Math 445 Class 2/11
Parabolas
We have seen parabolas show up in several guises so far.
We will settle on this the initial definition:
The parabola with focus F and directrix d (which d is a line and F is
a point not on d) is the set of points P equidistant from F and d.
Quadratic equation for a parabola
Let d be the x-axis and F be on the y-axis. Find the equation from the defintion.
A plane section of a cone
The intersection of a plane p with a cone is a parabola when the plane is parallel
to one of the generating lines of the cone. We have seen two ways of deducing
this.
- The general story of Dandelin spheres includes this case, but the details
of the parabola were not worked out, so this is a bit of a loose end. In
this case F is the point of tangency of the Dandelin sphere and the plane
p and d is the line of intersection of p and the plane containing the "tangent
circle" of points at which the Dandelin sphere is tangent to the cone.
- In Ogilvy is an alternative method due to Galileo of showing that such a
plane section of a cone is a parabola. This uses the quadratic equation for
a parabola, which must be derived from the definition.
Locus of centers of circles through F tangent to d
This definition was the most important one in lab. It can also be done with
Miras.
- From this definition, find that no tangent vectors are parallel to the axis
of symmetry of the parabola.
- All the reflections of F in tangent lines are on d
- Light rays parallel to the axis reflect to rays through F
- Also, all parabolas are similar.
Bezier curve "curve-stitching definition"
See handout and 444 lab.
Trajectory of a thrown object
Vectors are used in this approach.