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Suppose that we are viewing a surface where 
, then if the first
derivative exists at each point on the surface (i.e. the surface is of
class 
) then a tangent plane can be defined for each point. A
normal vector to the tangent plane at 
, representing the orientation
of the tangent plane (also called a surface normal), can be defined by
This may be written as 
where
A normal vector defined in this form has a 
-component of magnitude 1
pointing towards the viewer, so it can only correspond to visible
surface points.
The pair 
corresponding to represents the surface
orientation at , and is called the gradient of 
.
Thus, surface orientation for visible surfaces has two degrees of freedom.
A plane defined by 
and 
axes (representing slope in the 
and 
directions, respectively) is called a gradient space and every point on it
corresponds to a particular orientation of a visible surface.
Since a point with gradient space coordinates corresponds to the
the surface normal vector , gradients may be readily interpreted
in terms of spherical coordinates. Consider a unit sphere centered at
and an infinite plane with equation 
, which necessarily
touches the spehere at 
(see Figure 31). Any
vector extending from the center of the sphere to the plane has direction
numbers where is the gradient of the sphere where it
intersects the vector. The plane represents a gradient space.
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