Half-circle Parabolas: A Misconception?

[CosmicVoyager]

Greetings,

Is a half-circle a parabola? I am guessing yes because it is a conic section, the very bottom of the cone?

If yes, why aren't antenna dishes and telescope mirrors half-spheres? Wouldn't that be simpler?

Thanks

 

[arildno]

No, it is not.

Circles have constant curvature at all points, parabolas have non-constant curvature.

As for your conic section argument, that would make hyperbolas into circles as well...

As a reminder:
A circle is the locus of points equidistant from a single point.
A parabola is the locus of points that have, individually, the same distance from a fixed point (the focus) and a fixed line (the directrix).

Since the definitions are not logically equivalent, nor are any part of the curves "the same", either.

 

[Integral]

No, a circle is not the same as a parabola.

A parabola is formed by the intersection of a cone and a plane parallel to the axis of the cone. A circle is formed when the intersecting plane is perpendicular to the axis of the cone.

 

[rohit63]

This is a good question...most of the students may go baffle with this...
Actually, in conic sections, the main difference among parabaola, hyperbola and circle is their eccentricity. Eccentricity of parabola is 1 and that of the circle is 0.
For more info. about eccentricity.. check out this link:
http://en.wikipedia.org/wiki/Eccentricity_(mathematics )

 

[CosmicVoyager]

Thanks for the answers. I understand now. I was thinking a parabola was a curve formed by a *any* plane intersecting the base and the side. Now I see the plane must be parallel to the side of the cone.