Rick Parent - CIS681

Interpolation of a time-space curve

Interpolation (of a curve in space)

Speed control along curve

Reparameterizing according to arc length

Rick Parent - CIS681

Problem: generate a path through points at designated times with smooth motion

A

B

C

D

Time = 0

Time = 10

Time = 35

Time = 60

Rick Parent - CIS681

Solution: generate a space curve, distribute points evenly along curve, smoothly vary points temporally

A

B

C

D

Time = 0

Time = 10

Time = 35

Time = 60

Rick Parent - CIS681

Curves

Explicit form: y = f(x)

Implicit form: 0 = f(x,y)

x = f(u)

y = g(u)Parametric form:

Good for testing points or good for generating points?

Rick Parent - CIS681

CurvesParametric form: P = P(u) = (x,y,z)

x = f(u)y = g(u)z = h(u)

Space-curve P = P(u) 0.0 <=u<=1.0

u=1/3

u=2/3

u=0.0

u=1.0

Rick Parent - CIS681

Curves

Local v. global control

Computational complexity

Continuity

Interpolation v. approximation Match data v. design

cubic

local

1st order (tangential)

Rick Parent - CIS681

CurvesParametric form: P = P(u) = (x,y,z)

x = f(u)y = g(u)z = h(u)

Complexity: linear, cubic, higher order?

Cubic is lowest order that allows point of inflection between two points

Rick Parent - CIS681

Curves - continuity

none Zero order but not first order

1st but not 2nd order

2nd order

position tangent curvature

At junction of two circular arcs

Rick Parent - CIS681

Curves - local v. global control

Is area affected by editing operation restricted?Or does an edit affect the entire curve (even if slightly)?

Rick Parent - CIS681

Curvesu=1/3

u=2/3u=0.0

u=1.0

P(u) = a*u3 + b*u2 + c*u + d

P(0.0) = d = P0P(1/3) = a*(1/3)3 + b*(1/3)2 + c*(1/3) + d = P1P(2/3) = a*(2/3)3 + b*(2/3)2 + c*(2/3) + d = P2P(1.0) = a + b + c + d = P3

4-point formP0 P1

P2

P3

Rick Parent - CIS681

Curves

P(u) = a*u3 + b*u2 + c*u + d

P(0.0) = d = P0

P(1.0) = a + b + c + d = P1

P0

P1

P’0P’1

P’(0.0) = c = P’0

P’(1.0) = 3*a + 2*b + c = P’1

Hermite form

Rick Parent - CIS681

CurvesP(u) = u3*( m00*P1 + m01*P2 + m02*P3 + m03*P4) + u2*( m10*P1 + m11*P2 + m12*P3 + m13*P4) + …

P1P2P3P4

P(u) u3 u 1u2

m00 m01 m02 m03m10 m11 m12 m13m20 m21 m22 m23m30 m31 m32 m33

P(u) UMP

Rick Parent - CIS681

CurvesHermite

Bezier

Catmull-Rom

Blended parabolas

B-splines, NURBS

Rick Parent - CIS681

Hermite

Rick Parent - CIS681

Bezier

Rick Parent - CIS681

Catmull-Rom

Rick Parent - CIS681

Blended Parabolas

Rick Parent - CIS681

B-Spline / NURBS

Rick Parent - CIS681

Space-Time Curve

Given arclength* w, find u such that w = arclength(P(0.0), P(u))

w=0.3 w=0.6

w=0.0

w=1.0u=0.2 u=0.5

* relative arclength

Rick Parent - CIS681

Equal Parametric Increments

Rick Parent - CIS681

Arc Length

Rick Parent - CIS681

Arc Length - analytic

s dPduw1

w2

du

dPdu

((dx(u) /(du)),(dy(u) /(du)),(dz(u) /du)))

dPdu

((dx(u) /(du))2 (dy(u) /(du))2 (dz(u) /du))2)

dPdu

Au4 Bu3 Cu2 du E

A9(ax2 ay

2)

B12(axbx ayby )

C 6(axcx aycy ) 4(bx2 by

2)

D4(bxcx bycy )

E cx2 cy

2

Rick Parent - CIS681

Arc Length

P(0.0)

P(0.1)

P(0.2) P(0.3)

P(0.4)

Create table of parametric values and summed linear approximations to arclength

Rick Parent - CIS681

Arc Lengthdistance

Estimated u U Distance0.1 0.08

0.2 0.19

0.3 0.32

0.4 0.45

… …

Rick Parent - CIS681

Given arc length - find indexGiven arc length , d = 0.37

Find entries in table that bound given value:

U Distance0.1 0.080.2 0.190.3 0.320.4 0.45

… …

U U[i] (GivenDistance Distance[i]Distance[i 1] Distance[i]

)(U[i 1] U[i])

0.30.37 0.320.45 0.32

(0.3 0.4)

0.3385

Rick Parent - CIS681

Arc Length - Gaussian Quadrature

Weighted sum of samples - from table

See book

Adaptive - compare length of segment to sum of lengths of sub-segments

Rick Parent - CIS681

Controlling Motion Along a Curve

slowslowfast

Rick Parent - CIS681

Time-Distance Curve

time

distance

Slope indicates velocity

Usually use normalized distance (0:1) and normalized time (0:1) to reuse curves

Rick Parent - CIS681

Ease-in/ease-out

distance

time

(arclength)

Equally spaced samples in time specify arc length required for that frame

Rick Parent - CIS681

Time-Velocity Curve

velocity

time

Slope indicates acceleration

Area under curve indicates distance traveled

Use normalized distance and normalized time

Rick Parent - CIS681

Time- Acceleration Curve

acceleration

time

Signed area under curve is velocity (at end should be zero if object starts and ends at a stop)

This curves shows only constant acceleration and deceleration

Rick Parent - CIS681

Sine Ease-in / Ease-out

Ease in/ease out control for time-distance curve

Rick Parent - CIS681

Sinusoidal segment

distance

time

(arclength)

Sin(u)

2

u2

(Sin(t 2

)1)2

0 t 1

-1

1

Rick Parent - CIS681

Sinusoidal End Segments

distance

time

Linear segment(arclength)

Sinusoidal segments

Rick Parent - CIS681

Sinesoidal and linear segments

Rick Parent - CIS681

Sinesoidal and linear segments

ease(t) (k12

sin(( tk1

2

2

) 1)) / f

( k1

2

t k1) / f

( k1

2

k2 k1 ((1 k2) 2

)sin( t k2

1 k2

2

)) / f

f k12

k2 k1 (1 k2) 2

t k1

k1 t k2

k2 t

where

Rick Parent - CIS681

Constant Acceleration

acceleration

timet1 t2

Relationship between acceleration, deceleration, time accleration ends, time deceleration starts, and distance traveled

Assume begin and end with velocity = 0

A

D

A*t1 = - B*(1-t2)

Rick Parent - CIS681

Constant Acceleration

velocity

timet1 t2

Distance is area under the curve

Find v such that area = 1

v

Rick Parent - CIS681

Constant Acceleration

Parabolic segments

Linear segment

distance

time