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8.3 Planar Functions

A function \( z \mathrel{\overset{f}{\leftarrow}} (x, y) \) is planar if it can be written in the form
\[ \begin{align} z - z_p &= s_x \mult (x - x_p) + s_y \mult (y - y_p) \end{align} \]
where
\[ \begin{align} &x_p : \Number, & &y_p : \Number, & &z_p : \Number, \\ &s_x : \Number, & &s_y : \Number & & \end{align} \]
are all constants.
Important.
The graph of a planar function
\[ \begin{align} z - z_p &= s_x \mult (x - x_p) + s_y \mult (y - y_p) \end{align} \]
is the plane that
  • has height \( z_p \) at the point \( p = (x_p, y_p) \),
  • has constant \( x \)-slope \( s_x \), and
  • has constant \( y \)-slope \( s_y \).
Why?
We can solve for height \( z \).
\[ \begin{align} z &= z_p + s_x \mult (x - x_p) + s_y \mult (y - y_p) \end{align} \]
By localizing at \( p = (x_p, y_p) \), we find
\[ \begin{align} z &= z_p. \end{align} \]
Taking partials of \( z \), we find the constants
\[ \begin{align} \overset{x}{\diff} z &= s_x & \overset{y}{\diff} z &= s_y. \end{align} \]
You may find it helpful to compare the definition of a planar function to that of a linear function. Let's see an example of a planar function.
Example.
Consider the planar function \( z \depends (x, y) \) given by the equation
\[ \begin{align} z - 2 &= - 3 \mult (x - 1) + \frac{1}{2} \mult (y - 4). \end{align} \]
We can understand this plane by reading off the constants:
\[ \begin{align} &x_p = 1, & &y_p = 4, & &z_p = 2, \\ &s_x = -3, & &s_y = \frac{1}{2}. & & \end{align} \]
At the point \( (1, 4) \), our plane has height two. The \( x \)-slope is negative three everywhere, and the \( y \)-slope is one-half everywhere.

Horizontal Planes

We can describe a horizontal plane as a planar function by setting both slopes \( s_x \) and \( s_y \) equal to zero. We'll write
\[ \begin{align} z - z_p &= 0 \mult (x - x_p) + 0 \mult (y - y_p) \end{align} \]
or more simply
\[ \begin{align} z &= z_p. \end{align} \]
This is an equation we've seen before: we set the height \( z \) equal to a constant to find horizontal slices.
Unlike horizontal planes, a vertical plane is never the graph of a function \( z \depends (x, y) \). After all, a function must produce only a single height \( z \) for each point \( p = (x_p, y_p) \).
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