Evaluators provide a way to use polynomial or rational polynomial mapping to produce vertices, normals, texture coordinates, and colors. The values produced by an evaluator are sent on to further stages of GL processing just as if they had been presented using
glVertex, glNormal, glTexCoord, and glColor commands, except that the generated values do not update the current normal, texture coordinates, or color.All polynomial or rational polynomial splines of any degree (up to the maximum degree supported by the GL implementation) can be described using evaluators. These include almost all surfaces used in computer graphics, including B-spline surfaces, NURBS surfaces, Bezier surfaces, and so on.
Evaluators define surfaces based on bivariate Bernstein polynomials. Define $p ( u hat , v hat ) $ as
p ( u hat , v hat ) ~~=~~ down 30 {{size 18 {sum}} from {size 8 {i~=~0}} to {size 9 {n}} } ~ {{size 18 {sum}} from {size 8 {j~=~0}} to {size 9 {m}} } ~ up 15 { B sub i sup n ( u hat ) ~ B sub j sup m ( v hat ) ~ R sub ij }
where $R sub ij$ is a control point, $B sub i sup n ( u hat )$ is the $i$th Bernstein polynomial of degree
$n$ (
uorder = $n ~+~ 1$)
B sub i sup n ( u hat ) ~~=~~ left ( down 20 {cpile { n above i }} ~ right ) u hat sup i ( 1 - u hat ) sup { n - i }
and $B sub j sup m ( v hat )$ is the $j$th Bernstein polynomial of degree $m$ (
vorder = $m ~+~ 1$)
B sub j sup m ( v hat ) ~~=~~ left ( ^down 20 {cpile { m above j }} ~^ right ) v hat sup j ( 1 - v hat ) sup { m - j }
Recall that $0 sup 0 ~==~ 1 $ and $ left ( ^ down 20 {cpile { n above 0 }} ~^ right ) ~~==~~ 1 $
glMap2 is used to define the basis and to specify what kind of values are produced. Once defined, a map can be enabled and disabled by calling glEnable and glDisable with the map name, one of the nine predefined values for target, described below. When glEvalCoord2 presents values $u$ and $v$, the bivariate Bernstein polynomials are evaluated using $u hat$ and $v hat$, where
$u hat ~~=~~ {u ~-~ "u1"} over {"u2" ~-~ "u1"}$
$v hat ~~=~~ {v ~-~ "v1"} over {"v2" ~-~ "v1"}$
target is a symbolic constant that indicates what kind of control points are provided in points, and what output is generated when the map is evaluated. It can assume one of nine predefined values:
GL_MAP2_VERTEX_3
Each control point is three floating-point values representing $x$, $y$, and $z$. Internal glVertex3 commands are generated when the map is evaluated.
GL_MAP2_VERTEX_4
Each control point is four floating-point values representing $x$, $y$, $z$, and $w$. Internal glVertex4 commands are generated when the map is evaluated.
GL_MAP2_INDEX
Each control point is a single floating-point value representing a color index. Internal glIndex commands are generated when the map is evaluated but the current index is not updated with the value of these glIndex commands.
GL_MAP2_COLOR_4
Each control point is four floating-point values representing red, green, blue, and alpha. Internal glColor4 commands are generated when the map is evaluated but the current color is not updated with the value of these glColor4 commands.
GL_MAP2_NORMAL
Each control point is three floating-point values representing the $x$, $y$, and $z$ components of a normal vector. Internal glNormal commands are generated when the map is evaluated but the current normal is not updated with the value of these glNormal commands.
GL_MAP2_TEXTURE_COORD_1
Each control point is a single floating-point value representing the $s$ texture coordinate. Internal
glTexCoord1 commands are generated when the map is evaluated but the current texture coordinates are not updated with the value of these glTexCoord commands.
GL_MAP2_TEXTURE_COORD_2
Each control point is two floating-point values representing the $s$ and $t$ texture coordinates. Internal
glTexCoord2 commands are generated when the map is evaluated but the current texture coordinates are not updated with the value of these glTexCoord commands.
GL_MAP2_TEXTURE_COORD_3
Each control point is three floating-point values representing the $s$, $t$, and $r$ texture coordinates. Internal glTexCoord3 commands are generated when the map is evaluated but the current texture coordinates are not updated with the value of these glTexCoord commands.
GL_MAP2_TEXTURE_COORD_4
Each control point is four floating-point values representing the $s$, $t$, $r$, and $q$ texture coordinates. Internal
glTexCoord4 commands are generated when the map is evaluated but the current texture coordinates are not updated with the value of these glTexCoord commands.
ustride, uorder, vstride, vorder, and points define the array addressing for accessing the control points. points is the location of the first control point, which occupies one, two, three, or four contiguous memory locations, depending on which map is being defined. There are $ "uorder" ~times~ "vorder" $ control points in the array. ustride specifies how many float or double locations are skipped to advance the internal memory pointer from control point $R sub {i j} $ to control point $R sub {(i+1) j} $. vstride specifies how many float or double locations are skipped to advance the internal memory pointer from control point $R sub {i j} $ to control point $R sub {i (j+1) } $.