2. Quadric Surfaces
A trace of a surface is the intersection of the surface
with a plane. The traces of a given surface S that we
are interested in are traces parallel to the coordinate
planes,
3. Quadric Surfaces
A trace of a surface is the intersection of the surface
with a plane. The traces of a given surface S that we
are interested in are traces parallel to the coordinate
planes, i.e. the intersection of the surface with the
planes, x = c, y = c, or z = c as shown here.
4. Quadric Surfaces
A trace of a surface is the intersection of the surface
with a plane. The traces of a given surface S that we
are interested in are traces parallel to the coordinate
planes, i.e. the intersection of the surface with the
planes, x = c, y = c, or z = c as shown here.
x=c
S
A trace with x = c
5. Quadric Surfaces
A trace of a surface is the intersection of the surface
with a plane. The traces of a given surface S that we
are interested in are traces parallel to the coordinate
planes, i.e. the intersection of the surface with the
planes, x = c, y = c, or z = c as shown here.
x=c y=c
S S
A trace with x = c A trace with y = c
6. Quadric Surfaces
A trace of a surface is the intersection of the surface
with a plane. The traces of a given surface S that we
are interested in are traces parallel to the coordinate
planes, i.e. the intersection of the surface with the
planes, x = c, y = c, or z = c as shown here.
z=c
x=c y=c
S S S
A trace with x = c A trace with y = c A trace with z = c
7. Quadric Surfaces
A trace of a surface is the intersection of the surface
with a plane. The traces of a given surface S that we
are interested in are traces parallel to the coordinate
planes, i.e. the intersection of the surface with the
planes, x = c, y = c, or z = c as shown here.
z=c
x=c y=c
S S S
A trace with x = c A trace with y = c A trace with z = c
Because hand point–plotting is not useful in R3, one
method to visualize a surface is the trace-method.
9. Quadric Surfaces
Parallel traces are similar to the
successive ribs of a structure that
show the contour of its surface.
With the trace–method, using
parallel planes, we construct a
series of traces on the surface S
to reveal the shape of the S.
10. Quadric Surfaces
Parallel traces are similar to the
successive ribs of a structure that
show the contour of its surface.
With the trace–method, using
parallel planes, we construct a
series of traces on the surface S
to reveal the shape of the S.
The planes x = c, y = c, or z = c
produce traces that are
perpendicular to the x, y, and z
axes respectively.
11. Quadric Surfaces
Parallel traces are similar to the
successive ribs of a structure that
show the contour of its surface.
With the trace–method, using
parallel planes, we construct a
series of traces on the surface S
to reveal the shape of the S.
Traces with x = c.
The planes x = c, y = c, or z = c
produce traces that are
perpendicular to the x, y, and z
axes respectively. For example,
the traces of S with the x = c,
as c varies, are plane curves that
stacked perpendicularly to the x–axis.
12. Quadric Surfaces
Parallel traces are similar to the
successive ribs of a structure that
show the contour of its surface.
With the trace–method, using
parallel planes, we construct a
series of traces on the surface S
to reveal the shape of the S.
Traces with x = c.
The planes x = c, y = c, or z = c
produce traces that are
perpendicular to the x, y, and z
x
axes respectively. For example,
the traces of S with the x = c,
as c varies, are plane curves that
stacked perpendicularly to the x–axis.
13. Quadric Surfaces
Parallel traces are similar to the
successive ribs of a structure that
show the contour of its surface.
With the trace–method, using
parallel planes, we construct a
series of traces on the surface S
to reveal the shape of the S.
Traces with x = c.
The planes x = c, y = c, or z = c
produce traces that are
perpendicular to the x, y, and z
x
axes respectively. For example,
the traces of S with the x = c,
as c varies, are plane curves that
stacked perpendicularly to the x–axis.
14. Quadric Surfaces
Parallel traces are similar to the
successive ribs of a structure that
show the contour of its surface.
