How to Draw a Hyperbolic Paraboloid
Quick Guide
John Ganci1
Al Lehnen2
1 Richland
College
Dallas, TX
[email protected]
2 Madison
Area Technical College
Madison, WI
[email protected]
The steps
Identify the axis
Identify two parabolas
Draw the parabolas
Identify two hyperbolas
Draw the hyperbolas
Connect the hyperbolas
Shade the surface
Identify the axis
Write the equation in the form u =
v2
a2
−
w2
b2
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 x = y 2 − z 2 then
u = x, v = y , w = z, a = 1, b = 1
The axis is the x-axis
Identify two parabolas (1/2)
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 is u =
v2
a2
2
The lower parabola is u = − wb2
For x = y 2 − z 2
The upper parabola is x = y 2
The lower parabola is x = −z 2
Identify two parabolas (2/2)
The upper parabola is in the uv -plane
The lower parabola is in the uw -plane
For x = y 2 − z 2
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
For x = y 2 − z 2
Upper parabola is x = y 2 ; limit y to [ −2, 2 ] or [ −1, 1 ]
Lower parabola is x = −z 2 ; limit z to [ −2, 2 ] or [ −1, 1 ]
Draw the parabolas (1/2)
Draw the upper parabola: x = y 2
z
y
x
−2 ≤ y ≤ 2
0≤x ≤4
Note the upper bound for x
Draw the parabolas (2/2)
Draw the lower parabola: x = −z 2
z
y
x
Note the lower bound for x
−2 ≤ z ≤ 2
−4 ≤ x ≤ 0
Identify two 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 x = y 2 − z 2 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
For x = y 2 − z 2 this is the plane x = −4
Vertices are on the lower parabola
Draw the hyperbolas (1/2)
Draw the upper hyperbola: 4 = y 2 − z 2 or 1 =
y2
4
z
y
x
Plane: x = 4
−2√
≤z ≤2
√
−2 2 ≤ y ≤ −2 or 2 ≤ y ≤ 2 2
−
z2
4
Draw the hyperbolas (2/2)
Draw the lower hyperbola: −4 = y 2 − z 2 or 1 =
z2
4
z
y
x
Plane: x = −4
−2√
≤y ≤2
√
−2 2 ≤ z ≤ −2 or 2 ≤ z ≤ 2 2
−
y2
4
Connect the hyperbolas
Connect the hyperbolas by drawing four line segments
Connect the upper hyperbola, upper ends, to the lower
hyperbola, upper ends
2 Connect the upper hyperbola, lower ends, to the lower
hyperbola, lower ends
1
If the arcs of the two hyperbolas are appropriately matched
(see the document An Interesting Property of Hyperbolic
Paraboloids), then these line segments lie on the surface of
the hyperbolic parabolid
Connect the hyperbolas (1/2)
The pink line segments connect the upper ends.
z
y
x
Connect the hyperbolas (2/2)
The aqua line segments connect the lower ends.
z
y
x
Shade the surface
Draw additional hyperbolas (shown in blue) along the upper
parabola, each parallel to the fixed upper hyperbola. Do the same
(shown in green) for the lower parabola.
z
y
x
Shaded surface with Winplot graph
z
y
x