# Three-dimensional graph

This surface consists of points whose coordinates (x, y, z) satisfy the equation z = sin(x2) × cos(y2).

A three-dimensional graph is the graph of a function f(x, y) of two variables, or the graph of a relationship g(x, y, z) among three variables.

Provided that x, y, and z or f(x, y) are real numbers, the graph can be represented as a planar or curved surface in a three-dimensional Cartesian coordinate system. A three-dimensional graph is typically drawn on a two-dimensional page or screen using perspective methods, so that one of the dimensions appears to be coming out of the page.

## Examples

The graph of the trigonometric function on the real line

${\displaystyle f(x,y)=\sin {x^{2}}\cdot \cos {y^{2}}}$

is

${\displaystyle \{(x,y,\sin {x^{2}}\cdot \cos {y^{2}}):x,y\in \mathbb {R} \}}$.

If this set is plotted on a three dimensional Cartesian coordinate system, the result is a surface (see above figure).

A two-dimensional perspective projection of a sphere

A three-dimensional graph of a sphere, with equation ${\displaystyle x^{2}+y^{2}+z^{2}=r^{2}}$ is shown at left.

## Collapsing the information in a three-dimensional graph into a two-dimensional graph

From economics, an indifference map with three indifference curves shown. All points on a particular indifference curve have the same value of the utility function, whose values implicitly come out of the page in the unshown third dimension.

The information in a three-dimensional graph is often collapsed into a two-dimensional graph with the use of contour lines, as illustrated at right. The x and y axes are retained, but instead of depicting a z or f(x, y) axis as "coming out of the page (or screen)", all x, y combinations giving rise to the same z or f(x, y) value are connected with a contour line; an arbitrary number of these may be shown for various values of z or f(x, y).