With the trace–method, using
parallel planes, we construct a
series of traces on the surface S
to reveal the shape of the S.
Traces with x = c.
The planes x = c, y = c, or z = c
produce traces that are
perpendicular to the x, y, and z
x
axes respectively. For example,
the traces of S with the x = c,
as c varies, are plane curves that
stacked perpendicularly to the x–axis.
15. Quadric Surfaces
Parallel traces are similar to the
successive ribs of a structure that
show the contour of its surface.
With the trace–method, using
parallel planes, we construct a
series of traces on the surface S
to reveal the shape of the S.
Traces with x = c.
The planes x = c, y = c, or z = c
produce traces that are
perpendicular to the x, y, and z
x
axes respectively. For example,
the traces of S with the x = c, S
as c varies, are plane curves that
stacked perpendicularly to the x–axis.
16. Quadric Surfaces
We will examine quadratic surfaces, i.e. the graphs of
2nd degree equations of the form
Ax2 + By2 + Cz2 + Dx + Ey + Fz + G = 0. We will study
the ones that can be transformed to one of the following
forms (with x2 as the leading term):
x2 y2 z2 x2 y2
+ + =1 z =
2
+ 2
a2 b2 c 2 a2 b
All positive square terms Constant term = 0
x2 y2 z2
+ – =1 x2 y2
a 2
b 2
c 2
z = + Assuming no z2
One negative square term a2 b2
x2 y2 z2 x2
– – =1 y2 Assuming no z2
a2
b2
c 2 z = –
a2 b2
Two negative square terms
To sketch then surfaces, if possible, find the square-
sums who give traces that are ellipses or circles.
17. Quadric Surfaces
To graph the trace of f(x, y, z) = 0 with y = c,
for example, set y = c to obtain f(x, c, z) = 0,
an xz-equation.
18. Quadric Surfaces
To graph the trace of f(x, y, z) = 0 with y = c,
for example, set y = c to obtain f(x, c, z) = 0,
an xz-equation. The trace is the 2D graph of
f(x, c, z) = 0 in the plane y = c.
19. Quadric Surfaces
To graph the trace of f(x, y, z) = 0 with y = c,
for example, set y = c to obtain f(x, c, z) = 0,
an xz-equation. The trace is the 2D graph of
f(x, c, z) = 0 in the plane y = c. The traces of quadric
surfaces, after setting a variable to constants, are
conic (2nd degree) sections in the planes.
20. Quadric Surfaces
To graph the trace of f(x, y, z) = 0 with y = c,
for example, set y = c to obtain f(x, c, z) = 0,
an xz-equation. The trace is the 2D graph of
f(x, c, z) = 0 in the plane y = c. The traces of quadric
surfaces, after setting a variable to constants, are
conic (2nd degree) sections in the planes.
Circles: Ellipses: b
x2+y2=r2 r x2 + y2 a
a2 =1
b2
Hyperbolas: b
Parabolas:
x2 – y
2 a
a2 b2 = 1 y = ax2
Figures are shown centered at (0, 0), in gereral they centered at (h, k).
22. Quadric Surfaces
x2 + y2 + z2 = 1
The graphs of a2 b2 c2
Isolate the sum of two square terms, e.g. x2 and y2:
x2 y2 = 1 – z2
a2 + b2 c2
23. Quadric Surfaces
The graphs of a2 x2 + y2 + z2 = 1
b2 c2
Isolate the sum of two square terms, e.g. x2 and y2:
x2 y2 = 1 – z2
a2 + b2 c2
If z > |c|, we get
x2 + y2 = negative #,
a2 b2
so there is no graph in the
z = c plane.
24. Quadric Surfaces
The graphs of a2 x2 + y2 + z2 = 1
b2 c2
Isolate the sum of two square terms, e.g. x2 and y2:
x2 y2 = 1 – z2
a2 + b2 c2
If z > |c|, we get
x2 + y2 = negative #,
a2 b2
so there is no graph in the
z = c plane.
