Mtc124 lecture 6 interpolation

Baharul IslamSenior Lecturer

Department of MTCADaffodil International University

Computer graphics generated images that change with time

Any computer graphics parameters can be modified as functions of time• Location, direction, shape, pose, texture

coordinates, light parameters, camera parameters, etc

Key-frame method• Specify the parameters at key-frames and

interpolate the values of the parameters for other frames

Physically-based simulation• Give initial conditions, then modify the

values of the parameters by simulating physics, etc

Specify the parameters at key-frames and interpolate the values of the parameters for other frames

Interpolation methods: linear, spline, etc

t1

t2

t3

Designing appropriate interpolation functions for different applications

Parameterization of the interpolation functions based on distance traveled

Controllability of the interpolated values over time

Location of objects in a scene• Prefer smooth curves (non-linear) trajectory

rather than straight lines (linear)

Location of objects in a scene• Prefer smooth curves (non-linear) trajectory

rather than straight lines (linear)

Size of an object• Non-linear or linear, depends on the need

Size of an object• Non-linear or linear, depends on the need

Interpolation vs approximation

Parametric continuity

Global vs local control

Interpolating curve• Keys are the sample points of a curve• Keys represent actual locations that the

curve should pass through Approximating curve

• Only two end points (keys) are interpolated• Other points (keys) are meant to control the

shape of the curve

Keys are the sample points of a curve Keys represent actual locations that

the curve should pass through

Interpolating curve• Keys are the sample points of a curve• Keys represent actual locations that the

curve should pass through Approximating curve

• Only two end points (keys) are interpolated• Other points (keys) are meant to control the

shape of the curve

Only two end points (keys) are interpolated

Other points (keys) are meant to control the shape of the curve

Smoothness of a curve in a mathematical sense• Positional continuity C0 (zeroth-order)• Tangential continuity C1 (first-order)• Curvature continuity C2 (second-order)

A small change in the value of the parameter always results in a small change in the value of the curve function

u

Not C0

A small change in the value of the parameter always results in a small change in the value of the curve function

u

Is C0

A small change in the value of the parameter always results in a small change in the first derivative of the curve function

u

Not C1

A small change in the value of the parameter always results in a small change in the first derivative of the curve function

u

Is C1

A small change in the value of the parameter always results in a small change in the second derivative of the curve function

u

Not C2

Junction of two circular arcs

A small change in the value of the parameter always results in a small change in the second derivative of the curve function

u

Is C2

Junction of two circular arcs

Global control• Modifying a key has an effect on the overall

shape of the curve• Natural spline

Local control• Modifying a key has an effect on the limited

region of the curve• Bézier, B-spline, etc

Interpolation between two points Use a straight line to connect two

interpolation points

0.0u

0.1u

)0.0(P

)0.1(P

)(uP

)0.1()0.0()1()( uPPuuP

Use a straight line to connect two interpolation points

0.0u

0.1u

)0.0(P

)0.1(P

)(uP

)0.1()()0.0()()( PuGPuFuP

)0.1()0.0()1()( uPPuuP

Use a straight line to connect two interpolation points

)0.1()()0.0()()( PuGPuFuP

PMU LTuP

P

PuuP

P

PuGuFuP

)(

)0.1(

)0.0(

01

111)(

)0.1(

)0.0()()()(

Explicit form

Implicit form

Parametric form

)(xfy

0),( yxf

)(

)(

ugy

ufx

Parametric form

),,()( zyxuPP )(

)(

)(

uhz

ugy

ufx

0.10.0 u

0.0u

0.1u31u

32u

Four control points P0, P1, P2, P3

C1 continuity, convex-hull property

0P

1P

2P

3P

Four control points P0, P1, P2, P3

3

0

)()(i

ii ubPuP

33

22

21

30

)(

)1(3)(

)1(3)(

)1()(

uub

uuub

uuub

uub

0P

1P

2P

3P

Four control points P0, P1, P2, P3

PMU BTuP

P

P

P

P

uuuuP

)(

0001

0033

0363

1331

1)(

3

2

1

0

23

• Two end points P0, P1 and their tangents t0 and t1

• C1 continuity

0P

1P

0t

1t

PMU

t

t

1

0

HTuP

P

P

uuuuP

)(

0001

0100

1233

1122

1)( 1

0

23

• Two end points P0, P1 and their tangents t0 and t1

• Can be derived from Bézier curves– Compute the derivatives at the end points of a

Bézier curve– Substitute the results (tangents) to the Bézier

curves equation