INTERPOLATION VS. APPROXIMATION

CS 490.006/582.001INTERPOLATING

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1

2

3

4

5

6

71

2

3

4

5

6

7

DE CASTELJAU APPROXIMATION

CS 490.006/582.001INTERPOLATING

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1

2

3

4

MIDPOINT BETWEEN CONTROL POINTS 1

AND 2

MIDPOINT BETWEEN

SECOND TWO MIDPOINTS

MIDPOINT BETWEEN CONTROL POINTS 3

AND 4MIDPOINT BETWEEN FIRST TWO MIDPOINTS

BEZIER POINT AT T=0.5MIDPOINT BETWEEN CONTROL POINTS 2

AND 3

QUARTERPOINT BETWEEN

CONTROL POINTS 1

AND 2

QUARTERPOINT BETWEEN

CONTROL POINTS 2

AND 3

QUARTERPOINT BETWEEN

CONTROL POINTS 3

AND 4

QUARTERPOINT BETWEEN FIRST TWO

QUARTERPOINTS

QUARTERPOINT BETWEEN

SECOND TWO QUARTERPOIN

TSBEZIER POINT AT T=0.25

SPLINE CONTINUITY

CS 490.006/582.001INTERPOLATING

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POSITIONAL DISCONTINUITY

TANGENTIAL DISCONTINUITY

CURVATURE DISCONTINUITY

ARC LENGTH

CS 490.006/582.001INTERPOLATING

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𝑐𝑢𝑟𝑣𝑒 h𝑙𝑒𝑛𝑔𝑡 =∫𝑡 0

𝑡 1

|𝑑𝑃 /𝑑𝑡|𝑑𝑡

0.5

0.0

1.0

ADAPTIVE SUBDIVISION

0.5

0.0

1.0

FORWARD DIFFERENCING

0.25

0.75

SPEED CONTROL

CS 490.006/582.001INTERPOLATING

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𝑃 (𝑡 )= (1−𝑡 ) 𝑃0+𝑡 𝑃1

𝑃 (𝑡 )=𝑃0 cos2(𝜋 𝑡 /2)+𝑃1sin

2(𝜋𝑡 /2)

DISTANCE-TIME FUNCTIONS

CS 490.006/582.001INTERPOLATING

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Time

Distance

ZERO SLOPE

: SMOO

TH START

ZERO SLOPE

: SMOO

TH STOP

ZERO SLOPE: STALLPOSITIVE SLOPE:

FORWARD MOTION

NEGATIVE

SLOPE:

REVERSE

MOTION

CONCAVE UP:

ACCELERATING

CONCAVE DOWN:

DECELERATING

QUATERNION INTERPOLATION

CS 490.006/582.001INTERPOLATING

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INTERPOLATING ROTATION MAY BE IMPLEMENTED BY

PERFORMING LINEAR INTERPOLATION AND PROJECTING POINTS

ON THE INTERPOLATED LINE ONTO THE DESIRED ARC.

THE RESULTING ARC LENGTHS VARY IN SIZE, MAKING THE ROTATIONAL TRANSITION

RATES INCONSISTENT.