If z = |c|, we get 2x2 + y2 we get the two poles.
2 =
a b
0,
25. Quadric Surfaces
The graphs of a2 x2 + y2 + z2 = 1
b2 c2
Isolate the sum of two square terms, e.g. x2 and y2:
x2 y2 = 1 – z2 z
a2 + b2 c2
c
If z > |c|, we get
b
x + y = negative #,
2 2 a y
a2 b2 x
z=0
so there is no graph in
the
z = c plane. get x2
If z = |c|, we y2
a 2 +
b 2 = we get the two poles.
y2 0,
If z < |c|, we get x 2 + b2 = positive #,
2
a
which are ellipses, with the trace z = 0 being the
largest elliptic trace forming the equator.
27. Quadric Surfaces
x2 y2 z2 = 1
The graphs of + – 2
a 2
b 2
c
Example A. Graph x2 + y2 – z2 = 1
28. Quadric Surfaces
x2 y2 z2 = 1
The graphs of + – 2
a 2
b 2
c
Example A. Graph x2 + y2 – z2 = 1
Solve for the sum–of–two–squares: x2 + y2 = 1 + z2
29. Quadric Surfaces
x2 y2 z2 = 1
The graphs of + – 2
a 2
b 2
c
Example A. Graph x2 + y2 – z2 = 1
Solve for the sum–of–two–squares: x2 + y2 = 1 + z2
Let z varies, we get
circles in the z-planes.
30. Quadric Surfaces
x2 y2 z2 = 1
The graphs of + – 2
a 2
b 2
c
Example A. Graph x2 + y2 – z2 = 1
Solve for the sum–of–two–squares: x2 + y2 = 1 + z2
Let z varies, we get
circles in the z-planes.
When z = 0, we get the
smallest circle
31. Quadric Surfaces
x2 y2 z2 = 1
The graphs of + – 2
a 2
b 2
c
Example A. Graph x2 + y2 – z2 = 1
Solve for the sum–of–two–squares: x2 + y2 = 1 + z2
Let z varies, we get
circles in the z-planes.
When z = 0, we get the
smallest circle and as
|z| gets larger, the circles
get larger.
32. Quadric Surfaces
x2 y2 z2 = 1
The graphs of + – 2
a 2
b 2
c
Example A. Graph x2 + y2 – z2 = 1
Solve for the sum–of–two–squares: x2 + y2 = 1 + z2
Let z varies, we get
circles in the z-planes.
When z = 0, we get the
smallest circle and as
|z| gets larger, the circles
get larger. These circles
are stacked vertically.
33. Quadric Surfaces
x2 y2 z2 = 1
The graphs of + – 2
a 2
b 2
c
Example A. Graph x2 + y2 – z2 = 1
Solve for the sum–of–two–squares: x2 + y2 = 1 + z2
Let z varies, we get
circles in the z-planes.
When z = 0, we get the
smallest circle and as
|z| gets larger, the circles y –z
2 2
=1
get larger. These circles
are stacked vertically.
Set x = 0, we get the
hyperbolic outline of the
"stacked circles“.
34. Quadric Surfaces
x2 y2 z2 = 1
The graphs of + – 2
a 2
b 2
c
Example A. Graph x2 + y2 – z2 = 1
Solve for the sum–of–two–squares: x2 + y2 = 1 + z2
Let z varies, we get
circles in the z-planes.
When z = 0, we get the
smallest circle and as
|z| gets larger, the circles y –z =1
2 2
get larger. These circles
are stacked vertically.
Set x = 0, we get the
hyperbolic outline of the
"stacked circles“.
