Section 12.7 Quadric surfaces
12.71
Learning outcomes
After completing this section, you will inshaAllah be able to
1. know what are quadric surfaces
2. how to sketch quadric surfaces
3. how to identify different quadric surfaces
Key Idea
ƒ It is not helpful to plot points to sketch a surface
ƒ Mainly we use traces and intercepts to sketch
surfaces
Here we only do quadric surfaces
12.7 2
What are quadric surfaces?
A quadric surface is the graph of a 2nd degree equation in 3 variables.
Its most general form is
Ax 2 + By 2 + Cz 2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0
Main technique for sketching
Determine XY-trace, YZ-trace, XZtrace, other traces (if needed) and use
this information to complete the sketch
Most important traces
The trace of a surface S in a
ƒ XY-trace
plane is the intersection of S
ƒ YZ-trace
and the plane.
ƒ XZ-trace
ƒ traces in planes parallel to
coordinate planes
12.73
How to find traces?
• To find XY-trace put z = 0 in the equation
ƒ put z = k , to get trace in a plane parallel to
XY-plane at level k
• To find YZ-trace put x = 0 in the equation
ƒ put x = k , to get trace in a plane parallel to
YZ-plane at level k
• To find XZ-trace put y = 0 in the equation
ƒ put y = k , to get trace in a plane parallel to
XZ-plane at level k
Example
Find XY-trace, YZ-trace, XZ-trace of z = x 2 + y 2 .
What will be the traces in planes parallel to XY-plane.
Solution
Done in class.
12.7 4
Sketching standard quadric surfaces
Ellipsoid
x2 y 2 z 2
+
+ =1
a 2 b2 c2
What are
the traces in
planes z=k
for
ƒ
If you draw all
ƒ
traces on one
ƒ
k <c
k=c
k >c
coordinate axis,
can you guess
how the surface
should
look
like?
• See the Matlab illustration in class for
more understanding of different traces
& views
• download Matlab file eg_ellipsoid.m
from WebCT and experiment with it.
12.75
Sketching standard quadric surfaces
Hyperboloid of one sheet
x2 y 2 z 2
+
− =1
a 2 b2 c2
What are
the traces in
If you draw all
planes
traces on one
parallel to
coordinate axis,
XY-plane?
can you guess
i.e. in the
how the surface
planes z=k
should
like?
look
12.76
Sketching standard quadric surfaces
Hyperboloid of two sheets
x2 y2 z 2
− 2 − 2 + 2 =1
a
b
c
What are
If you draw all
the traces in
traces on one
planes z=k
coordinate axis,
for
can you guess
ƒ
how the surface
should
ƒ
look
ƒ
like?
• See the Matlab illustration in class for
more understanding of different traces
& views
• download Matlab file eg_hyper2.m
from WebCT and experiment with it.
k <c
k=c
k >c
12.77
Sketching standard quadric surfaces
Elliptic cone
x2 y 2 z 2
+
− =0
a 2 b2 c2
What are
the traces in
planes
parallel to
XY-plane?
If you draw all
i.e. in the
traces on one
planes z=k
coordinate axis,
can you guess
how the surface
should
look
like?
• See the Matlab illustration in class for
more understanding of different traces
& views
• download Matlab file eg_cone.m from
WebCT and experiment with it.
12.78
Sketching standard quadric surfaces
Elliptic paraboloid
x2 y 2
+
=z
a 2 b2
What are
the traces in
planes
parallel to
XY-plane?
i.e. in the
If you draw all
planes z=k
traces on one
coordinate axis,
can you guess
how the surface
should
look
like?
• See the Matlab illustration in class for
more understanding of different traces &
views
• download Matlab file eg_ paraboloid.m
from WebCT and experiment with it.
12.79
Sketching standard quadric surfaces
Hyperbolic paraboloid
As an example we first sketch z = y − x
2
2
ƒ Traces
ƒ Fitting the traces together
• See the Matlab illustration in class for more
understanding of different traces & views
• download Matlab file eg_ hyp_paraboloid.m
from WebCT and experiment with it.
