Applying a texture to a primitive implies a mapping from texture image
space to framebuffer image space.
In general,
this mapping involves a reconstruction of the sampled texture image,
followed by a homogeneous warping implied by the mapping to framebuffer
space,
then a filtering,
followed finally by a resampling of the filtered, warped,
reconstructed image before applying it to a fragment.
In the GL this mapping is approximated by one of two simple filtering schemes.
One of these schemes is selected based on whether the mapping from texture
space to framebuffer space is deemed to magnify or minify
the texture image.
The choice is governed by a scale factor 
and
;
if 
is less than or equal to some constant
(the selection of the constant is described below in section 3.8.2)
the texture is said to be magnified;
if it is greater,
the texture is minified. 
is called the level of detail.
Let 
be the function that associates an s texture coordinate
with each set of window coordinates 
that lie within a primitive;
define 
analogously.
Let 
and 
(for a one-dimensional texture, define 
).
For a polygon,
is given at a fragment with window coordinates 
by
where 
indicates the derivative of
u with respect to window x,
and similarly for the other derivatives.
For a line,
the formula is
where 
and 
with 
and 
being the
segment's window coordinate endpoints and 
.
For a point, pixel rectangle, or bitmap,
.
While it is generally agreed that equations 3.11 and 3.12
give the best results when texturing,
they are often impractical to implement.
Therefore,
an implementation may approximate the ideal 
with a function
subject to these conditions:
is continuous and monotonically increasing in each of
, 
,
, and 
,
Then 
.
When 
indicates minification,
the value assigned to TEXTURE_MIN_FILTER is used
to determine how the texture value for a fragment is selected.
When TEXTURE_MIN_FILTER is NEAREST,
the texel nearest (in Manhattan distance) to that specified by 
is obtained.
This means the texel at location 
becomes the texture value,
with i given by
(Recall that if TEXTURE_WRAP_S is REPEAT,
then 
.)
Similarly, j is found as
For a one-dimensional texture, j is irrelevant; the texel at location i becomes the texture value.
When TEXTURE_MIN_FILTER is LINEAR,
a 
square of texels is selected.
This square is obtained by first computing
and
Then
and
Let
where 
denotes the fractional part of x.
Let 
be the texel at location 
in the texture image.
Then the texture value, 
is found as
for a two-dimensional texture. For a one-dimensional texture,
where 
indicates the texel at location i in the one-dimensional
texture.
If any of the selected 
(or 
) in the above equations
refer to a border texel with
,
,
,
or 
,
then the border color given by the current setting of
TEXTURE_BORDER_COLOR is used instead of the unspecified value or values.
The RGBA values of the TEXTURE_BORDER_COLOR are interpreted to match the texture's
internal format in a manner consistent with Table 3.7.