35. Quadric Surfaces
x2 y2 z2
Summary for the graphs of a2 + b2 – c2 = 1
36. Quadric Surfaces
x2 y2 z2
Summary for the graphs of a2 + b2 – c2 = 1
x2 y2 z2
i. Solve for the sum of squares: a2 + b2 = 1 + c2
37. Quadric Surfaces
x2 y2 z2
Summary for the graphs of a2 + b2 – c2 = 1
x2 y2 z2
i. Solve for the sum of squares: a2 + b2 = 1 + c2
ii. Let z varies,
for a fixed k, we get
an ellipses in the z = k planes,
38. Quadric Surfaces
x2 y2 z2
Summary for the graphs of a2 + b2 – c2 = 1
x2 y2 z2
i. Solve for the sum of squares: a2 + b2 = 1 + c2
ii. Let z varies,
for a fixed k, we get
an ellipses in the z = k planes,
with z = 0 giving the smallest
ellipse x2/a2+y2/b2 =1 as the waist
and progressively larger ellipses
stacked vertically.
39. Quadric Surfaces
x2 y2 z2
Summary for the graphs of a2 + b2 – c2 = 1
x2 y2 z2
i. Solve for the sum of squares: a2 + b2 = 1 + c2
ii. Let z varies, z traces:
x 2 2
y k2
for a fixed k, we get a 2 + b = 1+
2 c2
an ellipses in the z = k planes, z = k
with z = 0 giving the smallest
ellipse x2/a2+y2/b2 =1 as the waist
and progressively larger ellipses
stacked vertically.
x2 y2 – z2 = 1
a2 + b2 c2
40. Quadric Surfaces
x2 y2 z2
Summary for the graphs of a2 + b2 – c2 = 1
x2 y2 z2
i. Solve for the sum of squares: a2 + b2 = 1 + c2
ii. Let z varies, z traces:
x 2 2
y k2
for a fixed k, we get a 2 + b = 1+
2 c2
an ellipses in the z = k planes, z = k
with z = 0 giving the smallest
ellipse x2/a2+y2/b2 =1 as the waist
and progressively larger ellipses
stacked vertically.
Setting x = 0 or y =0, we have the
hyperbolic outlines of the surface.
x2 y2 – z2 = 1
a2 + b2 c2
41. Quadric Surfaces
x2 y2 z2
Summary for the graphs of a2 + b2 – c2 = 1
x2 y2 z2
i. Solve for the sum of squares: a2 + b2 = 1 + c2
ii. Let z varies, z traces:
x 2 2
y k
2
for a fixed k, we get a 2 + b = 1+ c
2 2
an ellipses in the z = k planes, z = k
with z = 0 giving the smallest
ellipse x2/a2+y2/b2 =1 as the waist
and progressively larger ellipses
stacked vertically.
Setting x = 0 or y =0, we have the
hyperbolic outlines of the surface.
Exercise: Draw a few traces along the x2 y2 – z2
a2 + b2 c2 = 1
x and y axes and find their equations.
43. Quadric Surfaces
If we fix the a, b and c in
x2 y2 z2
+ 2 = 1+ 2
a2
b c
but we replace 1 by k so we have
x2 + y2 z2
= k + c2
a2 b2
x2 + y2 = 1 + z2
44. Quadric Surfaces
If we fix the a, b and c in
x2 y2 z2
+ 2 = 1+ 2
a2
b c
but we replace 1 by k so we have
x2 + y2 z2
= k + c2
a2 b2
then as k → 0 the waists of the
corresponding surfaces shrink, x2 + y2 = 1 + z2
45. Quadric Surfaces
If we fix the a, b and c in
x2 y2 z2
+ 2 = 1+ 2
a2
b c
but we replace 1 by k so we have
x2 + y2 z2
= k + c2
a2 b2
then as k → 0 the waists of the
corresponding surfaces shrink, x 2
+ y2 = 1 + z2
and if k = 0 we have the equation
z2 = x2 y2
+ 2
c 2
a2
b
whose waist is just one point as in
the case of z2 = x2 + y2 shown here.
x2 + y2 = 0 + z2
46. Quadric Surfaces
If we fix the a, b and c in
x2 y2 z2
+ 2 = 1+ 2
a2
b c
but we replace 1 by k so we have
x2 + y2 z2
= k + c2
a2 b2
then as k → 0 the waists of the
corresponding surfaces shrink, x 2
+ y2 = 1 + z2
and if k = 0 we have the equation
z2 = x2 y2
+ 2
c 2
a2
b
z = ±y
whose waist is just one point as in
the case of z2 = x2 + y2 shown here.
