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Old 2022-09-02, 17:50   #1
a1call
 
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Default Parallel Parabolas

It occurred to me, I thought about it, I googled it and found a lot of nonsensical "opinions"/definitions.
I think that any parallel parabola to any given parabola will have to superimpose it and hence be the same (Unless it is on a separate 2D Plane).
I think the only valid definition of parallel curves is that any straight line perpendicular to one will be perpendicular to any other.
Furthermore I think there can not exist any separate curve on a single plane (parabolic or otherwise) that is parallel to any given parabola.
What do you think?

https://www.jstor.org/stable/3027202

https://www.reddit.com/r/math/commen...to_each_other/
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Old 2022-09-02, 18:16   #2
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To generalize parallel to curves, consider concentric circles of differing radii in the same plane.
The shortest distance between any selected point on one circle and the nearest point on the other is a segment of a radial line.

https://www.dictionary.com/browse/parallel

One of the responses here indicates a curve parallel in the same plane to a parabola will be a sixth order polynomial.

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Old 2022-09-02, 19:36   #3
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See also ENVELOPE OF A FAMILY OF PLANE CURVES
Quote:
The curves parallel to a curve are the envelopes of circles with constant radii centred on this curve.
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Old 2022-09-02, 22:36   #4
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Quote:
Originally Posted by a1call View Post
I think the only valid definition of parallel curves is that any straight line perpendicular to one will be perpendicular to any other.
Furthermore I think there can not exist any separate curve on a single plane (parabolic or otherwise) that is parallel to any given parabola.
What do you think?
Two parabolas cannot be parallel to each other, that is true if they are not the same.

Bézier curves have been studied extensively. A degree two Bézier curve is a parabolic section, and constant distance offsetting from a parabola is known to need degree six polynomials.

Last fiddled with by retina on 2022-09-02 at 22:41
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Old 2022-09-03, 00:41   #5
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I am sorry but I take an issue with using Offset-Lines to be equivalent/interchangeable with Parallel-Lines, despite the Wikipedia article stating just that:
https://en.wikipedia.org/wiki/Parallel_curve
The examples there contradict each other. Please see the attached screenshot.
In the image EQD-170-A which I made using AutoCAD's "Offset" command only the outer 4 polylines are parallel to the outermost polyline. not the inner 3 rectangles. The issue with them is that if you offset them outwards after they loose their round corners they create rectangles which are different than the original fillet-ed polylines. so you can have 2 different offsets of the same "curve" one with the new sharp corners and the original rounded corners at the same distance (and on the same side). I don' think you can have 2 different parallel "curves" to a single curve at equal distances and on the same side. If you could then you could show that all closed/open curves on a 2D plane are parallel to all other closed/open curves on that plane. And that would be nonsense.
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Old 2022-09-03, 16:23   #6
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parabola 1: z=z1
parabola 2: z2 != z1
https://en.wikipedia.org/wiki/Parabo...er_Quadric.png

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Old 2022-09-03, 17:17   #7
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Yes those are all identical parabolas on parallel 2D planes. I also considered parallel cross sections of cones forming parabolas, see 4th image below:

https://www.mechamath.com/geometry/a...ons-of-a-cone/

Which raises the question:
If we define a perpendicular line to a curve at a given point by a perpendicular line to the tangent line to that curve at that point, how would you define a tangent line to a parabola on a plane other than the plane the parabola is drawn on?
ETA: I assume you could still define a perpendicular line as a perpendicular to the tangent on the original plane.

Re: Parallel == Offset,
Consider any (non square) rectangle and square on a plane whose all edges are parallel and do not cross. They Can not be obtained in a single Offset from one another. Are they not parallel?

Last fiddled with by a1call on 2022-09-03 at 17:34
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Old 2022-09-05, 02:31   #8
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FWIW:
So in conclusion:
* If the offset is not large enough and internal to eliminate the vertex curve there can parallel curves to a parabola
* Parabolas formed by parallel slicing planes to a cone do not create parallel Parabolas
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