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A Quadratic surfaces
In this appendix we will study several families of so-called quadratic surfaces, namely surfaces z = f (x, y) which are defined by equations of the type
Ax^2 + By^2 + Cz^2 + Dxy + Exz + F yz + Hx + Iy + Jz + K = 0, (A.1)
with A, B, C, D, E, F, H, I, J and K being fixed real constants and x, y, z being variables. These surfaces are said to be quadratic because all possible products of two of the variables x, y, z appear in (A.1). In fact, by suitable translations and rotations of the x, y and z coordinate axes it is possible to simplify the equation (A.1) and hence classify all the possible surfaces into the following ten types:
- Spheres
- Ellipsoids
- Hyperboloids of one sheet
- Hyperboloids of two sheets
- Cones
- Elliptic paraboloids
- Hyperbolic paraboloids
- Parabolic cylinders
- Elliptic cylinders
- Hyperbolic cylinders
It is a requirement of this calculus course that you should be able to recognize, classify and sketch at least some of these surfaces (we will use some of them when doing triple integrals). The best way to do that is to look for identifying signs which tell you what kind of surface you are dealing with. Those signs are:
- The intercepts: the points at which the surface intersects the x, y and z axes.
- The traces: the intersections with the coordinate planes (xy-, yz- and xz- plane).
- The sections: the intersections with general planes.
- The centre: (some have it, some not).
- If they are bounded or not.
- If they are symmetric about any axes or planes.
A.1 Spheres
A sphere is a quadratic surface defined by the equation:
(x − x 0 )^2 + (y − y 0 )^2 + (z − z 0 )^2 = r^2. (A.2)
the point (x 0 , y 0 , z 0 ) in the 3D space is the centre of the sphere. The points (x, y, z) in the sphere are all points whose distance to the centre is given by r. Therefore:
(a) The intercepts of the sphere with the x, y, z-axes are the points x 0 ± r, 0 , 0), (0, y 0 ± r, 0) and (0, 0 , z 0 ± r.
(b) The traces of the sphere are circles or radius r.
(c) The sections of the sphere are circles of radius r′^ < r.
(d) The sphere is bounded.
(e) Spheres are symmetric about all coordinate planes.
Rule: If we expand the square terms in equation (A.2) we obtain the following equation:
x^2 + y^2 + z^2 − 2 x 0 x − 2 y 0 y − 2 z 0 z + x^20 + y 02 + z 02 − r^2 = 0, (A.3)
Comparing this equation with the general formula (A.1) we see it has the form
x^2 + y^2 + z^2 + Hx + Iy + Jz + K = 0, (A.4)
with H, I, J and K being some real constants. Therefore, the rule to recognize a sphere is the following: any quadratic equation such that the coefficients of the x^2 , y^2 and z^2 terms are equal and that no other quadratic terms exist corresponds to a sphere.
Sphere centered at the origin.
The sphere is a perfect example of a surface of revolution. A surface of revolution is a surface which can be generated by rotating a particular curve about a particular coordinate axis. For example, one way to generate the sphere of the picture above is to take the circle x^2 + y^2 = 1 and rotate it about the z-axis.
A.3 Hyperboloids of one sheet
A hyperboloid of one sheet is parameterized by the equation
x^2 a^2
y^2 b^2
z^2 c^2
= 1. (A.9)
(a) The intercepts of the hyperboloid of one sheet with the x, y, z-axes are the points (±a, 0 , 0), (0, ±b, 0). Notice that there is no intersection with the z- axis. The reason is that if we set x = y = 0 in the equation (A.9) we obtain the condition z^2 = −c^2 which admits no real solution for real c.
(b) The traces of the hyperboloid of one sheet are ellipses in the xy-plane
x^2 a^2
y^2 b^2
= 1 for z = 0 (xy-plane), (A.10)
and hyperbolas in the xz- and yz-planes:
x^2 a^2
z^2 c^2
= 1 for y = 0 (xz-plane), (A.11)
y^2 b^2
z^2 c^2
= 1 for x = 0 (yz-plane). (A.12)
(c) The sections of the hyperboloid of one sheet are ellipses for planes parallel to the xy plane and hyperbolas for planes parallel to the yz- and xz-planes.
(d) The hyperboloid of one sheet is not bounded.
(e) The centre of the hyperboloid of one sheet in the picture is the origin of coordinates. It can be changed by shifting x, y, z by constant amounts.
(f ) The hyperboloid of one sheet is symmetric about all coordinate planes.
Hyperboloid of one sheet centered at the origin.
A.4 Hyperboloid of two sheets
A hyperboloid of two sheets is a surface generated by the points satisfying the equation x^2 a^2
y^2 b^2
z^2 c^2
= − 1. (A.13)
(a) The intercepts of the hyperboloid of two sheets with the x, y, z-axes are the points (0, 0 , ±c). There are no intersections with the x, y-axes. The reason is that if we set z = x = 0 in the equation (A.9) we obtain the condition y^2 = −b^2 which can not be fulfilled for any real values of y and b. Analogously if we set z = y = 0 we obtain x^2 = −a^2 which also has no real solutions.