The x = 0 traces give the outline
z = ±y in the yz–plane. x +y
2 2
= 0 + z2
48. Quadric Surfaces
z2 x2 y2
The graphs of a2 – b2 – c2 = 1
x2 y2 = z2 – 1
Solve for the sum of squares: 2 + 2
a b c2
49. Quadric Surfaces
z2 x2 y2
The graphs of a2 – b2 – c2 = 1
x2 y2 = z2 – 1
Solve for the sum of squares: 2 + 2
a b c2
x2 + y2 z2
it’s equation if we set k = –1 in 2 = k
+ 2
a2
b c
50. Quadric Surfaces
z2 x2 y2
The graphs of a2 – b2 – c2 = 1
x2 y2 = z2 – 1
Solve for the sum of squares: 2 + 2
a b c2
x2 + y2 z2
it’s equation if we set k = –1 in 2 = k
+ 2
a2
b c
In this case, we have
non–empty level curves if
z2 – 1 ≥ 0, i.e. |z| ≥ c.
c2
51. Quadric Surfaces
z2 x2 y2
The graphs of a2 – b2 – c2 = 1
x2 y2 = z2 – 1
Solve for the sum of squares: 2 + 2
a b c2
x2 + y2 z2
it’s equation if we set k = –1 in 2 = k
+ 2
a2
b c
In this case, we have
non–empty level curves if
z2 – 1 ≥ 0, i.e. |z| ≥ c.
c2
For |z| > c, we get large and
larger ellipses .
52. Quadric Surfaces
z2 x2 y2
The graphs of a2 – b2 – c2 = 1
x2 y2 = z2 – 1
Solve for the sum of squares: 2 + 2
a b c2
x2 + y2 z2
it’s equation if we set k = –1 in 2 = k
+ 2
a2
b c
In this case, we have
non–empty level curves if
z2 – 1 ≥ 0, i.e. |z| ≥ c.
c2
For |z| > c, we get large and
larger ellipses .The surface
x2 + y2 = z2 – 1
is shown here with
x +y
2 2
= z2 – 1
some x and z traces.
53. Quadric Surfaces
Summary of The graphs of z2 = x2 + y2 + k:
z2 = x2 + y2 + 1 z2 = x2 + y2 + 0 z2 = x2 + y2 – 1
x2 y2 z2 x2 y2
+ + =1 z =
2
+ 2
a 2
b2
c 2 a2 b
All positive square terms Constant term = 0
x2 y2 z2 Hence we have addressed
+ – =1
a2 b2 c2
One negative square term cover these cases.
x2 y2 z2 The following is a summary
– – =1
a2
b2
c 2 and the remaining cases.
Two negative square terms
54. Quadric Surfaces
x2 y2 z2 x2 y2 z2
+ + =1 + – =0
a2 b2 c 2
a2
b 2
c 2
All positive square terms Constant term = 0
x2 y2 z2 z2 x2 y2
+ – =1 – – =1
a2 b2 c 2 a2
b 2
c 2
One negative square term Two negative square terms
The remaining cases have a missing square term.
We will assume that there is no z2. We may in these
cases, express z as a function of x and y.
Again, there are two cases; the sum or the difference
of squares.
x2 – y2
z = x2 + y2
2 2
z= 2
a b a b2
Assuming no z2 Assuming no z2
56. Quadric Surfaces
Traces with z = c are called level curves since z
coordinates give the height of the poitnt.
Example A. a. Sketch the traces of z – x2 – y2 = 0
with z = –1, 0, 1, 4.
