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General Science => General Science => Topic started by: Democritus on 13/12/2009 14:51:45

Title: Are all parabolas congruent?
Post by: Democritus on 13/12/2009 14:51:45

Are all parabolas congruent? Are all conic sections congruent?

Title: Are all parabolas congruent?
Post by: Mr. Scientist on 13/12/2009 15:22:23

Congruently yes, as a curve. 

Title: Are all parabolas congruent?
Post by: Bored chemist on 13/12/2009 20:37:38

Assuming that you mena this sort of congruence
"In geometry, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections. Less formally, two figures are congruent if they have the same shape and size, but are in different positions (for instance one may be rotated, flipped, or simply placed somewhere else)."
(copied from Wiki)
then no, they are not all congruent.
If you were alowed to pick one up off the plane and project it then I think they would be but I'd not know where to start on proving it.

Title: Are all parabolas congruent?
Post by: Democritus on 14/12/2009 04:12:08

I mean congruent in the simple geometrical sense, in that take any parabola and scale it up and down, in dimension in direct proportional ratio point for point.

Will, during that exercise... during that transformation (term?), our parabola duplicate in turn all other parabolas?

If so, do all other conic sections behave similarly?   

Title: Are all parabolas congruent?
Post by: Soul Surfer on 14/12/2009 09:06:21

No, ellipses and hyperbolas are different, they have more variation in their parameters (notably eccentricity) in this sense a parabola is like a circle which has only one scalable solution. so all circles and all parabolas can be matched to each other just by altering the magnification and position of the curve.  this is not true for ellipses or hyperbolas which will not fit over each other without adjusting the relative scales between the two major axes of the curves.

Title: Are all parabolas congruent?
Post by: LeeE on 14/12/2009 18:17:54

All of these curves are types of conic section.  Have a look at:

http://en.wikipedia.org/wiki/Conic_sections (http://en.wikipedia.org/wiki/Conic_sections)

Title: Are all parabolas congruent?
Post by: Bored chemist on 14/12/2009 19:50:11

"Will, during that exercise... during that transformation (term?), our parabola duplicate in turn all other parabolas"
No.