1. How to draw a hyperbolic paraboloid
John Ganci Adjunct Math Faculty Richland CC, Dallas TX
Al Lehnen
Math Instructor
Madison Area Technical College, Madison WI

2. The steps Identify the axis Identify the parabolas Draw two parabolas
Draw two hyperbolas
Connect the hyperbolas

3. Identify the axis Write equation in the form
u, v, and w are x,y, and z
u = x, v = y, w = z u = x, v = z, w = y
u = y, v = x, w = z u = y, v = z, w = x
u = z, v = x, w = y u = z, v = y, w = x
The one of degree 1, u, is the axis
If the equation is,
Then u = x, v = y, w = z, a = 1, b = 1
axis is the x-axis

4. Identify the parabolas
Two parabolas are used for the sketch
The remaining two variables in the equation, v and w, are used for the parabolas
the “upper parabola”
the “lower parabola”
For
the “upper parabola” is
the “lower parabola” is

5. Draw the parabolas The upper parabola is in the uv-plane
The lower parabola is in the uw-plane
For
The upper parabola is in the xy-plane
The lower parabola is in the xz-plane
Determine “reasonable” limits for the domain values for the two parabolas
Upper: x = y^2; limit y to [-2,2] or [-1,1]
Lower: x = -z^2; limit z to [-2,2] or [-1,1]

6. Draw the parabolas (1/2) The upper parabola x = y^2 -2 ≤ y ≤ 2
Note upper
bound for x

7. Draw the parabolas (2/2) The lower parabola x = -z^2 -2 ≤ z ≤ 2
Note lower
bound for x

8. Draw the hyperbolas One hyperbola for each of the parabolas
Drawn in planes perpendicular to the axis
Upper hyperbola drawn with upper parabola
The plane is the upper bound for the u variable
For
this is the plane x = 4
Vertices are on the upper parabola
Lower hyperbola drawn with lower parabola
The plane is the lower bound for the u variable
this is the plane x = – 4
Vertices are on the lower parabola

9. Draw the hyperbolas (1/2)
The upper hyperbola
4 = y^2 – z^2
--- or ---
1 = y^2/4 – z^2/4
Plane: x = 4
-2 ≤ z ≤ 2
-2sqrt(2) ≤ y ≤ -2 or 2 ≤ y ≤ 2sqrt(2)

10. Draw the hyperbolas (2/2)
The lower hyperbola
-4 = y^2 – z^2
--- or ---
1 = z^2/4 – y^2/4
Plane: x = -4
-2 ≤ y ≤ 2
-2sqrt(2) ≤ z ≤ -2 or 2 ≤ z ≤ 2sqrt(2)

11. Connect the hyperbolas
Connect the upper hyperbola, upper ends, to the lower hyperbola, upper ends
Connect the upper hyperbola, lower ends, to the lower hyperbola, lower ends
If the two hyperbola arcs are appropriately matched (see the document “An Interesting Property of Hyperbolic Paraboloids”), then these line segments lie on the surface of the hyperbolic paraboloid.

12. Connect the hyperbolas (1/2)
The upper ends

13. Connect the hyperbolas (2/2)
The lower ends

14. Graph with sketch lines