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Interpolation Vs. Approximation. 7. 7. 1. 1. 4. 4. 2. 2. 6. 6. 3. 3. 5. 5. CS 490.006/582.001 Interpolating Values Page 21. De Casteljau Approximation. 4. Quarterpoint between control points 2 and 3. 2. Midpoint between control points 1 and 2. - PowerPoint PPT Presentation
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INTERPOLATION VS. APPROXIMATION
CS 490.006/582.001INTERPOLATING
VALUESPAGE 1
1
2
3
4
5
6
7 1
2
3
4
5
6
7
DE CASTELJAU APPROXIMATION
CS 490.006/582.001INTERPOLATING
VALUESPAGE 2
1
2
3
4
MIDPOINT BETWEEN CONTROL POINTS 1
AND 2MIDPOINT 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
VALUESPAGE 3
POSITIONAL DISCONTINUITY
TANGENTIAL DISCONTINUITY
CURVATURE DISCONTINUITY
ARC LENGTH
CS 490.006/582.001INTERPOLATING
VALUESPAGE 4
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
VALUESPAGE 5
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
VALUESPAGE 6
Time
Distanc
e
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
VALUESPAGE 7
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.