Conic Sections (2D)
Cylinders and Quadric Surfaces
What you will learn today
Conic Sections (in 2D coordinates)
Cylinders (3D)
Quadric Surfaces (3D)
Vectors and the Geometry of Space
1/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Parabolas
ellipses
Hyperbolas
Shifted Conics
Conic sections result from intersecting a cone with a plane.
Vectors and the Geometry of Space
2/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Vectors and the Geometry of Space
Parabolas
ellipses
Hyperbolas
Shifted Conics
3/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Parabolas
ellipses
Hyperbolas
Shifted Conics
A parabola is the set of points in a plane that are equidistant from
a fixed point F (called the focus) and a fixed line (called the
directrix). The point halfway between the focus and the directrix is
on the parabola, it is called the vertex. The line perpendicular to
the directrix and through the focus is called the axis.
Vectors and the Geometry of Space
4/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Parabolas
ellipses
Hyperbolas
Shifted Conics
Choose the origin O to be at the vertex, the y-axis as the axis of
the parabola, the focus F(0,p). For a point P(x,y) on the parabola,
have
q
|PF | = x 2 + (y − p)2 = |y + p| ⇒
x 2 = 4py
In general, it opens upward if p¿0 and downward if p¡0. It is
symmetric about the y-axis.
If we interchange x and y in the equations, y 2 = 4px is the
parabola with focus (p,0) and directrix x = −p. It opens to the
right if p¿0 and opens to the left if p¡0.
Vectors and the Geometry of Space
5/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Parabolas
ellipses
Hyperbolas
Shifted Conics
Find the focus and directrix of the parabola y 2 + 10x = 0 and
sketch the graph.
Reflection properties of parabola.
Vectors and the Geometry of Space
6/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Parabolas
ellipses
Hyperbolas
Shifted Conics
An ellipse is the set of points in a plane the sum of whose
distances from two fixed points F1 and F2 is a constant. F1 and F2
are called foci.
Kepler’s law 1 says that the orbits of the planets in the solar
system are ellipses with the sun at one focus.
Put the foci on the x-axis (−c, 0) and (c, 0), the sum of distance is
2a > 0. Therefore the vertices of ellipse are (−a, 0) and (a, 0) on
the x-axis. The line segment between the two vertices is called the
major axis. When c=0, the two foci coincide and the ellipse is a
circle.
P(x, y ) is on the ellipse,
|PF1 | + |PF2 | = 2a
Put b 2 = a2 − c 2 , the above equation can be written as
x2 y2
+ 2 =1
a2
b
Vectors and the Geometry of Space
7/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Parabolas
ellipses
Hyperbolas
Shifted Conics
On the other hand, the ellipse
x2 y2
+ 2 = 1, a ≥ b > 0
b2
a
has foci (0, ±c) and vertices (0, ±a) on the y-axis.
Vectors and the Geometry of Space
8/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Parabolas
ellipses
Hyperbolas
Shifted Conics
1. Sketch the graph of 9x 2 + 16y 2 = 144 and locate the foci.
2. Find an equation of the ellipse with foci (0, ±2) and vertices
(0, ±3).
3. Reflection properties of ellipse.
Vectors and the Geometry of Space
9/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Parabolas
ellipses
Hyperbolas
Shifted Conics
A hyperbola is the set of all points in a plane the difference of
whose distances from two fixed points F1 and F2 is a constant.
Vectors and the Geometry of Space
10/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Parabolas
ellipses
Hyperbolas
Shifted Conics
Put the vertices F1 , F2 = (±c, 0) along the x-axis, the vertices are
(±a, 0). Have
|PF1 | − |PF2 | = ±2a
Write b 2 = c 2 − a2 . The equation of the parabola is
x2 y2
− 2 =1
a2
b
A hyperbola has two branches. It has asymptotes
b
y =± x
a
Vectors and the Geometry of Space
11/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Parabolas
ellipses
Hyperbolas
Shifted Conics
On the other hand, the hyperbola with foci (0, ±c), vertices
(0, ±a) on the y-axis has form
y2 x2
− 2 =1
a2
b
It has asymptotes y = ± ba x
Vectors and the Geometry of Space
12/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Parabolas
ellipses
Hyperbolas
Shifted Conics
If we shift the origin h units to the right and k units up, we will
replace x and y by x-h and y-k in the equations.
Sketch the conic 9x 2 − 4y 2 − 72x + 8y + 176 = 0
Vectors and the Geometry of Space
13/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Quadric Surfaces
General surface:
To Sketch the graph of a surface, it is useful to determine the
curves of intersection if the surface with planes parallel to the
coordinate planes. These curves are called traces (or
cross-sections) of the surface.
Cylinders:
A cylinder is a surface that consists all lines (called rulings) that
are parallel to a given line and pass through a given plane curve.
The parabolic cylinder: x = ay 2 , a > 0
Vectors and the Geometry of Space
14/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Quadric Surfaces
When one of the variable x, y or z (say x) is missing from the
equation, then the surface is a cylinder, and the rulings are parallel
to (say x).
Example: Sketch x 2 + y 2 = 1.
Vectors and the Geometry of Space
15/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Quadric Surfaces
A quadric surface is the graph of a second-degree equation in three
variables x, y and z. The most general such equation is
Ax 2 + By 2 + Cz 2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0
where the capital letters are constants (some of them could be 0!).
By translation and rotation it can be brought into the standard
forms
Ax 2 + By 2 + Cz 2 + J = 0
or
Ax 2 + By 2 + Iz = 0
Vectors and the Geometry of Space
16/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Quadric Surfaces
Example: use traces to sketch the surface with equation
x2 y2 z2
+
+
=1
4
9
4
In general, an ellipsoid has equation
x2 y2 z2
+ 2 + 2 =1
a2
b
c
Vectors and the Geometry of Space
17/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Quadric Surfaces
Example:
z = 4x 2 + y 2
This is called an elliptic paraboloid.
Vectors and the Geometry of Space
18/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Quadric Surfaces
Example:
z = y2 − x2
This is called a hyperbolic paraboloid (or saddle),
Vectors and the Geometry of Space
19/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Quadric Surfaces
Example:
x2
z2
+ y2 −
=1
4
4
Hyperboloid of one sheet,
Vectors and the Geometry of Space
20/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Quadric Surfaces
Example:
4x 2 − y 2 + 2z 2 + 4 = 0
Hyperboloid of two sheets,
whether a hyperboloid is one sheet or two sheets depends on
whether z could be 0 or not.
Vectors and the Geometry of Space
21/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Quadric Surfaces
Example: Cones:
x2 y2 z2
+ 2 − 2 =0
a2
b
c
Vectors and the Geometry of Space
22/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Quadric Surfaces
Applications:
A satellite dish is a paraboloid.
Cooling towers in power stations is in the shape of hyperboloids.
Twin gears in a triturating juicer,
Vectors and the Geometry of Space
23/24
Conic Sections (2D)
Cylinders and Quadric Surfaces
Quadric Surfaces
What you have learned today
Conic Sections (in 2D coordinates)
Cylinders (3D)
Quadric Surfaces (3D)
Vectors and the Geometry of Space
24/24