12.710
Sketching standard quadric surfaces
Hyperbolic paraboloid
y 2 x2
z= 2 − 2
b
a
Similar to above example we have the following graph
See explanation
given in class
where the traces
ƒ
traces in planes parallel to YZ-plane are parabolas opening upwards
ƒ
traces in planes parallel to XZ-plane are parabolas opening downwards
ƒ
traces in planes parallel to XY-plane are hyperbolas
12.711
Techniques for identifying and sketching quadric surfaces
ƒ Bring the given equation in one of the
For
this,
Complete the
square (if needed)
you
standard forms
must
properly
ƒ Compare with standard equations to
understand
the
identify the surface
graphs
of
ƒ Use the graph of standard form to
standard quadric
complete the sketch
surfaces
Use the idea of
translation of
surfaces
(when needed)
Translation of a surface
A change from ( x, y , z ) to ( x − h, y − k , z − l ) means the surface has
translated
ƒ h units along X-axis
ƒ k units along Y-axis
ƒ l units along Z-axis
Note: h, k , l may be negative
Example:
( x − 1) 2 ( y + 2) 2 z 2
+
+ 2 = 1 means an ellipsoid with centre at (1, −2,0) .
2
2
a
b
c
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Identifying quadric surfaces (Examples)
Recall the graph of Ellipsoid
Observations
x2 y 2 z 2
• Equation 2 + 2 + 2 = 1
a
b
c
• Form f ( x, y , z ) = 1 with
ƒ all variables quadratic
ƒ all variables with positive
coefficient
ƒ Centre at (0,0,0)
ƒ X-intercept: ± a
ƒ Y-intercept: ±b
ƒ X-intercept: ±c
Example 12.7.1
Describe the surface 4 x + 4 y + z + 8 y − 4 z = −4
Solution
Done in class
2
2
2
12.713
Identifying quadric surfaces (Examples)
Recall the graph of Hyperboloid of one sheet
Observations
x2 y 2 z 2
• Equation 2 + 2 − 2 = 1
a
b
c
• Form f ( x, y , z ) = 1 with
ƒ all variables quadratic
ƒ one variable with negative
coefficient
ƒ Axis of symmetry corresponds to
variable with negative coefficient
ƒ Opens along axis of symmetry
Example 12.7.2
Sketch the graph of 16 x − 9 y + 36 z = 144 .
Solution
Done in class
2
2
2
12.714
Identifying quadric surfaces (Examples)
Recall the graph of Hyperboloid of two sheets
Observations
x2 y2 z 2
• Equation − 2 − 2 + 2 = 1
a
b
c
• Form f ( x, y , z ) = 1 with
ƒ all variables quadratic
ƒ two variable with negative
coefficient
ƒ Axis of symmetry corresponds to
variable with positive coefficient
ƒ Opens along axis of symmetry
Example 12.7.3
Identify the surface given by x − 4 y − 2 z = 4 .
Solution
Done in class
2
2
2
12.715
Identifying quadric surfaces (Examples)
Recall the graph of elliptic cone
Observations
x2 y 2 z 2
• Equation 2 + 2 − 2 = 0
a
b
c
• Form f ( x, y, z ) = 0 with
ƒ all variables quadratic
ƒ sign of coefficient of one
variable
other two.
ƒ Axis of symmetry corresponds to
variable with different sign
ƒ Opens along axis of symmetry
Example 12.7.4
Sketch the surface x =
Solution
Done in class
y2 + z2 .
different
from
12.716
Identifying quadric surfaces (Examples)
Recall the graph of elliptic paraboloid
Observations
x2 y 2
• Equation 2 + 2 = z with
a
b
ƒ one linear variable
ƒ two
quadratic
with same sign
ƒ Axis of symmetry corresponds to
variable which is linear
Example 12.7.5
Describe the surface x + 2 z − 6 x − y + 10 = 0 .
Solution
Done in class
2
2
variables
12.717
Identifying quadric surfaces (Examples)
Recall the graph of hyperbolic paraboloid
Observations
y 2 x2
• Equation z = 2 − 2 with
b
a
ƒ one linear variable
ƒ two
quadratic
variables
with opposite signs
ƒ Such surfaces are like a horse saddle
ƒ To get an idea about orientation
o imagine the horse saddle and the axis along which the
horse would stand
o e.g. in above figure the horse would stand along Y-axis
Example 12.7.6
x2 y2
Sketch the surface z =
− .
9
4
Solution
Done in class
12.718
Final remarks
The identification tricks learnt here are good short cuts
but the best approach is to sketch the surfaces by using
traces (as we did in the earlier part of these notes.)
Exercise:
For the following equations
• Find the (appropriate) traces
• Sketch these traces
• Use these traces to sketch the surface
(a)
z = − x2 − y 2
(b)
x2 y 2
− −
+ z2 = 0
4 16
Do Qs: 1-36
End of 12.7