approximate non-linear transformation of affine combination

approximate non-linear transformation of affine combination

Consider
applying an arbitrary

transformation

f

to an

affine combination

∑

i

=

1

n

w

i

⁢

x

i

of some points

x

i

,
with non-negative weights

w

i

that sum to unity.
Obviously, in general,

f

⁢

(

∑

i

=

1

n

w

i

⁢

x

i

)

≠

∑

i

=

1

n

w

i

⁢

f

⁢

(

x

i

)

.

However, sometimes it is desirable to compute the image

f

⁢

(

∑

i

w

i

⁢

x

i

)

as if

f

were a linear (or affine) transformation;
the result is then hoped to be a good

approximation

to the true value.
(See below for an application.)

Actually, it is possible to show that, provided that

f

is twice

continuously differentiable

, the approximation is good to first order,
despite the absence of any

derivatives

of

f

in the formula.
The domain and range of

f

may be any

normed vector spaces

.

First, we write:

f

⁢

(

∑

i

w

i

⁢

x

i

)

=

f

⁢

(

∑

i

w

i

⁢

(

x

i

-

x

1

)

+

x

1

)

=

f

⁢

(

x

1

)

+

∑

i

w

i

⁢

f

′

⁢

(

x

1

)

⁢

(

x

i

-

x

1

)

+

O

⁢

(

∥

∑

i

w

i

⁢

(

x

i

-

x

1

)

∥

2

)

.

If

h

=

max

i

⁡

∥

x

i

-

x

1

∥

,
then

∥

∑

i

w

i

⁢

(

x

i

-

x

1

)

∥

≤

∑

i

|

w

i

|

⁢

h

=

h

,
and so the error term in the

Taylor expansion

can be simplified to

O

⁢

(

h

2

)

.

Substituting another Taylor expansion

f

⁢

(

x

i

)

-

f

⁢

(

x

1

)

=

f

′

⁢

(

x

1

)

⁢

(

x

i

-

x

1

)

+

O

⁢

(

h

2

)

,

into the first, we obtain:

f

⁢

(

∑

i

w

i

⁢

x

i

)

=

f

⁢

(

x

1

)

+

∑

i

w

i

⁢

(

f

⁢

(

x

i

)

-

f

⁢

(

x

1

)

+

O

⁢

(

h

2

)

)

+

O

⁢

(

h

2

)

=

∑

i

w

i

⁢

f

⁢

(

x

i

)

+

O

⁢

(

h

2

)

.

Furthermore, it is not hard to see,
by accounting the error from the Taylor expansions more carefully,
that we have the bound:

∥

f

⁢

(

∑

i

w

i

⁢

x

i

)

-

∑

i

w

i

⁢

f

⁢

(

x

i

)

∥

≤

M

⁢

h

2

,

where

M

is the

maximum

,
as

ξ

ranges inside the

convex hull

formed by the points

x

i

,
of the quantity

∥

f

′′

⁢

(

ξ

)

∥

=

sup

u

≠

0

⁡

∥

f

′′

⁢

(

ξ

)

⁢

(

u

,

u

)

∥

∥

u

∥

2

.
Finally, the point

x

1

over which we performed Taylor expansions
can be replaced by any other point

x

k

,
and so correspondingly

h

can be replaced by

min

k

⁡

max

i

⁡

∥

x

i

-

x

k

∥

.

Application in computer graphics

The principle just derived is often applied in vector-based computer graphics
when curved objects are drawn by cubic Bézier curves:

γ

⁢

(

t

)

=

∑

i

=

0

3

w

i

⁢

(

t

)

⁢

x

i

,

w

i

⁢

(

t

)

=

(

3

i

)

⁢

(

1

-

t

)

n

-

i

⁢

t

i

,

which are affine combinations of the

control points

x

i

.
To compute and display a smooth transformation

f

of such curves,
it may be too much work to compute

f

⁢

(

γ

⁢

(

t

)

)

repeatedly
for many parameter values

t

.
Provided

γ

is not too wavy,
computing and displaying

∑

i

=

0

3

w

i

⁢

(

t

)

⁢

f

⁢

(

x

i

)

is vastly more efficient, and may result in little or
no visually perceptible difference.

As a concrete example, consider bending a

straight line

segment

into a circle.
Mathematically, we are mapping the

interval

[

0

,

2

⁢

π

]

via

t

↦

(

r

⁢

cos

⁡

t

,

r

⁢

sin

⁡

t

)

.
If the interval is split into sub-segments,
each considered as a
cubic Bézier curve with its interior control points
both set at the

midpoint

of the

line segment

,
then a circle can be approximated
by transforming these control points.
The following diagram shows the approximation for 24 segments (three Bézier
curves per

45

∘

arc).

Figure 1:

Circle drawn using approximate mapping of line segment

Title

approximate non-linear transformation of affine combination

Canonical name

ApproximateNonlinearTransformationOfAffineCombination

Date of creation

2013-03-22 16:50:35

Last modified on

2013-03-22 16:50:35

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

7

Author

stevecheng (10074)

Entry type

Example

Classification

msc 41A58

Classification

msc 51N20

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