels whose center falls
  inside a polygon P.

28
Motivation

• Smooth curves and surfaces are used for
  aesthetic, manufacturing, and analysis
  applications where discontinuities due to
  piecewise linear approximations would create
  misleading artifacts.
• Designers like to specify such curves by
  adjusting a control network with only a few
  vertices
• How to go from a crude 2D control polyloop
  (black) to a smooth curve (red)?
• Suggestions?

29
Different requirements
interpolating
smooth
30
Operators on polyloops
Simplification Subsampling
Smoothing Denoising
Subdivision Refinement
Adding noise
31
Motivation for curve smoothing

• Smooth curves and surfaces are used for
  aesthetic, manufacturing, and analysis
  applications where discontinuities due to
  triangulated approximations would create
  misleading artifacts.
• Smooth curves may be used to design trajectories
  and paths for moving objects and cameras in
  animations.
• Designers like to specify such curves by
  adjusting a crude control polygon with only a few
  vertices
• How to go from a crude control polygon (black)
• to a smooth curve (red)?
• Suggestions?

32
Local curve analysis

• We first investigate local differential
  properties of a polyloop L
• to measure forces, compare schemes, adjust
  speeds
• but these are trivial for a polyloop
• straight line segments (zero curvature)
• not differentiable at vertices (infinite
  curvature)
• So we really want the properties of the smooth
  curve or path J defined by L
• tangent, normal, acceleration, curvature, jerk
• but, we usually do not know J
• it may be defined as the limit of a smoothing
  process
• Therefore, we use discrete estimators for these
  properties

33
Velocity at a vertex V(B)(ABBC)/2

• Let A,B,C,D,E be consecutive vertices in L
• AB, BC, are the edges of L
• J is the unknown smooth curve passing through the
  vertices
• When traveling along J from A to B, it takes 1
  sec.
• Hence the average velocity vector for that
  segment is AB.
• Hence the velocity at B is estimated as
  V(B)(ABBC)/2

34
Implementing V(B)(ABBC)/2

• class vec float x,y
• vec average(vec U, vec V) return(new
  vec((U.xV.x)/2,(U.yV.y)/2))
• void unit() float nsqrt(sq(x)sq(y)) if
  (ngt0.000001) x/n y/n
• class pt float x,y
• vec vecTo(pt P) return(new vec(P.x-x,P.y-y))
  
• Write a method for the pt class that computes the
  velocity at point B
• vec velocity (pt A, pt C)
• tangent T(B) V(B).unit()

35
Normal at a vertex N(B)AC.left().unit()

• The normal N(B) at B is the unit vector
  orthogonal to AC.
• We pick the one pointing left N(B)AC.left().unit
  ()
• class vec float x,y
• void left() float wx x-y yw
• void unit() float nsqrt(sq(x)sq(y)) if
  (ngt0.000001) x/n y/n
• Write a method for the pt class that computes the
  normal at point B
• vec normal (pt A, pt C)

36
Acceleration at a vertex g(B)BCAB

• The acceleration at B is the velocity change
  g(B)BCAB
• It is useful for computing forces during
  animation (dynamics)
• Write a method for the pt class that computes the
  acceleration at point B
• vec acceleration (pt A, pt C)

37
Radius of curvature r(B)V2/(2AB?N(B))

• The radius r(B) of curvature (sharpness of turn
  in the path) at B could be estimated as the
  radius of the circle through A, B, and C, but
  this approach yields unexpected results when the
  angle at b is acute.
• I prefer to use the parabolic curvature,
  r(B)V2/(2h), where V is the magnitude of V(B)
  and where h is the distance from B to the
  Line(A,C).
• hAB?N(B) is the normal component of g(B)
• It measures the centrifugal force
• Curvature 1/r(B)
• Write a method for the pt class that computes the
  r(B) at point B
• vec radiusOfCurvature (pt A, pt C)

38
Jerk j(B,C)AD3CB

• It is the change of acceleration between B and C.
  It measures the change in the force felt by a
  person traveling along the curve
• j(B,C) g(C)g(B)
• (CDBC)(BCAB) CDCBCBAB
• ABBCCD CBCBCB AD3CB
• To show the second degree discontinuities in the
  curve,
• draw the normal component of j(B,C) at (BC)/2

39
Smoothing a polyloop

• Given a polyloop L, we wish to move its vertices
  to produce a curve J that resembles L, but is
  smoother (less jerk?)
• Smoothing does not change the number of vertices,
  
