Quadric surface
Quadric surfaces are the graphs of any equation that can be put into the general form
a 11x 2 + a 22 y 2 + a 33 z 2 + 2a 12 xy + 2a 13 xz + 2a 23 yz + 2a 10 x + 2a 20 y + 2a 30 z + a 00 = 0
where a ij ∈ R , i, j = 0,1,2,3 are real constants and
a 112 + a 22 2 + a 33 2 + a 12 2 + a 13 2 + a 23 2 ≠ 0.
There is no way that we can possibly list all of them, but there are some
standard equations so here is a list of some of the more common quadric surfaces.
However, in order to make the discussion in this section a little easier we have
chosen to concentrate on surfaces that are “centred” on the origin in one way or
another.
Ellipsoid
The general equation of an ellipsoid is given by:
x 2 y2 z2
+
+
= 1 , a,b,c>0
(1)
a 2 b2 c2
where a, b, c are fixed positive real numbers.
The points (a,0,0), (0,b,0) and (0,0,c) lie on the surface and the line segments from the
origin to these points are called the semi-principal axes. These correspond to the
semi-major axis and semi-minor axis of the appropriate ellipses.
Here is a sketch of a typical ellipsoid.
If a=b=c then we will have a sphere.
Notice that we only gave the equation for the ellipsoid that has been centred on the
origin.
Using spherical coordinates, where ϕ is the colatitude, or zenith, and θ is the
longitude in 2π , or azimuth, the ellipsoid (1) can be parameterized by:
⎧ x = a sin ϕ cos θ
⎪
⎨ y = b sin ϕ sin θ, 0 ≤ θ ≤ 2π; 0 ≤ ϕ ≤ π
⎪ z = c cos ϕ
⎩
The volume of an ellipsoid is given by the formula
V=
4
πabc .
3
Cone
The general equation of a cone is
x2
a2
Here is a sketch of a typical cone.
+
y2
b2
=
z2
c2
,
a , b, c >0.
(2)
The parametrized equations of the cone (2) are:
⎧ x = au cos v
⎪
u ∈ R , 0 ≤ v ≤ 2π
⎨ y = bu sin v,
⎪ z = cu
⎩
Note that this is the equation of a cone that will open along the z-axis. To get the
equation of a cone that opens along one of the other axes all we need to do is make a
slight modification of the equation. In the case of a cone the variable that sits by itself
on one side of the equal sign will determine the axis that the cone opens up along.
For instance, a cone that opens up along the x-axis will have the equation,
y2 z2 x 2
+
=
b2 c2 a 2
For most of the following surfaces we will not give the other possible formulas. We
will however acknowledge how each formula needs to be changed to get a change of
orientation for the surface.
Cylinder
A cylinder whose cross section is an ellipse, parabola, or hyperbola is called
an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder.
An elliptic cylinder is a cylinder with an elliptical cross section.
Here is the general equation of an elliptic cylinder.
x 2 y2
+
= 1 , a, b>0.
a 2 b2
A parametric representation for this surface is:
⎧ x = a cos u
⎪
⎨ y = b sin u , 0 ≤ u ≤ 2π .
⎪ z = v,
v∈R
⎩
(3)
If a =b we have a cylinder whose cross section is a circle. We’ll be dealing with those
kinds of cylinders more than the general form so the equation of a cylinder with a
circular cross section is,
x 2 + y2 = r 2
Be careful to not confuse this with a circle. In two dimensions it is a circle, but in
three dimensions it is a cylinder.
Here is a sketch of the cylinder (3) with an ellipse cross section.
The cylinder will be centred on the axis corresponding to the variable that does not
appear in the equation.
A parabolic cylinder:
x 2 + 2ay = 0 .
(4)
A hyperbolic cylinder:
x2
a2
−
y2
b2
= 1,
a, b>0.
(5)
Hyperboloid of One Sheet
The equation of a hyperboloid of one sheet is:
x 2 y2 z2
+
−
= 1,
a 2 b2 c2
a , b, c ≥ 0.
(6)
Here is a sketch of the hyperboloid of one sheet (6).
The variable with the positive in front of it will give the axis along which the graph is
centred.
The parametric equations of the hyperboloid of one sheet (6) are:
⎧ x = a 1 + u 2 cos v
⎪⎪
2
⎨ y = b 1 + u sin v, 0 ≤ v ≤ 2π, u ∈ R
⎪ z = cu
⎪⎩
Hyperboloid of Two Sheets
Notice that the only difference between the hyperboloid of one sheet and the
hyperboloid of two sheets is the signs in front of the variables. They are exactly the
opposite signs.
The equation of a hyperboloid of two sheets is:
−
x2
−
y2
+
z2
= 1,
a, b, c>0.
(7)
a 2 b2 c2
The variable with the positive in front of it will give the axis along which the graph is
centred.
Here is a sketch of a typical hyperboloid of two sheets (7).
The parametric equations of hyperboloid (7) are
⎧x = a sinh u cos v
⎪
⎨ y = b sinh u sin v
⎪ z = c cosh u
⎩
for
u∈R
and
v ∈ [0,2π) .
Elliptic Paraboloid
Here is the equation of an elliptic paraboloid.
x 2 y2 z
+
= , a, b>0
a 2 b2 c
(8)
In this case the variable that isn’t squared determines the axis upon which the
paraboloid opens up. Also, the sign of c will determine the direction that the
paraboloid opens. If c is positive then it opens up and if c is negative then it opens
down.
With a = b an elliptic paraboloid is a paraboloid of revolution: a surface
obtained by revolving a parabola around its axis. It is the shape of the parabolic
reflectors used in mirrors, antenna dishes, and the like; and is also the shape of the
surface of a rotating liquid, a principle used in liquid mirror telescopes. It is also
called a circular paraboloid.
The elliptic paraboloid (8) is parametrized simply as:
⎧
⎪
x=u
⎪
⎪
y = v,
⎨
2
⎪
⎛u
v 2 ⎞⎟
⎪z = c⎜
+
⎜ a 2 b2 ⎟
⎪⎩
⎝
⎠
u, v ∈ R
.
Hyperbolic Paraboloid
A quadric surface whose equation in a suitable coordinate system is
x 2 y2 z
−
= , a,b>0.
(10)
a 2 b2 c
are named a hyperbolic paraboloid.
These graphs are vaguely saddle shaped and as with the elliptic paraboloid the sign of
c will determine the direction in which the surface “opens up”. The graph above is
shown for c positive.
With the both of the types of paraboloids discussed above the surface can be easily
moved up or down by adding/subtracting a constant from the left side.
The hyperbolic paraboloid (10) has the parametric equations
⎧
⎪
⎪ x=u
⎪
u , v ∈ R.
⎨ y = v,
2
2
⎪
⎛u
v ⎞⎟
⎪z = c⎜
−
⎜ a 2 b2 ⎟
⎪⎩
⎝
⎠
The yz-plane and the zx-plane are planes of symmetry. Sections by planes
parallel to the xy-plane are hyperbolas, the section by the xy-plane itself being a pair
of straight lines. Sections by planes parallel to the other axial planes are parabolas.
Planes through the z-axis cut the paraboloid in parabolas with vertex at the origin. The
origin is a saddle-point.