(b) In this case we have two sheets, in contrast to all examples we have seen so far. The reason is that the equation (A.13) implies
x^2 a^2
y^2 b^2
z^2 c^2
− 1. (A.14)
This equation can only be solved if z^2 /c^2 − 1 ≥ 0, which implies that |z| ≥ c.
Hyperboloid of two sheets centered at the origin.
(c) The traces of the hyperboloid of two sheets are hyperbolas in the xz- and yz-planes:
x^2 a^2
z^2 c^2
= − 1 for y = 0 (xz-plane), (A.15)
y^2 b^2
z^2 c^2
= − 1 for x = 0 (yz-plane). (A.16)
(d) The sections of the hyperboloid of two sheets are hyperbolas for any planes parallel to the xz- or yz-planes and ellipses for planes parallel to the xy-plane with |z| > c.
Cone centered at the origin.
Rule: Any quadratic surface such that: the coefficients of x^2 , y^2 and z^2 are dif- ferent, one of the coefficients is negative and two of the coefficients are positive, no other quadratic terms appear and no constant term appears describes a cone.
A.6 Elliptic paraboloids
A quadratic surface is said to be an elliptic paraboloid is it satisfies the equation
x^2 a^2
y^2 b^2
= z. (A.21)
(a) The only intercept of the elliptic paraboloid with the x, y, z-axes is the origin of coordinates (0, 0 , 0).
(b) The traces of the paraboloid are parabolas in the xz- and yz-planes
z =
x^2 a^2
for y = 0 (xz-plane), (A.22)
z =
y^2 b^2
for x = 0 (yz-plane), (A.23)
and the origin (0, 0) in the xy-plane corresponding to the equation:
x^2 a^2
y^2 b^2
= 0 for z = 0 (xy-plane), (A.24)
(c) The sections of the elliptic paraboloid with any planes parallel to the xz- or yz-planes are parabolas. The sections with planes parallel to the xy-plane are ellipses (or circles if a = b).
(d) The paraboloid is not bounded from above.
(e) The centre of the paraboloid in the picture is the origin of coordinates. It can be changed by shifting x, y, z by constant amounts.
(f ) The elliptic paraboloid is symmetric about the xz- and yz-planes.
Rule: Any quadratic surface which contains: only linear terms in one of the vari- ables (in our example z), quadratic terms in the other two variables with coefficients of the same sign and no constant term is an elliptic paraboloid.
Elliptic paraboloid centered at the origin.
A.7 Hyperbolic paraboloids
A hyperbolic paraboloid is defined by the equation
x^2 a^2
y^2 b^2
= z, (A.25)
(a) The only intercept of the hyperbolic paraboloid with the x, y, z-axes is the origin of coordinates (0, 0 , 0).
(b) The traces of the paraboloid are parabolas in the xz- and yz-planes
z =
x^2 a^2
for y = 0 (xz-plane), (A.26)
z = −
y^2 b^2
for x = 0 (yz-plane), (A.27)
and two lines in the xy-plane corresponding to the equation:
x^2 a^2
y^2 b^2
= 0 for z = 0 (xy-plane), (A.28)
Parabolic cylinder centered at the origin.
A.9 Elliptic cylinders
An elliptic cylinder is a quadratic surface described by the equation
x^2 a^2
y^2 b^2
= 1. (A.30)
As in the previous case, this surface does not depend explicitly on the coordinate z. Equation (A.30) describes an ellipse in the xy-plane (or a circle if a = b). Therefore an elliptic cylinder is the surface generated by the translation of the ellipse (A.30) along the z direction.
Rule: Again we have here a surface easy to recognize, since its equation does not depend explicitly on one of the variables and is the equation of an ellipse in the other two variables.
Elliptic cylinder centered at the origin.
A.10 Hyperbolic cylinders
A hyperbolic cylinder is a quadratic surface parameterized by the equation
x^2 a^2
y^2 b^2
= 1. (A.31)
The equation above describes a hyperbola in the xy-plane. Therefore this surface is simply generated by the translation of the hyperbola (A.31) along the z-direction.
Rule: The rule to recognize this kind of surface will be the same as in the previous case, with the difference that instead of an ellipse we have now a hyperbola.
Hyperbolic cylinder centered at the origin.
General rule: In general, cylinders are surfaces whose corresponding quadratic equation does not involve z explicitly. Therefore we must be told that they are in 3D to recognize a cylindrical surface.
References
[1] R. A. Adams, Calculus: A complete course (Addison Wesley).
[2] T. Apostol, Calculus (Wiley).
[3] S. N. Salas and E. Hille, Calculus: One and several variables (Wiley).
[4] H. Anton, Calculus (Wiley).