• it only moves them

40
Tuck smoothing

• tuck(s) moves the vertices by executing the
  following two steps
• For each vertex B, compute M(B)(BABC)/2
• A (r. C) is the previous (r. next) vertex in the
  cyclic order
• Moving B to BM(B) brings it to the average of
  its neighbors
• For each vertex B, move B to BsM(B)
• Repeating tuck(s) tends to eliminate sharp
  features, but shrinks the curve

41
Avoid oscillations

• Keep s in 0,2/3 to avoid oscillations

tuck(1)
tuck(1)
42
Avoid shrinking Tuck-tuck smoothing

• Alternate tuck(s), which shrinks, and tuck(s),
  which expands
• untuck cannot recover the sharp features lost
  during the tuck

43
Tuck(1/2) dual of dual

• Tuck(.5) is the dual of the dual
• refine introduces a new vertex in the middle of
  each edge
• kill deletes the old vertices
• dual (refine, kill)
• tuck(.5) (dual, dual)

refine
kill
44
Cubic fit tuck-tuck (A4B4DE)/6

• Compute a cubic polynomial curve,
  C(t)at3bt2ctd, through vertices A, B, D, E
  satisfying C(2)A, C(2)E, C(1)B, and C(1)D
• We want to insert a point CC(0)d between B and
  E. Solve for d
• C(2)8a4b2cdA and C(2)8a4b2cdE yields
  AE8b2d
• C(1)abcdB, and C(1)abcdD yields
  AE8b2d
• AE8b2d and AE8b2d yields 6d4(BD)(AE)
• d (A4B4DE)/6 MMM/3, with M(BD)/2,
  M(AE)/2)
• To insert such a cubic fit for each edge
  tuck(2/?6), tuck(2/?6)
• Note that 2/?60.82, which may be unstable
• Move towards C

45
Smoothing vs subdivision

• Smoothing should be performed when the desired
  curve has been densely sampled but the resulting
  polyloop has undesired sharp features
  (acquisition noise, too many details, sharp
  changes in direction or curvature)
• Subdivision refines the polyloop (adds more
  vertices). It should be performed when the
  polyloop is a crude outline of the desired curve.

subdivision
smoothing
46
Polyloop refine/smooth operators

• We will use the following operators
• r (refine) introduces a new vertex in the middle
  of each edge
• k (kill) deletes the old vertices (those present
  before r)
• d (dual) performs (r, k)
• t (tuck) performs tuck(1/2), which is (d, d)
• us (parameterized untuck) performs tuck(s)
• u (untuck) performs tuck(1/2), same as u.5
• Write a polyloops smoothing scheme using these

47
Quadratic uniform B-spline (r,d)

• Repeating (r,d) converges to a piecewise
  quadratic B-spline curve that interpolates the
  mid-edge points
• r rd rdrd rdrdrdrd

48
Cubic uniform B-spline (r,t)

• Repeating (r,t), which is (r,d,d) converges to a
  piecewise cubic B-spline curve that has second
  degree continuity (no jerk)
• rt rtr rtrt rtrtrt

49
Splittweak B-spline subdivision

• Given a control polygon, for example (A,B,C,D),
  repeat the following sequence of two steps, until
  all consecutive 4-tuples of control points are
  nearly co-linear.
• 1. Split insert a new control point in the
  middle of each edge (2,4,6,8)
• 2. Tweak move the old control points
  half-way towards the average of their new
  neighbors (1,3,5,7)
• The polyloop converges rapidly to a smooth curve,
  which happens to be a cubic B-spline curve.

50
Can the limit curve self-intersect?

• Can it self-intersect when P does not?

51
Where do the control vertices converge to?

• The original control vertex B converges to
  B(A4BC)/6
• The tangent at B to the B-spline curve is
  parallel to AC

b
c
b
a
52
Local control for B-spline curves

• Trace the original vertices during subdivision
• Do they move along straight lines?
• They subdivide the final curve into spans
• Each span is influenced by 4 control vertices
• Each control vertex influences 4 spans
• Each span converges to a cubic Bezier curve

53
Cubic B-splinetoBezier conversion

• Introduce 2 new vertices in each edge to split it
  in 3 equal parts
• Move the old vertices to the average of their new
  neighbors
• Make groups of 4 consecutive vertices (each
  sharing its old vertices with consecutive groups)
• Each group is the control polygon (A,B,D,E) of a
  Bezier curve