57. Quadric Surfaces
Traces with z = c are called level curves since z
coordinates give the height of the poitnt.
Example A. a. Sketch the traces of z – x2 – y2 = 0
with z = –1, 0, 1, 4.
Set z = x2 + y2, let z = –1, 0, 1, 4 and we get the
following xy-equations:
z = –1: –1 = x2 + y2
z = 0: 0 = x2 + y2
z = 1: 1 = x2 + y2
z = 4: 4 = x2 + y2
58. Quadric Surfaces
Traces with z = c are called level curves since z
coordinates give the height of the poitnt.
Example A. a. Sketch the traces of z – x2 – y2 = 0
with z = –1, 0, 1, 4.
Set z = x2 + y2, let z = –1, 0, 1, 4 and we get the
following xy-equations:
z = –1: –1 = x2 + y2 (no solution)
z = 0: 0 = x2 + y2
z = 1: 1 = x2 + y2
z = 4: 4 = x2 + y2
59. Quadric Surfaces
Traces with z = c are called level curves since z
coordinates give the height of the poitnt.
Example A. a. Sketch the traces of z – x2 – y2 = 0
with z = –1, 0, 1, 4.
Set z = x2 + y2, let z = –1, 0, 1, 4 and we get the
following xy-equations:
z = –1: –1 = x2 + y2 (no solution)
z = 0: 0 = x2 + y2
z = 1: 1 = x2 + y2
z = 4: 4 = x2 + y2 z=0
y
x
60. Quadric Surfaces
Traces with z = c are called level curves since z
coordinates give the height of the poitnt.
Example A. a. Sketch the traces of z – x2 – y2 = 0
with z = –1, 0, 1, 4.
Set z = x2 + y2, let z = –1, 0, 1, 4 and we get the
following xy-equations:
z = –1: –1 = x2 + y2 (no solution) z=4
z = 0: 0 = x2 + y2
z=1
z = 1: 1 = x2 + y2
z = 4: 4 = x2 + y2 z=0
y
x
61. Quadric Surfaces
Traces with z = c are called level curves since z
coordinates give the height of the poitnt.
Example A. a. Sketch the traces of z – x2 – y2 = 0
with z = –1, 0, 1, 4.
Set z = x2 + y2, let z = –1, 0, 1, 4 and we get the
following xy-equations:
z = –1: –1 = x2 + y2 (no solution) z=4
z = 0: 0 = x2 + y2
z=1
z = 1: 1 = x2 + y2
z = 4: 4 = x2 + y2 z=0
We note that for z < 0, there is y
no trace and as z →∞, we have
successive higher and larger x
circles.
62. Quadric Surfaces
Next we reconstruct the traces along the x–axis for
the outline of the vertically stacked circles.
63. Quadric Surfaces
Next we reconstruct the traces along the x–axis for
the outline of the vertically stacked circles.
b. Sketch the traces of z – x2 – y2 = 0 with
x = –2, –1, 0, 1, 2.
64. Quadric Surfaces
Next we reconstruct the traces along the x–axis for
the outline of the vertically stacked circles.
b. Sketch the traces of z – x2 – y2 = 0 with
x = –2, –1, 0, 1, 2.
Set z = x2 + y2, let x = –2, –1, 0, 1, 2 and get the
following yz-equations:
65. Quadric Surfaces
Next we reconstruct the traces along the x–axis for
the outline of the vertically stacked circles.
b. Sketch the traces of z – x2 – y2 = 0 with
x = –2, –1, 0, 1, 2.
Set z = x2 + y2, let x = –2, –1, 0, 1, 2 and get the
following yz-equations:
x = –2: z = 4 + y2
x = –1: z = 1 + y2
x = 0: z = y2
x = 1: z = 1 + y2
x = 2: z = 4 + y2
66. Quadric Surfaces
Next we reconstruct the traces along the x–axis for
the outline of the vertically stacked circles.
b. Sketch the traces of z – x2 – y2 = 0 with
x = –2, –1, 0, 1, 2.
