Graphics Reference
In-Depth Information
but I need to interpolate colors along the edges and at the interior points”). Unfor-
tunately, this isn't easy. For colors that are very similar (e.g., when their difference
is just a little more than the just-noticeable difference), almost any interpolation
scheme will work, including interpolating the RGB coefficients (or whatever other
tuples you might be using to represent the colors). But for distant colors (e.g., a
saturated green and a medium-brown), there are many possibilities, and no one of
them is right for all circumstances.
You can even show that certain reasonable assumptions about color inter-
polation cannot be met by
any
interpolation scheme on any three-dimensional
color space.
If, for instance, one insists
• That the color
C
(
α
,
C
1
,
C
2
)
, interpreted as “
α
of the way between colors
C
1
and
C
2
,” be a
continuous
function of
α
,
C
1
, and
C
2
, and should be
C
1
=
1
• That two colors of equal saturation and brightness should be interpolated
by intermediate colors of equal saturation
• And that the color interpolated as “
when
α
=
0 and
C
2
when
α
α
of the way from
C
1
to
C
1
” should
always be
C
1
then one has a contradiction. If we restrict our attention to colors of saturation
1 and brightness 1, we have a circle, which we'll denote
S
1
, and which we'll
parameterize by hue, ranging from 0 to 1 (so a hue of 0 and a hue of 1 both denote
totally saturated red, for example). Restricting the function
C
to just this circle
gives a function from
[
0, 1
]
S
1
S
1
to
S
1
; the properties above translate to
×
×
S
1
S
1
S
1
is continuous
•
C
:[
0, 1
]
×
×
→
S
1
•
C
(
0,
x
,
y
)=
x
for all
x
,
y
∈
S
1
•
C
(
1,
x
,
y
)=
y
for all
x
,
y
∈
S
1
C
(
α
,
x
,
x
)=
x
for all
x
∈
•
Now consider the functions
S
1
:
t
p
0
:[
0, 1
]
→
→
C
(
0, 0,
t
)
, and
S
1
:
t
p
1
:[
0, 1
]
→
→
C
(
1,
t
,
t
)
.
The second property tells us that
p
0
is a constant, that is, its winding number
around the circle is zero. The final property tells us that
p
1
wraps once around
the circle. But these two loops can be joined together by a family of intermediate
curves,
S
1
:
t
p
s
:[
0, 1
]
→
→
C
(
s
,
st
,
t
)
,
s
∈
[
0, 1
]
each of which starts and ends at the same place. This is impossible, for
t
p
s
(
t
)
is a continuous function of both
s
and
t
, and hence the winding number of
p
s
is a
continuous function of
s
. But a continuous integer-valued function is constant, so
the winding number cannot change from 0 to 1 as we vary
s
.
This is not to say that color interpolation is impossible or wrong; it merely
indicates that many seemingly natural constraints cannot be met. It's therefore
often best to carefully consider the application domain, and ask what's actually
needed: Do you
really
need to be able to interpolate between colors of opposite
hue without passing through white? Will the colors you're interpolating between
→