54
How to evaluate the Bezier control polygon

• Let t be the parameter in 0,1.
• Let C(t) be the point on the Bezier curve
  corresponding to t
• You can compute t as
• C(t)s( s(s(A,t,B),t,s(B,t,C)) , t , s(s(
  B,t,C),t,s(C,t,D) )
• where s(A,t,B) returns AtAB
• Note that this is a cubic polynomial in t

B
D
C(t)
tAB
E
A
55
How to subdivide the Bezier control polygon

• Perform the evaluation of C(0.5) and keep the
  construction points

56
Convex hull

• Each Bezier curve lies in the convex hull of 4
  control vertices
• The convex hull of 4 points in 3D?
• A tetrahedron
• The convex hull of 4 points in 2D?
• Contained in the union of 3 triangles ABD, ABE,
  BDE

57
How can we use the convex hull property?

• Display uncertainty (region guaranteed to contain
  the limit curve)
• Shade triangles (in 2D) or tetrahedra (in 3D)
• Decide when to stop the subdivision
• All triangles or tetrahedra are nearly linear
• Quickly decide when two B-spline curves in 2D are
  not intersecting
• The convex hulls do not overlap
• Useful for accelerating collision detection and
  for computing the exact collision time

58
Open curves

• If you want to subdivide an open polygon
• Trace the original vertices through the
  subdivision process
• They split the curve into spans (one per original
  edge)
• Delete the span (vn1,vn) and the two adjacent
  spans
• v0 should not influence the head (beginning of
  the curve)
• vn1 should not influence the tail (end)

59
B-spline curves are not interpolating

• The cubic B-spline is approximating
• It does not go through the control vertices
• You may prefer the 4-point, which is interpolating

B-spline (approximating)
4-point (interpolating)
60
4-point subdivision

• We want to leave the original vertices of L where
  they are.
• Hence, we displace (bulge-out) each inserted
  mid-edge point C.
• By how much? Fit a cubit CC(t) through A, B, D,
  and E
• C(t)at3bt2ctd, and thus C(0)d
• We use different constraints than the cubic fit
  tuck-tuck!
• BC(1) abcd
• DC( 1) abcd
• AC(3) 27a9b3cd
• EC( 3) 27a9b3cd
• CC(0)dMMM/8

M(BD)/2bd
M9M8d
M(AE)/29bd
8MMM8d
subdivision bulge MM/8
61
Implementing 4-point subdivision

• Design the construction of CMMM/8 using
  s(A,t,B)
• Avoid computing M
• instead extrapolate AB and ED and take the
  mid-point
• We will use this construction for smoothing
  key-frames motions

Compare with your neighbor
s( s(A,9/8,B) ,.5, s(E,9/8,D) )
62
4-point (r,t,u1)

• r rt rtu1 rtu1rt
  rtu1rtu1

Refine (r) puts a mid-edge point C at
M. Surprise! (t,u1) produces the cubic-fit bulge
MM/8 Verify by inspection
63
Compare cubic B-spline 4-point

• Which one interpolates the original vertices?
• Which one produces the smoothest curve?
• When is this important?
• Can both have cusps (non smooth points)?
• How can you force the B-spline curve to go
  through a point?
• Which one has a convex-hull property?
• When is this important?

64
Cubic B-spline and four-points

• From James Koch

65
Which one is smoother?

• By Stevie Strickland

66
Making a sharp corner in a B-spline curve

• By Stevie Strickland
• HOW?

67
A compromise

• B-spline subdivision rounds the corners and thus
  shrinks a convex shape
• 4-points subdivision bulges the edges out and
  hence grows a convex shape
• Both produce curves that are relatively far from
  the initial control polygon
• We may want a compromise
• Closer to the polygon
• Bulge less
• Cut less

68
Jareks compromise J4

• Split each edge as before (in both schemes)
• Tweak old vertices by half of the B-spline
  displacement and the new vertices by half of the
  4-point displacement

69
3 subdivision schemes
Bezier
J4
4-points
70
Implementing J4
1 - For each vertex vi, compute the average ai of
its neighbors and store a displacement vector
Di(aivi)/8.
2 - For each edge (vi,vi1), insert a new vertex
mi(vivi1)/2, and displace it by (DiDi1)/2.
3 - Displace each old vertex vi by Di.
71
Three subdivision schemes