Set z = x2 + y2, let x = –2, –1, 0, 1, 2 and get the
following yz-equations:
x = –2: z = 4 + y2
x = –1: z = 1 + y2
x = 0: z = y2
x = 1: z = 1 + y2
x = 2: z = 4 + y2
x=0
x
67. Quadric Surfaces
Next we reconstruct the traces along the x–axis for
the outline of the vertically stacked circles.
b. Sketch the traces of z – x2 – y2 = 0 with
x = –2, –1, 0, 1, 2.
Set z = x2 + y2, let x = –2, –1, 0, 1, 2 and get the
following yz-equations:
x = –2: z = 4 + y2
x = –1: z = 1 + y2
x = 0: z = y2
x = 1: z = 1 + y2
x = 2: z = 4 + y2
x=0
x=1
x
68. Quadric Surfaces
Next we reconstruct the traces along the x–axis for
the outline of the vertically stacked circles.
b. Sketch the traces of z – x2 – y2 = 0 with
x = –2, –1, 0, 1, 2.
Set z = x2 + y2, let x = –2, –1, 0, 1, 2 and get the
following yz-equations:
x = –2: z = 4 + y2
x = –1: z = 1 + y2
x = 0: z = y2
x = 1: z = 1 + y2
x = 2: z = 4 + y2
x=0
x=1
x x=2
69. Quadric Surfaces
Next we reconstruct the traces along the x–axis for
the outline of the vertically stacked circles.
b. Sketch the traces of z – x2 – y2 = 0 with
x = –2, –1, 0, 1, 2.
Set z = x2 + y2, let x = –2, –1, 0, 1, 2 and get the
following yz-equations:
x = –2: z = 4 + y2
x = –1: z = 1 + y2
x = 0: z = y2
x = 1: z = 1 + y2 x = –2
x = 2: z = 4 + y2 x = –1
x=0
x=1
x x=2
70. Quadric Surfaces
Next we reconstruct the traces along the x–axis for
the outline of the vertically stacked circles.
b. Sketch the traces of z – x2 – y2 = 0 with
x = –2, –1, 0, 1, 2.
Set z = x2 + y2, let x = –2, –1, 0, 1, 2 and get the
following yz-equations:
x = –2: z = 4 + y2
x = –1: z = 1 + y2
x = 0: z = y2
x = 1: z = 1 + y2 x = –2
x = 2: z = 4 + y2 x = –1
We put parts a and b together x=0
to visualize the surface. x=1
x x=2
76. Quadric Surfaces
x2 y2 z2 x2 y2 z2
+ + =1 + – =0
a2 b2 c 2
a2
b c 2 2
All positive square terms Constant term = 0
x2 y2 z2 z2 x2 y2
+ – =1 – – =1
a2 b2 c 2 a2
b 2
c 2
One negative square term Two negative square terms
x2 y2 x2 y2
z = + z = –
a 2
b 2 a 2
b2
Assuming no z2 Assuming no z2
77. Quadric Surfaces
x2 y2 z2 x2 y2 z2
+ + =1 + – =0
a2 b2 c 2
a2
b c 2 2
All positive square terms Constant term = 0
x2 y2 z2 z2 x2 y2
+ – =1 – – =1
a2 b2 c 2 a2
b 2
c 2
One negative square term Two negative square terms
x2 y2 x2 y2
z = + z = –
a 2
b 2 a 2
b2
Assuming no z2 Assuming no z2
The graph of z = x2 – y2
78. Quadric Surfaces
x2 y2 z2 x2 y2 z2
+ + =1 + – =0
a2 b2 c 2
a2
b c 2 2
All positive square terms Constant term = 0
x2 y2 z2 z2 x2 y2
+ – =1 – – =1
a2 b2 c 2 a2
b 2
c 2
One negative square term Two negative square terms
x2 y2 x2 y2
z = + z = –
a 2
b 2 a 2
b2
Assuming no z2 Assuming no z2
The graph of z = x2 – y2
There is no ellipse to track.
