CS 551/651
Advanced Graphics
Arc Length
Assignment 1
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Due Week from Thursday
Building a Cubic Bézier curve
OpenGL/Glut
Insert up to 100 points
Render Bézier curve using recursive
subdivision
• Left mouse button to add, middle to move,
and right to remove
• FLTK is extra credit
Papers for Next Tuesday
Presenters Needed
• Tour Into the Picture
• William T. Reeves, Particle systems - a
technique for modelling a class of fuzzy
objects. In SIGGRAPH '83, Computer
graphics 17, 3, July 1983.
• " Animation of plant development,"
Przemyslaw Prusinkiewicz, Mark S.Hammel,
and Eric Mjolsness. Siggraph 1993.
Arc Length
• Arc length is the distance along the curve
• To ensure constant speed of curve
evaluation, perform arc length
parameterization of curve
– Compute a mapping offline of arc length
values to parameter values
– Analytically compute arc length
Arc Length
• Given parameters u1 and u2, find LENGTH(u1,
u 2)
• Given an arc with length s and a parameter
u1, find u2 s.t. LENGTH (u1, u2) is s
• Can we compute s = G(u) = distance from
start of curve to point at u?
• If so, G-1 is used to build arc length
parameterized curve: P(G-1(s))
Analytic Computation
x  P(u )
u2
u2
dP
length(u1, u 2)  s  
du  
du
u1
u1
Cubic curve example:
P(u )  au 3  bu 2  cu  d
dP / du  3au 2  2bu  c
 3au
u2
s
u1
2

2
 2bu  c du
dP / du  du
2
Forward Differencing
• Sample curve a many parameter values
• Create piecewise linear representation
of curve from parameter evaluations
• Compute linear distance between samples
• Store distances in table
• Limitations/Shortcomings?
Adaptive Approach
• Adaptively subdivide
• First step, compare:
– length (start, middle) + length (middle, end)
– length (start, end)
• If error is too large, subdivide
• Use link list to store data
Numerical Computation
• Numerical methods exist to approximate
integral of curve given sample
points/derivatives
• Instead of using sum of linear
segments, use numerical method to
compute sum of curved segments
Gaussian Quadrature
• Commonly used to integrate function
between –1 and +1
• Function to be integrated is evaluated at
fixed points within [-1, 1] and multiplied
by a precalculated weight  f (u)   w f (u )
• More sample points leads
to better accuracy
1
i
1
i
i
Adaptive Gaussian Integration
• Gaussian quadrature uses sampling to preseve
accuracy
– How many samples are necessary for given curve?
• Adaptive gaussian integration monitors errors
to subsample when necessary
• Start trying few samples and add more as
necessary
• Recursive like spline subdivision
Adaptive Gaussian Integration
• Build a table that maps parameter
values to arc length values
• To solve arc length at u, find ui and ui+1
that bound u
• Use gaussian quadrature to compute
intermediate arc length between that at
ui and ui+1
Finding u given s
• Must compute u s.t. length (u1, u) = s
• Solve: s – length(u1, u) = 0
– Find the zeros (roots) of the function
• Newton-Raphson does this for us:
f ( pn 1 )
pn  pn 1 
f ( pn 1 )
• f = s – length(u1, pn-1)
f’= dp/du evaluated
at pn-1
Newton-Raphson
• Be aware that Newton-Raphson could
set pn to a value not defined by curve
• f’(pn-1) could be 0 (cannot divide by 0)
Speed Control
• Given arc-length parameterized curve
• Let s(t) define amount the curve has traveled
given t
– For simplicity, normalize total distance to 1
– Monotonic (can’t go backwards) and continuous
• To ease in/out, make s(t) = ease (t) that looks
like:
s(t)
t
Speed Control
• Easing in/out:


sin  t      1
2
s(t )  ease(t )  
2
• -pi/2 shifts curve rightward
• +1 shifts curve upward
• Divide by 2 scales from 0 to 1
Speed Control
• Transcendental functions are expensive
– Table lookups are one possibility\
– Roll your own
• User specifies acceleration period,
deceleration period, and max velocity
vel(t)
v0
t1
t2
t
v0
t1
t2 t
Speed Control
• Distance traveled is area under curve
1
1
1.0   v0  t1  v0  (t 2  t1 )   v0  (1.0  t 2 )
2
2
• User selects two of three unknowns and
distance traveled constraint determines
third
• Integrate velocity function to find s(t)
d  v0 
d  v0 
t2
2  t1
0.0  t  t1
t1
 v0  (t  t1 )
2
t1  t  t 2
SLERPing
• Quaternions are points on the unit sphere
in four dimensions
– Unit sphere in three dimensions plus a fourth
dimension representing the rotation about
the normal at a point on the sphere
• Interpolating quats
– Linearly interpolate each of four components
• Doesn’t quite work right
SLERPing
• Linear interpolation doesn’t produce
constant velocity interpolation of quats
• Instead, interpolate along the arc on
sphere between two quats
SLERPing
• Which way do we go?
– Long way or short way around sphere
• q = [s,v] = [-s,-v] = -q
• Compute angle between q1 and q2
– Four dimensional dot product
• cos(Q q1qq2 = s1*s2 + v1 q v2
– If positive, this is shortest,
– Else, interpolate between q1 and -q2
SLERPing
• SLERP: slerp (q , q , u)  ((sin(( 1 u)  )) /(sin  )  q  (sin( u  )) /(sin  )  q
• Unit quaternion is not guaranteed
1
2
1
• What about continuity of interpolation?
2
Continuity of SLERP
• Interpolate between points
– [p0, …, pn-1, pn, pn+1, …]
• Bezier interpolation would compute
intermediate points for each pn
– To define tangent at pn
– Define bn to be point before pn
– Define an to be point after pn
Continuity of SLERP
• an = (pn – pn+1 + pn * ½) + (½ * pn+1)
pn
pn-1
pn+ pn – pn+1
an
pn+1
• pn + pn – an = bn
Continuity of SLERP
• To compute midway (average) points
between quats, SLERP halfway
• Instead of adding vectors, concatenate
rotations
P1 = slerp(qn, qn, 1/3)
an
bn+1
P2 = slerp(an, bn+1, 1/3)
P3 = slerp (bn+1, qn+1, 1/3)
P12 = slerp (p1, p2, 1/3)
qn
qn+1
P23 = slerp (p2, p3, 1/3)
P = slerp (p12, p23, 1/3)