• B-spline (uniform cubic B-spline), most popular
• Introduces new vertices at edge mid-points
  (split)
• Tucks in the old vertices by 1/2 the displacement
  towards the average of their new neighbors
  (tweak)
• Very smooth result piecewise cubic parametric
  formulation
• Does not interpolate the original vertices
• Shrinks convex regions
• 4-points
• Leaves the old vertices unchanged
• Bulges the edges by displacing their mid-point
  away from the average of the neighbors (split
  tweak)
• Interpolates the original vertices
• Grows convex regions
• J4
• Combines half of the tweaks of the other two
• Final curve is closer to the original polygon

72
A unifying theory Js refinements

• Js ( r , t , u(s/8) )
• J0 cubic B-spline
• J2 osculates edges (smoother than quadratic
  B-spline)
• J4 in general preserves area well
• J5 reduces discrepancy with L (good for LoD)
• J8 4-point (interpolating)

J8
J4
J2
J0
73
Comparing Js schemes

• Jm J2, J0, J2, J2, J2

L
J0
J2
J4
J5
J6
J8
Jm
74
Retrofitting

• A new control polygon may be computed so that the
  subdivision curve passes through, or close to,
  the original control vertices.

75
Continuity measures for paths and curves

• Parametric (path)
• C0 Continuous trajectory at finite speed (?
  acceleration?)
• C1 Velocity is a continuous function of
  parameter t
• C2 Acceleration is a continuous function of
  parameter t
• C3 Jerk is a continuous function of parameter t
• Geometric (curve)
• G0 Continuous curve, no gap
• G1 Tangent is a continuous function of arclength
• G2 Curvature is a continuous function of
  arclength

76
Smoothness

• Visualize (lack of smoothness) using jerk
• J0 (no visible jerk) second degree (curvature)
  continuous, I.e., C2
• J8 (total jerk) first degree (tangent)
  continuous, I.e. C1
• Jm (brown) is C3

77
Open curves

• Temporarily add (guesses of) 2 extra points on
  each end
• Subdivide as before
• Render curve, except 5 spans
• Remove the extra points

78
Spans

• Subdivision bends each original edge into a span

79
Per span subdivision

• You do not need to subdivide the entire curve at
  the same time
• You can subdivide one span at a time (need 6
  control vertices)
• Or use a sliding window (ringing), which requires
  storing a ring of only 4 vertices per level of
  subdivision (see paper)

80
Multiresolution design

• At each level of subdivision, let the user
  specify adjustment vectors for some of the
  vertices
• Retain these vectors and apply them during
  rendering
• This lets the user control large and small
  features with few strokes

81
What about curves in 3D?

• The smoothing and subdivision procedures work for
  any dimension
• 1D for speed control in P1a
• 2D for curves, paths, and trajectories of cars
• 3D for curves, paths, and trajectories of
  airplanes
• 2D for designing and smoothing trajectories on
  surfaces (skier)
• In the space of screws for smoothing key-frames
  motions

82
What about animation?

• At each frame, draw subdivision curve
• Its control vertices move with time
• Their trajectories are each defined as a
  subdivision curve

83
What about surfaces?

• Same as curve animation, but draw quads between
  consecutive frames (curves)

84
Adaptive resolution

• The level of subdivision of a curve or surface
  should be adjusted to the resolution and the size
  of its projection on the screen
• When the size changes, as we switch resolution, a
  popping may be perceived.
• Jarek subdivision exhibits less popping that
  four-point and cubic B-spline
• Geomorphs are also used to smoothen the popping
  over several frames

85
Examples of quiz/exam questions

• Define/illustrate curve, path, motion, trajectory
• Explain interpolating and approximating
• Constructions of velocity, normal, acceleration,
  radius of curvature, jerk
• Implementation of r, k, d, t, u, u(x) steps
• Write polyloop smoothing using them
• Why tuck-tuck is better then tuck
• Cubic fit formulation for tuck-tuck
• Difference between smoothing and subdivision
• Subdivision formula for quadratic B-spline
• Subdivision formula for cubic B-spline
• Spans. local control, open curves
• Bezier control polygons from polyloop
• Cubic Bezier evaluation and subdivision
• Convex hull of cubic bezier curve
• 4-point subdivision as cubic fit
• Draw 4-points, jarek, and cubic B-spline for a
  simple polyloop by hand
• General formulation for jarek subdivision Js
• Properties of J0, J3, J4, J5, J8