79. Quadric Surfaces
x2 y2 z2 x2 y2 z2
+ + =1 + – =0
a2 b2 c 2
a2
b c 2 2
All positive square terms Constant term = 0
x2 y2 z2 z2 x2 y2
+ – =1 – – =1
a2 b2 c 2 a2
b 2
c 2
One negative square term Two negative square terms
x2 y2 x2 y2
z = + z = –
a 2
b 2 a 2
b2
Assuming no z2 Assuming no z2
The graph of z = x2 – y2
There is no ellipse to track.
We may track parabolas by
letting x = c, we get z = c2 – y2
which are parabolas with
maximum at (0, c) in the x planes.
80. Quadric Surfaces
x2 y2 z2 x2 y2 z2
+ + =1 + – =0
a2 b2 c 2
a2
b c 2 2
All positive square terms Constant term = 0
x2 y2 z2 z2 x2 y2
+ – =1 – – =1
a2 b2 c 2 a2
b 2
c 2
One negative square term Two negative square terms
x2 y2 x2 y2
z = + z = –
a 2
b 2 a 2
b2
Assuming no z2 Assuming no z2
The graph of z = x2 – y2 Traces with x = c.
There is no ellipse to track.
We may track parabolas by
x
letting x = c, we get z = c – y
2 2
which are parabolas with
maximum at (0, c) in the x planes.
81. Quadric Surfaces
x2 y2 z2 x2 y2 z2
+ + =1 + – =0
a2 b2 c 2
a2
b c 2 2
All positive square terms Constant term = 0
x2 y2 z2 z2 x2 y2
+ – =1 – – =1
a2 b2 c 2 a2
b 2
c 2
One negative square term Two negative square terms
x2 y2 x2 y2
z = + z = –
a 2
b 2 a 2
b2
Assuming no z2 Assuming no z2
The graph of z = x2 – y2 Traces with x = c.
There is no ellipse to track.
We may track parabolas by
x
letting x = c, we get z = c – y
2 2
which are parabolas with
maximum at (0, c) in the x planes.
82. Quadric Surfaces
x2 y2 z2 x2 y2 z2
+ + =1 + – =0
a2 b2 c 2
a2
b c 2 2
All positive square terms Constant term = 0
x2 y2 z2 z2 x2 y2
+ – =1 – – =1
a2 b2 c 2 a2
b 2
c 2
One negative square term Two negative square terms
x2 y2 x2 y2
z = + z = –
a 2
b 2 a 2
b2
Assuming no z2 Assuming no z2
The graph of z = x2 – y2 Traces with x = c.
There is no ellipse to track.
We may track parabolas by
x
letting x = c, we get z = c – y
2 2
which are parabolas with
maximum at (0, c) in the x planes.
84. Quadric Surfaces
If c = 0, the vertex is at z = 0. The larger the |x| is,
the higher the vertex z = c thus the parabola is.
These parabolas are stacked in the x-direction.
x
Traces with x = c.
85. Quadric Surfaces
If c = 0, the vertex is at z = 0. The larger the |x| is,
the higher the vertex z = c thus the parabola is.
These parabolas are stacked in the x-direction.
x
Traces with x = c.
86. Quadric Surfaces
If c = 0, the vertex is at z = 0. The larger the |x| is,
the higher the vertex z = c thus the parabola is.
These parabolas are stacked in the x-direction.
We obtain the saddle with the origin as a saddle point.
x
Traces with x = c.
87. Quadric Surfaces
If c = 0, the vertex is at z = 0. The larger the |x| is,
the higher the vertex z = c thus the parabola is.
These parabolas are stacked in the x-direction.
We obtain the saddle with the origin as a saddle point.
The saddle point is the pass between two hills.
x
Traces with x = c.
The surface z = x2 – y2 with (0, 0, 0) as the saddle point
