### Parametric Surfaces (3-D Graphing)

Graphs surfaces in three dimensions specified parametrically, as x=f(s,t), y=g(s,t), and z=h(s,t), with parameters s and t
How to use   ||   Examples   ||   Other Notes

How to use
• Enter functions f1(s,t), g1(s,t), and h1(s,t) in the text input fields marked "f1(s,t)=", "g1(s,t)=", and "h1(s,t)=" (Example: f1(s,t)=t+4 cos t, g1(s,t)=4 sin t, h1(s,t)=t)
• Click the "Graph" button (this button also refreshes the graph)
• Rotate the graph by clicking and dragging the mouse on the graph.
• To see two surfaces graphed simultaneously:
• Enter the second set of functions f2(s,t), g2(s,t), and h2(s,t) in the text input fields marked "f2(s,t)=", "g2(s,t)=", and "h2(s,t)="
• Click (to its "on" state) the check box next to this set of input fields
• Click the "Graph" button
(Example: f1(s,t)=t+4 cos s, f1(s,t)=4 sin s, and f1(s,t)=t; f2(s,t)=5 cos t sin s, g2(s,t)=5 sin t sin s, and h2(s,t)=5 cos s)
Up to 2 surfaces can be graphed simultaneously
• To remove a function from the graph, click (to its "off" state) the check box next to the associated text input fields and click the "Graph" button to refresh
• To erase the graph and all input fields, click the "Clear" button
• There are four input fields which specify bounds on the variables for the function:
• The variable s takes values between smin and smax, specified by the fields marked "smin=" and "smax=".
• The variable t takes values between tmin and tmax, specified by the fields marked "tmin=" and "tmax=".
• These four text input fields can accept any decimal number input.
The text input fields for functions can accept a wide variety of expressions to represent functions, and the buttons under the graph allow various manipulations of the graph coordinates.
For another way to view surfaces, try the "wireframe" representation.
Examples
 Ellipsoid: f1(s,t)=5 cos t sin s g1(s,t)=3 sin t sin s h1(s,t)=2 cos s Plane: f1(s,t)=s g1(s,t)=t h1(s,t)=s+t Cylinder: f1(s,t)=5 cos t g1(s,t)=5 sin t h1(s,t)=s Slant Cylinder: f1(s,t)=s+5 cos t g1(s,t)=5 sin t h1(s,t)=s

Other Notes
Surfaces in three dimensions are represented in "faceted hidden surface" form. The facets are not subdivided at intersections of surfaces if more than one surface is drawn, so intersections of surfaces are not precise. The "wireframe" represenation for surfaces, in which the surface is transparent, only draws one surface at a time.